Holt McDougal
Florida
Larson Algebra 1
Chapter Resources
Volume 2: Chapters 7-12
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ISBN 13: 978-0-547-25018-2
ISBN 10: 0-547-25018-5
1 2 3 4 5 6 7 8 9 XXX 15 14 13 12 11 10 09
Contents
Chapter 7
Systems of Equations and Inequalities
Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1–4
7.1
Solve Linear Systems by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5–15
7.2
Solve Linear Systems by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–21
7.3
Solve Linear Systems by Adding or Subtracting . . . . . . . . . . . . . . . . . . . . . . 22–28
7.4
Solve Linear Systems by Multiplying First . . . . . . . . . . . . . . . . . . . . . . . . . . 29–35
7.5
Solve Special Types of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36–44
7.6
Solve Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45–55
Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 8
Exponents and Exponential Functions
Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57–60
8.1
Apply Exponent Properties Involving Products . . . . . . . . . . . . . . . . . . . . . . . 61–67
8.2
Apply Exponent Properties Involving Quotients . . . . . . . . . . . . . . . . . . . . . . 68–73
8.3
Define and Use Zero and Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . 74–83
8.4
Use Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84–90
8.5
Write and Graph Exponential Growth Functions . . . . . . . . . . . . . . . . . . . . . 91–101
8.6
Write and Graph Exponential Decay Functions . . . . . . . . . . . . . . . . . . . . . 102–114
Copyright © Holt McDougal. All rights reserved.
Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter 9
Polynomials and Factoring
Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117–120
9.1
Add and Subtract Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121–128
9.2
Multiply Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129–134
9.3
Find Special Products of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135–140
9.4
Solve Polynomial Equations in Factored Form . . . . . . . . . . . . . . . . . . . . . 141–147
9.5
Factor x 2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148–153
9.6
Factor ax 2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154–160
9.7
Factor Special Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161–166
9.8
Factor Polynomials Completely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167–173
Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Algebra 1
Chapter Resource Book
iii
Chapter 10
Quadratic Equations and Functions
Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175–178
10.1
Graph y 5 ax 2 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179–187
10.2
Graph y 5 ax 2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188–201
10.3
Solve Quadratic Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . 202–211
10.4
Use Square Roots to Solve Quadratic Equations . . . . . . . . . . . . . . . . . . . 212–218
10.5
Solve Quadratic Equations by Completing the Square . . . . . . . . . . . . . . 219–227
10.6
Solve Quadratic Equations by the Quadratic Formula . . . . . . . . . . . . . . . 228–234
10.7
Interpret the Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235–240
10.8
Compare Linear, Exponential, and Quadratic Models . . . . . . . . . . . . . . 241–251
Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
Chapter 11
Radicals and Geometry Connections
Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253–256
11.1
Graph Square Root Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257–266
11.2
Simplify Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267–276
11.3
Solve Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277–283
11.4
Apply the Pythagorean Theorem and Its Converse . . . . . . . . . . . . . . . . . 284–289
11.5
Apply the Distance and Midpoint Formulas . . . . . . . . . . . . . . . . . . . . . . 290–296
Chapter 12
Rational Equations and Functions
Chapter Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299–302
12.1
Model Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303–312
12.2
Graph Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313–321
12.3
Divide Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322–331
12.4
Simplify Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332–338
12.5
Multiply and Divide Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . 339–347
12.6
Add and Subtract Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 348–354
12.7
Solve Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355–361
Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Gridded Response Answer Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Resource Book Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1–A63
iv
Algebra 1
Chapter Resource Book
Copyright © Holt McDougal. All rights reserved.
Chapter Review Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Contents
Descriptions of Resources
This Chapter Resource Book is organized by lessons within the chapter in order to make
your planning easier. The following materials are provided:
Family Letter This guide helps families contribute to student success by providing
an overview of the chapter along with questions and activities for families to work on
together.
Graphing Calculator Activities with Keystrokes Keystrokes for two models of
calculators are provided for each Graphing Calculator Activity in the Student Edition.
Activity Support Masters These blackline masters make it easier for students to
record their work on selected activities in the Student Edition.
Practice A, B, and C These exercises offer additional practice for the material in each
lesson, including application problems. There are three levels of practice for each lesson:
A (basic), B (average), and C (advanced).
Review for Mastery These two pages provide additional instruction, worked-out
examples, and practice exercises covering the key concepts and vocabulary in each lesson.
Problem Solving Workshops These blackline masters provide extra problem solving
opportunities in addition to the workshops given in the textbook. There are three types of
workshops: Alternative Methods, Worked-Out Examples, and Mixed Problem Solving.
Challenge Practice These exercises offer challenging practice on the mathematics of
each lesson.
Chapter Review Game This worksheet offers fun practice at the end of the chapter
Copyright © Holt McDougal. All rights reserved.
and provides an alternative way to review the chapter content in preparation for the
Chapter Test.
Gridded Response Answer Sheet This page provides 12 answer grids for the teacher
to copy and distribute as needed for use with the Gridded Response questions in the Problem
Solving Workshops.
Algebra 1
Chapter Resource Book
v
Name ———————————————————————
CHAPTER
7
Date ————————————
Family Letter
For use with Chapter 7
Copyright © Holt McDougal. All rights reserved.
Lesson Title
Lesson Goals
Key Applications
7.1: Solve Linear Systems by
Graphing
Graph and solve systems of
linear equations.
• Rental Business
• Television
• Fitness
7.2: Solve Linear Systems by
Substitution
Solve systems of linear equations
by substitution.
• Websites
• Antifreeze
• Fundraising
7.3: Solve Linear Systems by
Adding or Subtracting
Solve linear systems using
elimination.
• Kayaking
• Rowing
• Cellular Phones
7.4: Solve Linear Systems by
Multiplying First
Solve linear systems by
multiplying first.
• Book Sale
• Music
• Farm Products
7.5: Solve Special Types of
Linear Systems
Identify the number of solutions
of a linear system.
• Art
• Recreation
• Photography
7.6: Solve Systems of Linear
Inequalities
Solve systems of linear
inequalities in two variables.
• Baseball
• Competition Scores
• Fish
CHAPTER SUPPORT
Chapter Overview One way you can help your student succeed in Chapter 7 is
by discussing the lesson goals in the chart below. When a lesson is completed, ask your
student the following questions. “What were the goals of the lesson? What new words
and formulas did you learn? How can you apply the ideas of the lesson to your life?”
Key Ideas for Chapter 7
In Chapter 7, you will apply the key ideas listed in the Chapter Opener (see page 437)
and reviewed in the Chapter Summary (see page 489).
1. Solving linear systems by graphing
2. Solving linear systems using algebra
3. Solving systems of linear inequalities
Algebra 1
Chapter 7 Resource Book
1
Name ———————————————————————
CHAPTER
Family Letter
continued
For use with Chapter 7
Key Ideas Your student can demonstrate understanding of key concepts by
working through the following exercises with you.
Lesson
Exercise
7.1
Solve the system by graphing. Check the solution.
x 1 2y 5 8
2x 2 y 5 6
7.2
Each day you either carpool to school, which takes 18 minutes, or ride the bus,
which takes 35 minutes. After 20 days of school you have spent 598 minutes
getting to school. How many days did you carpool? How many days did you ride
the bus?
7.3
Solve the system using elimination. Check the solution.
3x 1 5y 5 9
5y 5 3x 1 21
7.4
You and a friend are playing in a basketball tournament. You buy 4 sports drinks
and 5 power bars for $13. Your friend buys 3 sports drinks and 2 power bars for
$7.65. How much did each sports drink cost? How much did each power bar
cost?
7.5
Tell whether the linear system has one solution, no solution, or infinitely many
solutions. Explain.
(a) 3x 2 y 5 9
6x 2 2y 5 10
7.6
(b) 26x 1 8y 5 12
9x 2 12y 5 218
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
7
Date ————————————
Graph the system of inequalities. x 1 y ≥ 1
x2y≤4
y>2
Home Involvement Activity
Directions Write systems of inequalities for triangular shaded regions that would
be located solely within each of the four quadrants.
21
21
1
1
3
7.1:
5
y
5
; (4, 2) 7.2: 14 bus rides, 6 carpool rides 7.6:
7.3: (22, 3) 7.4: $1.75, $1.20
7.5: (a) no solution; same slope,
different y-intercept
x
(b) many solutions; same
slope and y-intercept
22
22
6
x
6
y
Answers
2
Algebra 1
Chapter 7 Resource Book
Nombre ——————————————————————
CAPÍTULO
7
Fecha ———————————
Carta para la familia
Usar con el Capítulo 7
Copyright © Holt McDougal. All rights reserved.
Título de la lección
Objetivos de la lección
Aplicaciones clave
7.1: Resolver sistemas
lineales con gráficas
Graficar y resolver sistemas de
ecuaciones lineales
• Negocio de alquiler
• Televisión
• Salud
7.2: Resolver sistemas
lineales con la
sustitución
Resolver sistemas de ecuaciones
lineales con la sustitución
• Sitios web
• Anticongelante
• Recaudación de fondos
7.3: Resolver sistemas
lineales con la suma o
la resta
Resolver sistemas lineales usando
la eliminación
• Hacer kayak
• Remar
• Teléfonos celulares
7.4: Resolver sistemas
lineales multiplicando
primero
Resolver sistemas lineales
multiplicando primero
• Venta de libros
• Música
• Productos de la granja
7.5: Resolver tipos especiales
de sistemas lineales
Identificar la cantidad de
soluciones de un sistema lineal
• Arte
• Recreo
• Fotografía
7.6: Resolver sistemas de
desigualdades lineales
Resolver sistemas de
desigualdades lineales con dos
variables
• Béisbol
• Puntajes de concursos
• Peces
CHAPTER SUPPORT
Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener éxito
en el Capítulo 7 es hablar sobre los objetivos de la lección en la tabla a continuación.
Cuando se termina una lección, pregúntele a su hijo lo siguiente: “¿Cuáles fueron los
objetivos de la lección? ¿Qué palabras y fórmulas nuevas aprendiste? ¿Cómo puedes
aplicar a tu vida las ideas de la lección?”
Ideas clave para el Capítulo 7
En el Capítulo 7, aplicarás las ideas clave enumeradas en la Presentación del capítulo
(ver la página 437) y revisadas en el Resumen del capítulo (ver la página 489).
1. Resolver sistemas lineales con gráficas
2. Resolver sistemas lineales usando álgebra
3. Resolver sistemas lineales de desigualdades lineales
Algebra 1
Chapter 7 Resource Book
3
Nombre ——————————————————————
CAPÍTULO
Carta para la familia
continúa
Usar con el Capítulo 7
Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer
los siguientes ejercicios con usted.
Lección
Ejercicio
7.1
Resuelve el sistema al graficar. Comprueba la solución.
x 1 2y 5 8
2x 2 y 5 6
7.2
Cada día o vas en carro a la escuela, que toma 18 minutos, o vas en autobús, que
toma 35 minutos. Después de 20 días escolares, has pasado 598 minutos en llegar
a la escuela. ¿Cuántos días fuiste en carro? ¿Cuántos días tomaste el autobús?
7.3
Resuelve el sistema usando la eliminación. Comprueba la solución.
3x 1 5y 5 9
5y 5 3x 1 21
7.4
Tú y un amigo juegan en un torneo de básquetbol. Compras 4 bebidas deportivas
y 5 barras de nutrición por $13. Tu amigo compra 3 bebidas deportivas y 2 barras
de nutrición por $7.65. ¿Cuánto costó cada bebida deportiva? ¿Cuánto costó cada
barra de nutrición?
7.5
Indica si el sistema lineal tiene una solución, ninguna solución o muchas
soluciones infinitas. Explica.
(a) 3x 2 y 5 9
6x 2 2y 5 10
(b) 26x 1 8y 5 12
9x 2 12y 5 218
Grafica el sistema de desigualdades. x 1 y ≥ 1
x2y≤4
y>2
7.6
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
7
Fecha ———————————
Actividad para la familia
Instrucciones Escribe sistemas de desigualdades para regiones triangulares
sombreadas que se localizarían dentro de cada uno de los cuatro cuadrantes.
21
21
1
1
3
7.1:
5
y
5
; (4, 2) 7.2: 14 días en autobús, 6 días en carro 7.6:
7.3: (22, 3) 7.4: $1.75, $1.20
7.5: (a) ninguna solución; misma
pendiente, diferente
x
intercepto en y
(b) muchas soluciones; misma
pendiente y mismo intercepto en y
22
22
6
x
6
y
Respuestas
4
Algebra 1
Chapter 7 Resource Book
Name ———————————————————————
Graphing Calculator Activity Keystrokes
LESSON
7.1
For use with page 446
Casio CFX-9850GC Plus
TI-83 Plus
Y=
(
3
ENTER
)
5
(
()
ENTER
Date ————————————
1
ENTER
ENTER
2
3
)
)
ZOOM
ENTER
X,T,,n
X,T,,n
6
2nd
3
(
[CALC] 5
5
From the main menu, choose GRAPH.
( 5 2 )
3
()
X,,T
( 1 3 )
( 5
X,,T
)
F5
EXE
SHIFT
F3
F3
EXIT
F6
EXE
3
SHIFT
F5
Copyright © Holt McDougal. All rights reserved.
LESSON 7.1
Algebra 1
Chapter 7 Resource Book
5
Name ———————————————————————
Date ————————————
Practice A
LESSON
7.1
For use with pages 439–445
Tell whether the ordered pair is a solution of the linear system.
1. (0, 24);
3. (1, 22);
2. (3, 3);
x 1 y 5 24
x 2 5y 5 20
x 1 2y 5 9
4x 2 y 5 15
4. (24, 26);
2x 2 3y 5 8
3x 1 2y 5 21
5. (4, 21);
23x 1 y 5 6
22x 1 y 5 28
6. (2, 26);
x 2 4y 5 8
23x 1 5y 5 23
4x 1 3y 5 210
3x 1 2y 5 26
LESSON 7.1
Match the linear system with its graph.
7. x 2 y 5 2
8. x 1 y 5 2
9. x 1 y 5 22
x1y55
x2y55
x 2 y 5 25
10. x 2 y 5 22
11. 2x 1 y 5 2
2x 2 y 5 5
y
A.
12. x 2 y 5 2
x1y55
B.
2x 2 y 5 5
C.
y
y
1
1
3
5
x
3
3
1
1
23
1
21
D.
y
23
x
E.
3
23
x
5
x
F.
y
y
21
21
1
23
23
3
1
x
21
21
23
x
21
1
3
5
Use the graph to solve the linear system. Check your solution.
13. 4x 1 3y 5 5
14. 2x 1 3y 5 9
2x 2 y 5 5
15. 5x 2 y 5 24
4x 2 y 5 8
22x 1 y 5 1
y
y
y
5
1
21
21
1
3
5 x
1
23
6
Algebra 1
Chapter 7 Resource Book
21
21
23
1
3
5 x
21
1
3 x
Copyright © Holt McDougal. All rights reserved.
21
Name ———————————————————————
Practice A
LESSON
7.1
Date ————————————
continued
For use with pages 439–445
Solve the linear system by graphing. Check your solution.
16. y 5 2x 1 6
17. y 5 22x 1 1
y5x22
y5x25
y
2x 2 y 5 3
y
5
1
3
21
21
1
23
21
21
18. 4x 2 y 5 212
1
3
5
x
9
5 x
3
1
29
23
23
25
9 x
3
29
20. y 5 22x 1 2
y 5 4x 2 9
3
21. 3x 1 y 5 7
y5x15
LESSON 7.1
19. y 5 x
y
22x 1 y 5 28
y
y
y
9
5
3
3
6
29
23
23
29
3
9
2
x
1
25
23
21
21
26
1
22
22
x
6
2
x
26
Population (thousands)
(in thousands of people) of the Buffalo, New York area and
the Charlotte, North Carolina area. Use the graph to find the
year in which the populations of these two areas were the same.
What was the population?
23. Juice You bought 15 one-gallon bottles of apple juice and
orange juice for a school dance. The apple juice was on sale
for $1.50 per gallon bottle. The orange juice was $2 per gallon
bottle. You spent $26. Write algebraic models for the situation.
Then graph the algebraic models. How many bottles of each
type of juice did you buy?
Bottles of orange juice
Copyright © Holt McDougal. All rights reserved.
22. City Populations The graph shows the estimated populations
y
1250
1200
1150
1100
1050
1000
950
0
y
16
14
12
10
8
6
4
2
0
Buffalo
Charlotte
0 1 2 3 4 5 6 7 x
Years since 1990
0 2 4 6 8 10 12 14 16 x
Bottles of apple juice
Algebra 1
Chapter 7 Resource Book
7
Name ———————————————————————
Date ————————————
Practice B
LESSON
7.1
For use with pages 439–445
Tell whether the ordered pair is a solution of the linear system.
1. (4, 1);
3. (4, 23);
2. (22, 1);
x 1 2y 5 6
5x 2 2y 5 212
23x 1 2y 5 218
3x 1 y 5 11
x 1 3y 5 1
6x 2 y 5 27
4. (24, 26);
6. (22, 25);
5. (24, 3);
3x 2 y 5 6
4x 1 3y 5 212
2x 1 y 5 23
2x 1 2y 5 8
x 1 2y 5 26
2x 1 3y 5 213
Use the graph to solve the linear system. Check your solution.
8. 5x 2 y 5 29
x 1 y 5 22
y 1 2x 5 2
22x 1 y 5 6
y
y
3
1
21
9. 2x 1 3y 5 2
5 x
y
5
23
3
25
1
23
10. 3x 2 2y 5 16
21
11. 2x 2 y 5 213
5x 1 y 5 18
y 1 3x 5 212
23x 1 4y 5 16
y
y
x
3
1
x
1
12. 6x 1 2y 5 8
y
21
21
3 x
Copyright © Holt McDougal. All rights reserved.
LESSON 7.1
7. x 2 y 5 8
5
23
3
3
25
1
1
25
23
21
23
x
21
3 x
1
Solve the linear system by graphing. Check your solution.
13. y 5 3x
14. 2x 1 y 5 24
y 5 4x 2 1
15. 23x 2 y 5 21
x 2 y 5 28
y
2x 1 4y 5 216
y
y
1
23
21
3
6
1
2
1
3
x
26
25
2 x
23
21
21
23
25
26
8
Algebra 1
Chapter 7 Resource Book
1
x
Name ———————————————————————
Practice B
LESSON
7.1
Date ————————————
continued
For use with pages 439–445
16. 2x 1 2y 5 26
17. 26x 1 y 5 33
25x 1 y 5 15
18. 29x 1 6y 5 26
2x 2 8y 5 234
2x 2 3y 5 8
y
y
y
35
15
1
23
21
21
1
x
3
215
29
3
x
215
19. 3x 1 2y 5 3
20. x 2 y 5 9
21. 6x 1 y 5 19
3x 1 2y 5 2
y
LESSON 7.1
5x 1 y 5 29
25
x
5
5x 2 2y 5 24
y
y
21
9
22
22
29
23
9 x
6
15
10 x
9
26
3
29
29
Non-blooming annuals
baskets. The plants you have picked out are blooming annuals
and non-blooming annuals. The blooming annuals cost $3.20
each and the non-blooming annuals cost $1.50 each. You bought
a total of 24 plants for $49.60. Write a linear system of equations
that you can use to find how many of each type of plant you
bought. Then graph the linear system and use the graph to find
how many of each type of plant you bought.
23. Baseball Outs In a game, 12 of a baseball team’s 27 outs
were fly balls. Twenty-five percent of the outs made by infielders
and 100% of the outs made by outfielders were fly balls.
a. Write a linear system you can use to find the number of outs
made by infielders and the number of outs made by outfielders.
(Hint: Write one equation for the total number of outs and
another equation for the number of fly ball outs.)
b. Graph your linear system.
c. How many outs were made by infielders? How many outs
were made by outfielders?
Outs made by outfielders
Copyright © Holt McDougal. All rights reserved.
22. Hanging Flower Baskets You will be making hanging flower
y
32
28
24
20
16
12
8
4
0
y
35
30
25
20
15
10
5
0
23
9 x
0 5 10 15 20 25 30 35 x
Blooming annuals
0 4 8 12 16 20 24 28 32 x
Outs made by infielders
Algebra 1
Chapter 7 Resource Book
9
Name ———————————————————————
Date ————————————
Practice C
LESSON
7.1
For use with pages 439–445
Tell whether the ordered pair is a solution of the linear system.
2. (7, 26);
1. (28, 4);
3. (4, 26);
2x 1 4y 5 28
3x 1 2y 5 9
3x 1 y 5 26
3x 2 5y 5 3
24x 2 3y 5 210
2x 1 2y 5 8
4. (4, 22);
6. (22.5, 2.5);
5. (23, 5);
3
7
4
2
3
61
4x 1 }8 y 5 }
4
1
2
}x 2 }y 5 }
21.5x 1 3.2y 5 11.5
6x 2 8y 5 235
4.1x 2 2y 5 222.3
4x 1 2y 5 25
LESSON 7.1
Solve the linear system by graphing. Check your solution.
7. 25x 1 8y 5 222
8. 210x 2 4y 5 64
3y 2 2x 5 29
9. 3x 2 7y 5 50
2x 1 2y 5 16
24x 1 2y 5 230
y
y
y
3
12
1
21
1
21
3
5
7 x
4
1
3
5
7 x
220
24
24
212
4
x
23
25
23
212
3
2
11. } x 1 } y 5 2
5
5
2
1
11
10. } x 2 } y 5 2}
3
3
3
1
1
2
x 1 }2 y 5 2}2
19
1
12. 4x 2 } y 5 2}
3
3
2
y
23
2
2}3 x 1 y 5 }3
2}3 x 1 y 5 }
3
y
y
21
10
3
6
1
2
21
21
15
210
10
26
22
22
Algebra 1
Chapter 7 Resource Book
2
x
1
3
9
5 x
3
23
215
29
23
3 x
Copyright © Holt McDougal. All rights reserved.
27
Name ———————————————————————
Practice C
LESSON
7.1
Date ————————————
continued
For use with pages 439–445
13. 1.8x 2 2.2y 5 24.2
14. 21.4x 1 6y 5 24.6
0.5x 1 3.2y 5 21.7
15. 3.2x 2 y 5 8.8
0.2x 1 y 5 0.2
5x 2 2.5y 5 10
y
y
y
7
10
6
5
6
2
3
2
22
22
210
1
21
1
3
26
22
22
2 x
2
6
10 x
26
5 x
LESSON 7.1
3
16. Find the values for m and b so that the system y 5 } x 2 2 and y 5 mx 1 b has (8, 4)
4
as a solution.
y
7
17. The graphs of the three lines given below form a triangle.
Use a graph to find the coordinates of the vertices
of the triangle.
2x 1 y 5 7
Line 2: x 1 2y 5 2
Line 3: 2x 1 y 5 4
5
3
Line 1:
1
23
21
21
1
3
x
6.5% annual interest. The combined annual interest is $2725. How much of the
$45,000 is invested in each type of fund? (Hint: Write one equation for the
amount invested in each fund and another for the interest earned.)
19. Umbrella Sales The table shows the number of automatic and
manual opening umbrellas sold at a shop in 2000 and 2005.
Use a linear model to represent the sales of each type of
umbrella. Let t 5 0 correspond to 2000. Sketch the graphs
and estimate when the number of automatic umbrellas sold
equaled the number of manual umbrellas sold.
Year
2000
2005
Automatic
15
25
Manual
25
15
Number of umbrellas
Copyright © Holt McDougal. All rights reserved.
18. Investments A total of $45,000 is invested into two funds paying 5.5% and
y
35
30
25
20
15
10
5
0
0 1 2 3 4 5 6 7 t
Years since 2000
20. Credit Account With a minimum purchase of $100, you can open a credit account
at a music store. The store is offering either $25 or 20% off your purchase if you
open a credit account. You make a purchase of $135 and decide to open a credit
account. Should you choose $25 or 20% off your purchase? Explain.
Algebra 1
Chapter 7 Resource Book
11
Name ———————————————————————
LESSON
7.1
Date ————————————
Review for Mastery
For use with pages 439–445
GOAL
Graph and solve systems of linear equations.
Vocabulary
A system of linear equations, or simply a linear system, consists of
two or more linear equations in the same variables.
A solution of a system of linear equations in two variables is an
ordered pair that satisfies each equation in the system.
Check the intersection point
LESSON 7.1
Use the graph to solve the system. Then
check your solution algebraically.
2x 1 y 5 4
Equation 1
3x 2 5y 5 6
Equation 2
y
3
1
23
Solution
The lines appear to intersect at the point (2, 0).
CHECK
21
21
3
x
23
Substitute 2 for x and 0 for y in each equation.
Equation 1
2x 1 y 5 4
Equation 2
3x 2 5y 5 6
2(2) 1 0 0 4
3(2) 2 5(0) 0 6
41004
62006
454✓
656✓
Because the ordered pair (2, 0) is a solution of each equation, it is a solution of
the system.
EXAMPLE 2
Use the graph-and-check method
x 2 3y 5 2
Equation 1
25x 1 y 5 4
Equation 2
Solve the linear system:
STEP 1
Graph both equations.
STEP 2
y
1
23
3
23
12
Algebra 1
Chapter 7 Resource Book
x
Estimate the point of the intersection.
The two lines appear to intersect at
(21, 21).
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 1
Name ———————————————————————
LESSON
7.1
Review for Mastery
Date ————————————
continued
For use with pages 439–445
STEP 3
Check whether (21, 21) is a solution by substituting 21 for x and 21 for
y in each of the original equations.
Equation 1
x 2 3y 5 2
Equation 2
25x 1 y 5 4
21 2 3(21) 0 2
25(21) 1 (21) 0 4
21 1 3 0 2
52104
252✓
454✓
Because the ordered pair (21, 21) is a solution of each equation, it is a solution of
the system.
Solve a multi-step problem
Delivery Service The Rosebud Flower Shop has a basic delivery charge of $5 plus a
rate of $.25 per mile. The Beautiful Bouquets Shop has a basic delivery charge of $7
plus a rate of $.20 per mile. Determine the number of miles a delivery must be for the
charges to be equal.
LESSON 7.1
EXAMPLE 3
Solution
Write a linear system. Let x be the number of miles driven and y be the total
cost of the delivery.
y 5 5 1 0.25x
Equation for Rosebud Flower Shop
y 5 7 1 0.20x
Equation for Beautiful Bouquets Shop
STEP 2
Graph both equations.
STEP 3
Estimate the point of intersection. The two lines
appear to intersect at (40, 15).
STEP 4
Check whether (40, 15) is a solution.
Delivery Service
Equation 1
y 5 5 1 0.25x
Equation 2
y 5 7 1 0.20x
15 0 5 1 0.25(40)
15 0 7 1 0.20(40)
15 5 15 ✓
15 515 ✓
Total cost (dollars)
Copyright © Holt McDougal. All rights reserved.
STEP 1
y
18
15
12
9
6
3
0
0 10 20 30 40 50 60 x
Miles driven
Exercises for Examples 1, 2, and 3
Solve the linear system by graphing.
1. 23x 1 y 5 4
5x 2 2y 5 27
2.
1
x 1 }2 y 5 4
5x 1 2y 5 18
3. 2x 2 6y 5 4
7x 2 4y 5 220
4. In Example 3, suppose Rosebud Flower Shop increases its basic charge to
$10, and Beautiful Bouquets raises its basic charge to $13. Determine when
the costs will be equal.
Algebra 1
Chapter 7 Resource Book
13
Name ———————————————————————
LESSON
7.1
Date ————————————
Problem Solving Workshop:
Worked Out Example
For use with pages 439–445
PROBLEM
Aerobics A fitness club offers two aerobics classes. There are currently 28 people
going to the afternoon class and attendance is increasing at a rate of 2 people per
month. There are currently 16 people going to the night class and attendance is
increasing at a rate of 4 people per month. Predict when the number of people in
each class will be the same.
STEP 1
Read and Understand
What do you know? The number of people that go to each aerobic class and
the increase each month.
STEP 2
Make a Plan Use what you know to write and solve a linear system.
STEP 3
Solve the Problem Let x be the number of months and y be the number of
people in the class.
Number of people
An equation that models the afternoon
class is y 5 2x 1 28.
An equation that models the night class
is y 5 4x 1 16.
Graph both equations. The point of
intersection occurs at the point (6, 40).
After 6 months, both the afternoon class
and the night class have the same number
of people, 40.
STEP 4
y
50
40
30
20
10
0
y 5 2x 1 28
y 5 4x 1 16
0 1 2 3 4 5 6 7 8 9 x
Number of months
Look Back Check whether (6, 40) is a solution.
y 5 2x 1 28
y 5 4x 1 16
40 0 2(6) 1 28
40 0 4(6) 1 16
40 5 40 ✓
40 5 40 ✓
The answer is correct.
PRACTICE
1. Carpet Store A charges $4 per square
foot for carpeting and $95 for installation. Store B charges $6 per square foot
for carpeting and $75 for installation.
Find the square footage for which the
total cost is the same for each store.
2. Football You are selling tickets to a
football game. Student tickets cost
$4 and general admission tickets cost
$7. You sell 213 tickets and collect
$1146. How many of each type of
ticket did you sell?
14
Algebra 1
Chapter 7 Resource Book
3. What If? For the next football game,
you sell 241 tickets and collect $1315.
How many of each type of ticket did
you sell?
4. Bowling Alley A charges $2.25 per
game of bowling and $1.75 for shoe
rentals. Alley B charges $2 per game
of bowling and $2.75 for shoe rentals.
Find the number of games for which
the total cost is the same to bowl at
each alley.
Copyright © Holt McDougal. All rights reserved.
LESSON 7.1
What do you want to find out? When each class has the same number of
people.
Name ———————————————————————
LESSON
7.1
Date ————————————
Challenge Practice
For use with pages 439–445
Tell whether the ordered pair is a solution of the system of linear
equations.
1.
1 }45, }45 2;
2.
11 9
,} ;
1}
4 42
2x 1 3y 5 4
x1y55
3x 1 2y 5 4
x 2 y 5 }2
1
3. (4, 1);
4.
x 1 2y 5 6
2x 2 3y 5 4
3a 1 2b 3b 2 2a
,} ;
1}
b 1a b 1a 2
2
2
2
2
ax 1 by 5 3
bx 2 ay 5 2
1990
2000
Bayside
100,000
105,000
Coal Flats
105,000
85,000
LESSON 7.1
In Exercises 5 and 6, use the table that shows the numbers of households
in two cities in the years 1990 and 2000.
5. For each city, write a linear model to represent the number of households at time t,
where t represents the number of years since 1990.
Copyright © Holt McDougal. All rights reserved.
6. Use a graph to estimate when the two cities had the same number of households.
In Exercises 7–9, use the table that shows the annual number of
spectators for three sports in a small town in the years 1950 and 2000.
1950
2000
Hockey
20,000
80,000
Soccer
0
100,000
90,000
40,000
Baseball
7. For each sport, write a linear model to represent the annual number of spectators at
time t, where t represents the number of years since 1950.
8. Use a graph to estimate when the annual number of spectators of soccer overtook the
annual number of spectators of hockey.
9. Use a graph to estimate when the annual number of spectators of soccer overtook the
annual number of spectators of baseball.
Algebra 1
Chapter 7 Resource Book
15
Name ———————————————————————
LESSON
7.2
Date ————————————
Practice A
For use with pages 447– 453
Solve for the indicated variable.
1. 9x 1 y 5 7; y
2. 3x 2 y 5 10; y
3. x 2 4y 5 1; x
4. 3x 1 6y 5 9; x
5. 2x 2 2y 5 8; y
1
6. } x 2 3y 5 7; x
2
Tell which equation you would use to isolate a variable. Explain your
reasoning.
7. x 5 5y 2 8
4x 1 3y 5 5
10. 2x 1 y 5 8
2y 2 3x 5 5
8. 23x 1 2y 5 7
y 5 6x 1 1
11. x 1 4y 5 22
3x 2 y 5 1
9. 4 1 8x 5 y
6x 2 y 5 2
12. 2x 5 4y 1 2
25x 1 5y 5 13
Solve the linear system by using substitution.
y 5 2x 2 2
16. y 5 22x 1 4
5y 2 2x 5 216
19. 28x 1 3y 5 233
5x 1 y 5 35
LESSON 7.2
22. x 1 y 5 3
3x 2 4y 5 219
14. x 5 4y 1 14
y 5 23x 1 3
17. 4x 2 2y 5 14
x 5 10 2 6y
20. x 1 2y 5 11
3x 2 4y 5 217
23. x 2 y 5 0
2x 1 4y 5 18
15. y 5 23x 2 1
4x 1 3y 5 2
18. x 1 2y 5 6
27x 1 3y 5 28
21. 23x 1 y 5 8
x 1 2y 5 25
24. 2x 1 2y 5 6
3x 2 5y 5 25
25. Driving Your brother and sister took turns driving on a 635-mile trip that took
11 hours to complete. Your brother drove at a constant speed of 60 miles per hour
and your sister drove at a constant speed of 55 miles per hour. Let x be the number
of miles your brother drove and let y be the number of miles your sister drove. Solve
the linear system x 1 y 5 11 and 60x 1 55y 5 635 to find the number of miles each
of your siblings drove.
26. Fundraising A wilderness group is selling cans of nuts and boxes of microwaveable
popcorn to raise money for a trip. A can of nuts sells for $4.50 and a box of
microwaveable popcorn sells for $3. The group sells $252 in nuts and popcorn and
they sell twice as many boxes of popcorn as cans of nuts.
a. Let x be the number of boxes of popcorn and let y be the number of cans of nuts
sold. Write an equation that relates the number of boxes of popcorn sold to the
number of cans of nuts sold.
b. Write an equation that gives the total amount of money made in terms of x and y.
c. How many boxes of popcorn did the group sell? How many cans of nuts did the
group sell?
16
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
13. x 5 1 2 y
Name ———————————————————————
LESSON
7.2
Date ————————————
Practice B
For use with pages 447–453
Solve for the indicated variable.
1. 8x 1 4y 5 12; y
2. 3x 2 4y 5 12; y
3. 6x 2 4y 5 8; x
Tell which equation you would use to isolate a variable. Explain your
reasoning.
4. x 5 8y 2 3
3x 2 4y 5 1
5. 24x 1 5y 5 11
6. 9 2 3x 5 y
y 5 4x 2 1
3x 2 y 5 22
Solve the linear system by using substitution.
7. x 5 6 2 4y
2x 2 3y 5 1
10. 6x 2 y 5 235
5x 2 2y 5 235
13. 2x 1 2y 5 6
23x 1 5y 5 233
16. 3x 1 2y 5 5
5x 2 9y 5 24
8. 4x 1 3y 5 0
9. 2x 1 2y 5 26
2x 1 y 5 22
11. 2x 1 3y 5 29
8x 1 y 5 31
12. 3x 1 3y 5 218
8x 2 4y 5 32
4x 2 y 5 214
14. 5x 1 2y 5 43
15. 4x 2 2y 5 24
26x 1 3y 5 230
17. 4x 2 3y 5 28
7x 2 5y 5 219
18. 8x 1 8y 5 24
2x 1 3y 5 24
x 1 5y 5 11
19. Drum Sticks A drummer is stocking up on drum sticks and brushes. The wood
Copyright © Holt McDougal. All rights reserved.
LESSON 7.2
sticks that he buys are $10.50 a pair and the brushes are $24 a pair. He ends up
spending $90 on sticks and brushes and buys two times as many pairs of sticks
as brushes. How many pairs of sticks and brushes did he buy?
20. Mowing and Shoveling Last year you mowed grass and shoveled snow for
12 households. You earned $225 for mowing a household’s lawn for the entire year
and you earned $200 for shoveling a household’s walk and driveway for an entire
year. You earned a total of $2600 last year.
a. Let x be the number of households you mowed for and let y be the number of
households you shoveled for. Write an equation in x and y that shows the total
number of households you worked for. Then write an equation in x and y that
shows the total amount of money you earned.
b. How many households did you mow the lawn for and how many households did
you shovel the walk and driveway for?
21. Dimensions of a Metal Sheet A rectangular hole 3 centimeters wide
and x centimeters long is cut in a rectangular sheet of metal that is
4 centimeters wide and y centimeters long. The length of the hole is
1 centimeter less than the length of the metal sheet. After the hole
is cut, the area of the remaining metal sheet is 20 square centimeters.
Find the length of the hole and the length of the metal sheet.
4 cm
3 cm
x
y
Algebra 1
Chapter 7 Resource Book
17
Name ———————————————————————
Date ————————————
Practice C
LESSON
7.2
For use with pages 447– 453
Tell which equation you would use to isolate a variable. Explain your
reasoning.
1. 6x 2 y 5 9
2. 22x 1 4y 5 10
5x 2 3y 5 2
3. 15 2 3x 5 2y
9y 5 5x 2 7
9x 2 3y 5 26
Solve the linear system by using substitution.
4. 13x 2 4y 5 38
5. 10x 2 20y 5 0
x 2 6y 5 254
x 1 5y 5 228
7. 10x 1 y 5 285
1
9. 4x 1 7y 5 8
0.2x 1 y 5 10.4
x 1 11y 5 76
11. 5x 1 y 5 41
2x 1 3y 5 4
3
1
1
13. } x 1 } y 5 }
2
3
4
y2x54
8. 4x 2 3y 5 222
0.1x 1 2.5y 5 11.6
10. 3x 1 2y 5 29
6. 3.5x 1 0.5y 5 14
12. 210x 1 3y 5 21
3x 2 y 5 23
x 2 6y 5 15
7
1
14. x 1 } y 5 2}
5
5
13
7
15. 6x 1 5y 5 2}
3
3
x 2 }4 y 5 }
16
5
9
1
3
23x 2 6y 5 }2
} x 2 y 5 2}
ax 1 by 5 210
Equation 1
ax 2 by 5 230
Equation 2
17. Painting and Cleaning During the spring and summer, you do a spring yard
cleanup for households and you also paint houses. You earn $8 an hour doing the
cleanups and $12 an hour painting. Last spring and summer, you worked a total of
400 hours and earned $3800. How many hours did you spend doing yard cleanups?
How many hours did you spend painting?
18. Room Dimensions The area of the room shown is
4 ft
224 square feet. The perimeter of the room is 64 feet.
Find x and y.
12 ft
8 ft
y ft
x ft
19. Potting Soil You are creating a potting mix for your window boxes that is 20% peat
moss and 80% potting soil. You add 100% potting soil to your mix that is currently
50% peat moss and 50% potting soil. You have 4 buckets of the mix that is half and
half. Do you have enough of the half and half mix to make 8 buckets of the mix that
is 20% peat moss and 80% potting soil? Explain.
18
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 7.2
16. Find the values of a and b so that the linear system shown has a solution of (4, 25).
Name ———————————————————————
LESSON
7.2
Date ————————————
Review for Mastery
For use with pages 447–453
GOAL
EXAMPLE 1
Solve systems of linear equations by substitution.
Use the substitution method
Solve the linear system: 2x 1 y 5 1
Equation 1
x 1 2y 5 5
Equation 2
Solution
STEP 1
Solve Equation 1 for y.
2x 1 y 5 1
Write original Equation 1.
y 5 22x 1 1
STEP 2
Subtract 2x from each side.
Substitute 22x 1 1 for y in Equation 2 and solve for x.
x 1 2y 5 5
Write Equation 2.
x 1 2(22x 1 1) 5 5
Substitute 22x 1 1 for y.
x 2 4x 1 2 5 5
Distributive property
23x 1 2 5 5
Simplify.
23x 5 3
Subtract 2 from each side.
x 5 21
STEP 3
Substitute 21 for x in the original Equation 1 to find the value of y.
2x 1 y 5 1
Write original Equation 1.
2(21) 1 y 5 1
LESSON 7.2
Copyright © Holt McDougal. All rights reserved.
Divide each side by 23.
Substitute 21 for x.
22 1 y 5 1
Simplify.
y53
Solve for y.
The solution is (21, 3).
CHECK
Substitute 21 for x and 3 for y in each of the original equations.
Equation 1
2x 1 y 5 1
Equation 2
x 1 2y 5 5
2(21) 1 3 0 1
21 1 2(3) 0 5
151✓
555✓
Algebra 1
Chapter 7 Resource Book
19
Name ———————————————————————
LESSON
7.2
Review for Mastery
Date ————————————
continued
For use with pages 447– 453
EXAMPLE 2
Use the substitution method
Solve the linear system: 2x 1 5y 5 5
x 2 4y 5 9
Equation 1
Equation 2
Solution
Solve Equation 2 for x.
x 2 4y 5 9
Write original Equation 2.
x 5 4y 1 9
STEP 2
Revised Equation 2
Substitute 4y 1 9 for x in Equation 1 and solve for y.
2x 1 5y 5 5
Write Equation 1.
2(4y 1 9) 1 5y 5 5
Substitute 4y 1 9 for x.
8y 1 18 1 5y 5 5
Distributive property
13y 1 18 5 5
Simplify.
13y 5 213
y 5 21
LESSON 7.2
STEP 3
Subtract 18 from each side.
Divide each side by 13.
Substitute 21 for y in the revised Equation 2 to find the value of x.
x 5 4y 1 9
Revised Equation 2
x 5 4(21) 1 9
Substitute 21 for y.
x55
Simplify.
The solution is (5, 21).
CHECK
Substitute 5 for x and 21 for y in each equation.
Equation 1
2x 1 5y 5 5
Equation 2
x 2 4y 5 9
2(5) 1 5(21) 0 5
5 2 4(21) 0 9
555✓
959✓
Exercises for Examples 1 and 2
Solve the linear system using the substitution method.
1. x 1 3y 5 210
2.
7x 2 5y 5 34
4. 6x 1 y 5 26
5x 2 2y 5 21
20
Algebra 1
Chapter 7 Resource Book
8x 1 5y 5 6
5x 2 y 5 221
5.
x 1 3y 5 11
5x 1 6y 5 1
3. 6x 2 7y 5 22
x 2 4y 5 22
3
6. } x 1 y 5 8
2
1
4x 2 }2 y 5 15
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
Date ————————————
Challenge Practice
LESSON
7.2
For use with pages 447–453
Solve the linear system by using the substitution method.
1. 2x 1 y 5 1
3
2
}x 1 y 5 6
1
4
2. 2} x 1 } y 5 22
2
3
3
4
2
1
y 5 }2
}x 1 }
3
In Exercises 3 and 4, use the method shown in the following example to
solve the system of equations.
Example:
x2 1 y2 5 4
1
2
} x 2 1 3y 2 5 8
Solution:
Let u 5 x 2 and v 5 y 2.
Using substitution, the system becomes
u1v54
1
2
.
} u 1 3v 5 8
8
12
.
Solving this system by substitution gives u 5 }5 and v 5 }
5
Î
Î
8
8 12
8
8 12
12
12
, 2Î}5 , Î }
, Î} , 2Î }
, and 1 Î}5 , Î }
.
are 1 2Î}5 , 2Î}
5 2 1
5 2 1 5
5 2
5 2
}
}
12
8
Because u 5 x 2 and v 5 y 2, x 5 6 }5 and y 5 6 }
. So, the solutions
5
}
}
}
}
}
}
}
LESSON 7.2
Copyright © Holt McDougal. All rights reserved.
}
3. 2x 2 1 4y 2 5 11
x 2 1 5y 2 5 8
4. 3x 1 5y 2 5 8
x 1 2y 2 5 6
Algebra 1
Chapter 7 Resource Book
21
Name ———————————————————————
LESSON
7.3
Date ————————————
Practice A
For use with pages 456–462
Rewrite the linear system so that the like terms are arranged in columns.
1. 3x 2 y 5 23
2. 8x 5 y 1 1
y 1 8x 5 11
3y 1 8x 5 7
4. 7x 2 y 5 13
5. 14 5 x 2 3y
y 5 14x 2 3
x 1 10y 5 23
3. 7x 2 4y 5 8
4y 5 27x 1 9
6. 8x 1 1 5 4y
4y 1 3 5 14x
Describe the first step you would use to solve the linear system.
7. x 1 4y 5 1
6x 2 4y 5 23
10. 24x 2 4y 5 7
4y 2 x 5 2
8. 2x 1 3y 5 21
3y 5 22x 1 3
11. 6x 2 4y 5 5
26x 2 5y 5 7
9. 5x 1 y 5 8
x 1 y 5 26
12. 3x 5 y 2 9
25x 1 y 5 8
Solve the linear system by using elimination.
14. x 1 4y 5 9
3x 1 y 5 4
2x 2 2y 5 3
16. 2x 1 y 5 7
17. 4x 1 3y 5 18
x1y51
19. 3x 5 y 1 5
2x 1 y 5 5
22. 6x 2 3y 5 36
5x 5 3y 1 30
4x 2 2y 5 8
20. x 2 4y 5 219
3y 2 15 5 x
23. 24x 1 y 5 227
2y 1 6x 5 43
15. 5x 2 3y 5 214
x 1 3y 5 2
18. 25x 1 2y 5 22
3x 1 2y 5 210
21. y 2 3 5 22x
2x 1 3y 5 13
24. 9x 2 4y 5 255
3x 5 24y 2 21
25. Rollerblading One day, you are rollerblading on a trail while it is windy. You travel
LESSON 7.3
along the trail, turn around and come back to your starting point. On your way out
on the trail, you are rollerblading against the wind. On your return trip, which is the
same distance, you are rollerblading with the wind. You can only travel 3 miles an
hour against the wind, which is blowing at a constant speed. You travel 8 miles an
hour with the wind. Use the models below to write and solve a system of equations
to find the average speed when there is no wind and the speed of the wind.
22
Against the wind: Your speed with no wind 2 Speed of wind 5 Your speed
With the wind: Your speed with no wind 1 Speed of wind 5 Your speed
26. Car Wash A gas station has a car wash. If you get your gas tank filled, then you are
charged a lower flat fee x in dollars for a car wash plus y dollars per gallon for the
gasoline. Two cars fill up with regular gasoline and both get a car wash. One car uses
8 gallons of gasoline and pays $22.80 for the gas and car wash and the other car uses
6 gallons of gasoline and pays $18.60 for the gas and car wash. Find the fee for the
car wash and the cost of one gallon of regular gasoline.
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
13. 6x 2 y 5 5
Name ———————————————————————
LESSON
7.3
Date ————————————
Practice B
For use with pages 456–462
Rewrite the linear system so that the like terms are arranged in columns.
1. 8x 2 y 5 19
y 1 3x 5 7
2. 4x 5 y 2 11
3. 9x 2 2y 5 5
6y 1 4x 5 23
2y 5 211x 1 8
Describe the first step you would use to solve the linear system.
4. 22x 2 y 5 24
5. 25 5 x 2 7y
y 5 6x 2 5
x 1 12y 5 28
7. x 1 9y 5 2
8. 4x 1 3y 5 26
14x 2 9y 5 24
6. 3x 1 7 5 2y
22y 2 1 5 10x
9. 4x 1 y 5 210
3y 5 25x 1 1
x 1 y 5 214
Solve the linear system by using elimination.
10. x 1 5y 5 28
2x 2 2y 5 213
13. 3x 5 y 2 20
27x 2 y 5 40
16. 23x 5 y 2 20
2y 5 25x 1 4
3
5
19. } x 1 y 5 2}
2
2
12. 6x 1 y 5 39
3x 1 4y 5 10
14. 2x 2 6y 5 210
22x 1 y 5 217
15. x 2 3y 5 6
4x 5 10 1 6y
11
1
17. x 2 } y 5 }
2
2
22x 5 3y 1 33
2
18. 2} x 1 6y 5 38
3
x 2 6y 5 233
2x 1 4y 5 26
1
20. 7x 2 } y 5 229
3
29
3
1
21. } x 2 } y 5 2}
2
2
2
1
2x 2 }3 y 5 29
1
2}2 x 1 3y 5 33
22. Fishing Barge A fishing barge leaves from a dock and moves upstream (against
the current) at a rate of 3.8 miles per hour until it reaches its destination. After the
people on the barge are done fishing, the barge moves the same distance downstream
(with the current) at a rate of 8 miles per hour until it returns to the dock. The speed
of the current remains constant. Use the models below to write and solve a system
of equations to find the average speed of the barge in still water and the speed of the
current.
Upstream: Speed of barge in still water 2 Speed of current 5 Speed of barge
Downstream: Speed of barge in still water 1 Speed of current 5 Speed of barge
23. Floor Sander Rental A rental company charges a flat fee of x dollars for a floor
sander rental plus y dollars per hour of the rental. One customer rents a floor sander
for 4 hours and pays $63. Another customer rents a floor sander for 6 hours and
pays $87.
a. Find the flat fee and the cost per hour for the rental.
LESSON 7.3
Copyright © Holt McDougal. All rights reserved.
4x 1 y 5 25
11. 7x 2 4y 5 230
b. How much would it cost someone to rent a sander for 11 hours?
Algebra 1
Chapter 7 Resource Book
23
Name ———————————————————————
LESSON
7.3
Date ————————————
Practice C
For use with pages 456–462
Solve the linear system by using elimination.
1. 4x 2 y 5 221
2. 22x 1 5y 5 14
24x 1 7y 5 51
8x 1 5y 5 94
4. 10y 2 2x 5 238
6. 215x 1 4y 5 43
8x 1 4y 5 128
7. 6x 2 3y 5 54
4y 5 23x 1 25
8. 2y 2 3x 5 10
6x 5 8y 2 36
9. 9x 5 235 2 5y
7x 5 22y 2 50
10. 1.8x 2 4.2y 5 215.6
1.8x 1 7.5y 5 42.9
2.8y 5 2.2x 1 6.4
14. 4.5x 1 0.5y 5 48.5
5
3
2.5x 5 0.5y 1 14.5
}y 1 }x 5 }
5y 2 10x 5 250
11. 27.4y 2 2.2x 5 47.2
2
1
2
13. } x 1 } y 5 }
3
3
3
1
3
x 5 6y 2 28
5. 8x 2 6y 5 212
22x 5 8y 1 52
1
3
3. 2y 2 x 5 7
12. 9.5x 2 7.4y 5 15.7
7.4y 2 4.2x 5 42.6
15. 3.2x 5 4.8y 1 8
6.4y 5 3.2x 2 19.2
16. For b Þ 0, what is the solution of the system 2x 1 by 5 22 and 4x 2 by 5 8?
x 1 3y 1 2z 5 9
Equation 1
2z 1 x 2 5y 5 27
Equation 2
6y 5 15 2 3x
Equation 3
18. Car Rental A car rental company charges a daily rental fee plus a per mile fee
over 150 miles. Two different people rent the same style of car for the same number
of days. The total bill for one person’s rental is $207.50 for a 5-day rental and
180 miles. The total bill for the other person’s rental is $212.50 for a 5-day rental
and 200 miles.
a. Write a linear system that you can use to find the daily rental fee and the per mile
fee over 150 miles. Explain how you got your linear system.
b. What is the daily rental fee? What is the fee per mile over 150 miles?
LESSON 7.3
19. Greeting Cards Two friends are making their own greeting cards. They already
24
have ink, but they will buy the stamps and cards. The table shows the numbers of
stamps and packages of cards each person is buying. Another friend, George, wants
to buy 3 stamps and 3 packages of cards. How much will it cost him? Explain.
Customer
Stamps
Packages of cards
Total cost (dollars)
Stan
4
2
22.98
Leeza
7
2
32.73
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
17. Solve for x, y, and z in the system of equations below. Explain your steps.
Name ———————————————————————
LESSON
7.3
Date ————————————
Review for Mastery
For use with pages 456–462
GOAL
EXAMPLE 1
Solve linear systems by elimination.
Use addition to eliminate a variable
Solve the linear system: 2x 1 4y 5 2
4x 2 4y 5 16
Equation 1
Equation 2
Solution
STEP 1
Add the equations to
eliminate one variable.
STEP 2
Solve for x.
STEP 3
Substitute 3 for x in either equation and solve for y.
2x 1 4y 5 2
4x 2 4y 5 16
6x
5 18
x53
2x 1 4y 5 2
Write Equation 1.
2(3) 1 4y 5 2
Substitute 3 for x.
Solve for y.
y 5 21
The solution is (3, 21).
Substitute 3 for x and 21 for y in each equation.
CHECK
Equation 1
2x 1 4y 5 2
Equation 2
4x 2 4y 5 16
2(3) 1 4(21) 0 2
4(3) 2 4(21) 0 16
EXAMPLE 2
16 5 16 ✓
Use subtraction to eliminate a variable
Solve the linear system: 7x 1 5y 5 18
Equation 1
7x 2 3y 5 34
Equation 2
Solution
STEP 1
Subtract the equations
to eliminate one variable.
STEP 2
Solve for y.
STEP 3
Substitute 22 for y in either equation and solve for x.
7x 1 5y 5 18
7x 1 5(22) 5 18
x54
7x 1 5y 5 18
7x 2 3y 5 34
8y 5 216
y 5 22
LESSON 7.3
Copyright © Holt McDougal. All rights reserved.
252✓
Write Equation 1.
Substitute 22 for y.
Solve for x.
The solution is (4, 22).
Algebra 1
Chapter 7 Resource Book
25
Name ———————————————————————
LESSON
7.3
Review for Mastery
Date ————————————
continued
For use with pages 456–462
EXAMPLE 3
Arrange like terms
Solve the linear system: 6x 2 4y 5 10
Equation 1
13y 5 6x 1 8
Equation 2
Solution
STEP 1
Rewrite Equation 1 so that the like terms are arranged in columns.
6x 2 4y 5 10
13y 5 6x 1 8
STEP 2 Add the equations.
STEP 3 Solve for y.
STEP 4
6x 2 4y 5 10
26x 1 13y 5 8
9y 5 18
y52
Substitute 2 for y in either equation and solve for x.
6x 1 4y 5 10
Write Equation 1.
6x 2 4(2) 5 10
Substitute 2 for y.
x53
Solve for x.
The solution is (3, 2).
Exercises for Examples 1, 2, and 3
Solve the linear system.
7x 2 8y 5 12
3. 9x 2 8y 5 7
9x 1 2y 5 213
5. 9x 1 8y 5 230
LESSON 7.3
9x 5 4y 1 42
26
Algebra 1
Chapter 7 Resource Book
2. 4x 1 5y 5 8
24x 2 3y 5 0
4. 24x 1 7y 5 11
2x 1 7y 5 47
6. 5y 5 4x 1 3
7x 5 36 2 5y
Copyright © Holt McDougal. All rights reserved.
1. 5x 1 8y 5 36
Name ———————————————————————
LESSON
7.3
Date ————————————
Problem Solving Workshop:
Using Alternative Methods
For use with pages 4562462
Another Way to Solve Example 4 on page 458
Multiple Representations In Example 4 on page 458, you saw how to
solve a problem about average speed using an inequality. You can also solve the problem
by substitution.
PROBLEM
Kayaking During a kayaking trip, a kayaker travels 12 miles upstream (against the
current) and 12 miles downstream (with the current), as shown on page 458. The
speed of the current remained constant during the trip. Find the average speed of the
kayak in still water and the speed of the current.
METHOD
Substitution You can solve the problem by substitution.
STEP 1
STEP 2
Write the system of equations from page 458.
Going upstream:
x2y54
Going downstream:
x1y56
Solve Equation 1 for x.
x2y54
x5y14
STEP 3
y141y56
y51
Write Equation 2.
Substitute y 1 4 for x.
Solve for y.
Substitute 1 for y in the revised Equation 1 to find the value of x.
x5y14511455
The average speed of the kayak in still water is 5 miles per hour, and the speed of
the current is 1 mile per hour.
PRACTICE
1. Running Running into the wind,
Calvin takes 56 minutes to run 7 miles.
The return run takes 50 minutes. The
wind speed remains constant during the
trip. Find the average speed (in miles
per hour) of Calvin in still air and the
speed (in miles per hour) of the wind.
2. What If? Suppose in Exercise 1 it takes
Calvin 70 minutes to run 7 miles into
the wind and 50 minutes on the return
run. Find the average speed of Calvin in
still air and the speed of the wind.
3. Boating James and Bret take a boat
out on a river. It takes them 15 minutes
to travel 5 miles upstream (against the
current). The return trip downstream
(with the current) takes 10 minutes. The
speed of the current remained constant
during the trip. Find the average speed
(in miles per hour) of the boat in still
water and the speed of the current.
Algebra 1
Chapter 7 Resource Book
LESSON 7.3
Copyright © Holt McDougal. All rights reserved.
Solve for x.
Substitute y 1 4 for x in Equation 2 and solve for y.
x1y56
STEP 4
Write Equation 1.
27
Name ———————————————————————
LESSON
7.3
Date ————————————
Challenge Practice
For use with pages 456–462
In Exercises 1–3, use the method shown in the following example to solve
the system of equations.
1
1
31 }x 2 1 2 }y 2 5 4
1
Example:
1
1
61 }x 2 2 2 }y 2 5 5
1
1
x
1
Let u 5 } and v 5 }y .
Solution:
Using substitution, the system becomes
3u 1 2v 5 4
.
6u 2 2v 5 5
Adding the equations results in the equation 9u 5 9.
1
1
1
So, u 5 1 5 }x and v 5 }2 5 }y . So, x 5 1 and y 5 2.
1
1
1. 4 } 1 7 } 5 3
x
y
1 2
12
1
1
241 }x 2 2 31 }y 2 5 5
1
2. 4 1 1 } 1 7y 5 3
x
1
2
1
241 1 1 }x 2 2 3y 5 5
1
3. 22(1 1 y 3) 1 7 }2 5 5
x
1
1x 2
4(1 1 y 3) 1 7 }2 5 27
Solve the system for x and y in terms of a and b.
4. 3ax 1 2by 5 4
6ax 1 2by 5 7
5. ax 1 by 5 10
LESSON 7.3
2ax 1 5by 5 13
28
6. 4ax 2 11y 5 b
2ax 1 2y 5 b
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
1 2
Name ———————————————————————
LESSON
7.4
Date ————————————
Practice A
For use with pages 463–469
1. 5x 2 2y 5 8
2. 7x 1 8y 5 3
3. 5x 1 2y 5 8
7x 1 8y 5 3
8x 2 2y 5 5
7x 1 8y 5 3
A. 220x 2 8y 5 232
7x 1 8y 5 3
B. 32x 2 8y 5 20
7x 1 8y 5 3
C. 20x 2 8y 5 32
LESSON 7.4
Match the linear system with an equivalent linear system.
7y 1 8y 5 3
Describe the first step you would use to solve the linear system.
4. x 1 y 5 4
3x 2 7y 5 10
7. 5x 2 2y 5 25
10x 2 3x 5 7
5. 2x 1 6y 5 21
24x 1 7y 5 8
8. 23x 1 9y 5 13
7x 2 3y 5 14
6. 3x 2 6y 5 21
x1y54
9. 4x 2 y 5 7
10x 1 2y 5 8
Solve the linear system by using elimination.
10. x 1 y 5 3
22x 1 4y 5 6
13. 5x 2 4y 5 42
x 2 6y 5 24
16. 3x 2 5y 5 250
12x 1 2y 5 246
Copyright © Holt McDougal. All rights reserved.
19. 4x 1 5y 5 100
3x 2 2y 5 6
11. 4x 1 y 5 28
3x 1 3y 5 3
14. 2x 1 3y 5 210
24x 1 5y 5 22
17. 26x 2 5y 5 243
7x 1 15y 5 41
20. 23x 1 11y 5 238
2x 1 9y 5 240
12. 3x 2 y 5 10
2x 1 5y 5 35
15. 5x 1 6y 5 100
2x 1 3y 5 46
18. 8x 2 6y 5 8
4x 1 5y 5 36
21. 5x 2 8y 5 235
27x 2 3y 5 222
22. Baseball Game Two families go to a baseball game. One family purchases two
adult tickets and three youth tickets for $33. Another family purchases three adult
tickets and two youth tickets for $37. Let x represent the cost in dollars of one adult
ticket and let y represent the cost in dollars of one youth ticket. The linear system
given by 2x 1 3y 5 33 and 3x 1 2y 5 37 represents this situation.
a. Solve the linear system to find the cost of one adult and one youth ticket.
b. How much would it cost two adults and five youths to attend the game?
23. Electricians Two different electrical businesses charge different rates.
Business A charges $30 for a service call, plus an additional $45 per hour for labor.
Business B charges $45 for a service call, plus an additional $40 per hour for labor.
a. Let x represent the number of hours of labor and let y represent the total charge
in dollars. Write a linear system that you could use to find the lengths of a
service call for which both businesses charge the same amount.
b. Solve the linear system.
c. When will the businesses charge the same amount?
Algebra 1
Chapter 7 Resource Book
29
Name ———————————————————————
LESSON
LESSON 7.4
7.4
Date ————————————
Practice B
For use with pages 463–469
Describe the first step you would use to solve the linear system.
1. 3x 2 4y 5 7
5x 1 8y 5 10
4. 7x 2 4y 5 6
3x 2 2y 5 215
2. 9x 1 4y 5 13
3x 1 5y 5 9
5. 7x 1 9y 5 26
25x 1 14y 5 11
3. 5x 1 7y 5 23
15x 1 4y 5 25
6. 9x 2 5y 5 14
26x 1 8y 5 13
Solve the linear system by using elimination.
7. x 1 3y 5 1
25x 1 4y 5 224
10. 8x 2 4y 5 276
5x 1 2y 5 216
13. 4x 2 3y 5 16
16x 1 10y 5 240
16. 23x 2 4y 5 27
5x 2 6y 5 27
8. 23x 2 y 5 215
8x 1 4y 5 48
11. 23x 1 10y 5 23
5x 1 2y 5 55
14. 20x 1 10y 5 100
25x 1 4y 5 53
17. 2x 1 7y 5 2
5x 2 2y 5 83
9. x 1 7y 5 237
2x 2 5y 5 21
12. 9x 2 4y 5 26
18x 1 7y 5 22
15. 3x 2 10y 5 225
5x 2 20y 5 255
18. 3x 2 5y 5 216
2x 2 3y 5 28
19. Hockey Game Two families go to a hockey game. One family purchases two adult
tickets and four youth tickets for $28. Another family purchases four adult tickets
and five youth tickets for $45.50. Let x represent the cost in dollars of one adult
ticket and let y represent the cost in dollars of one youth ticket.
a. Write a linear system that represents this situation.
c. How much would it cost two adults and five youths to attend the game?
20. Travel Agency A travel agency offers two Chicago outings. Plan A includes hotel
accommodations for three nights and two pairs of baseball tickets worth a total
of $557. Plan B includes hotel accommodations for five nights and four pairs of
baseball tickets worth a total of $974. Let x represent the cost in dollars of one
night’s hotel accommodations and let y represent the cost in dollars of one pair
of baseball tickets.
a. Write a linear system you could use to find the cost of one night’s hotel accommodations
and the cost of one pair of baseball tickets.
b. Solve the linear system to find the cost of one night’s hotel accommodations and
the cost of one pair of baseball tickets.
21. Highway Project There are fifteen workers employed on a highway project, some
at $180 per day and some at $155 per day. The daily payroll is $2400. Let x represent
the number of $180 per day workers and let y represent the number of $155 per day
workers. Write and solve a linear system to find the number of workers employed at
each wage.
30
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
b. Solve the linear system to find the cost of one adult and one youth ticket.
Name ———————————————————————
LESSON
7.4
Date ————————————
Practice C
For use with pages 463–469
1. 23x 1 5y 5 28
2. 2x 1 7y 5 213
9x 1 4y 5 68
4. 8x 2 6y 5 2140
23x 1 14y 5 25
23x 1 6y 5 269
5. 4x 1 9y 5 253
6. 26x 1 12y 5 48
26x 2 4y 5 32
27x 1 18y 5 84
3x 1 5y 5 20
7. 3x 1 9y 5 27
3. 4x 1 7y 5 243
8. 28x 1 5y 5 6
14x 1 6y 5 18
6x 2 3y 5 6
10. 6x 2 11y 5 293
11. 215x 1 4y 5 22
15x 1 13y 5 132
13x 2 10y 5 244
LESSON 7.4
Solve the linear system by using elimination.
9. 10x 2 8y 5 28
12x 1 5y 5 92
12. 9x 2 8y 5 23
14x 2 12y 5 26
Solve the linear system by using any algebraic method.
13. 0.4x 1 0.1y 5 0.7
14. 4x 2 3y 5 7
15. 1.5x 1 2.6y 5 212.7
x2y53
1.5x 1 y 5 9
24.5x 1 0.3y 5 21.9
16. x 1 y 5 7
7
17. 4x 1 y 5 2}
4
1
4
1
4
5
4
}x 2 }y 5 }
5x 2 2y 5 23
11
2
1
18. } x 2 } y 5 2}
3
3
4
1
3
3
5
16
15
}x 1 }y 5 }
19. Find the values of a and b so that the linear system has a solution of (2, 4).
Copyright © Holt McDougal. All rights reserved.
ax 2 by 5 0
bx 2 ay 5 26
Equation 1
Equation 2
20. Lift Tickets Two families go skiing on a Saturday. One family purchases two adult
lift tickets and four youth lift tickets for $166. Another family purchases four adult
lift tickets and five youth lift tickets for $263. Let x represent the cost in dollars of
one adult lift ticket and let y represent the cost in dollars of one youth lift ticket.
a. Write a linear system that represents this situation.
b. Solve the linear system to find the cost of one adult and one youth lift ticket.
c. How much would it cost two adults and five youths to ski for a day?
21. Asian Cuisine A group of your friends goes to a restaurant that features different
Asian foods. There are eight people in your group. Some of the group order the Thai
special for $14.25 and the rest of the group order the Szechwan special for $13.95.
If the total bill was $113.10, how many people ordered each dinner?
22. Getting to School You walk 1.75 miles to school at an average speed r (in miles per
hour). On the way back home, you are walking with a friend and your average speed
3
is }4 r. The round trip took a total of 90 minutes. Find the average speed for each leg
of your trip.
Algebra 1
Chapter 7 Resource Book
31
Name ———————————————————————
LESSON
LESSON 7.4
7.4
Date ————————————
Review for Mastery
For use with pages 463– 469
GOAL
EXAMPLE 1
Solve linear systems by multiplying first.
Multiply one equation, then add
Solve the linear system: 3x 2 2y 5 24
Equation 1
7x 2 4y 5 26
Equation 2
Solution
Multiply Equation 1 by 22 so that the coefficients of y are opposites.
STEP 1
3x 2 2y 5 24
3 (22)
7x 2 4y 5 26
STEP 2 Add the equations.
26x 1 4y 5 8
7x 2 4y 5 26
x
52
Substitute 2 for x in either equation and solve for y.
STEP 3
3x 2 2y 5 24
Write Equation 1.
3(2) 2 2y 5 24
Substitute 2 for x.
y55
Solve for y.
The solution is (2, 5).
Substitute 2 for x and 5 for y in each equation.
Equation 1
3x 2 2y 5 24
Equation 2
7x 2 4y 5 26
3(2) 2 2(5) 0 24
7(2) 2 4(5) 0 26
24 5 24 ✓
26 5 26 ✓
Exercises for Example 1
Solve the linear system using elimination.
1. 15x 1 4y 5 25
5x 2 3y 5 30
2. 5x 1 3y 5 18
9y 5 27x 1 6
3. 4x 5 7y 1 14
14y 5 3x 1 7
32
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
CHECK
Name ———————————————————————
LESSON
7.4
Review for Mastery
Date ————————————
continued
For use with pages 463– 469
Multiply both equations, then add
Solve the linear system: 5x 1 2y 5 218
LESSON 7.4
EXAMPLE 2
Equation 1
7y 5 3x 1 19
Equation 2
Solution
STEP 1
Arrange the equations so that like terms are in columns.
5x 1 2y 5 218
Write Equation 1.
23x 1 7y 5 19
STEP 2
Multiply Equation 1 by 3 and Equation 2 by 5 so that the coefficients of x in
the equations are the least common multiple of 5 and 3, or 15.
5x 1 2y 5 218
23x 1 7y 5 19
STEP 3 Add the equations.
STEP 4 Solve for y.
STEP 5
Rewrite Equation 2.
33
15x 1 6y 5 254
35
215x 1 35y 5 95
41y 5 41
y51
Substitute 1 for y in either of the original equations and solve for x.
5x 1 2y 5 218
Write Equation 1.
5x 1 2(1) 5 218
Substitute 1 for y.
x 5 24
Solve for x.
Copyright © Holt McDougal. All rights reserved.
The solution is (24, 1).
CHECK
Substitute 24 for x and 1 for y in each equation.
Equation 1
5x 1 2y 5 218
Equation 2
7y 5 3x 1 19
5(24) 1 2(1) 0 218
7(1) 0 3(24) 1 19
218 5 218 ✓
757✓
Exercises for Example 2
Solve the linear system using elimination.
4. 9x 1 5y 5 33
12x 2 7y 5 3
5. 3x 1 7y 5 20
5x 5 24y 1 41
6. 9y 5 10x 1 4
12x 5 5y 1 30
Algebra 1
Chapter 7 Resource Book
33
Name ———————————————————————
LESSONS
7.1–7.4
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
tickets to a high school play. Student tickets
cost $5 and general admission tickets cost
$8. You sell 556 tickets and collect $3797.
a. Write a system of linear equations that
represent the situation.
b. How many of each type of ticket did
you sell?
2. Multi-Step Problem Biking into the
wind on a flat path, a bicyclist takes 5 hours
to travel 30 miles. The return bike takes
3 hours. The wind speed remains constant
during the trip.
a. Find the bicyclist’s average speed for
each leg of the trip.
b. Write a system of linear equations that
represent the situation.
c. What is the bicyclist’s average speed
in still air? What is the speed of the
wind?
3. Multi-Step Problem A total of $30,000
is invested in two accounts paying 3% and
4% annual interest. The combined annual
interest is $1020.
a. Write a system of linear equations that
represent the situation. (Hint: Write
one equation for the amount invested
in each account and another for the
interest earned.)
b. How much of the $30,000 is invested
in each account?
4. Gridded Response A bag contains dimes
and nickels. There are 18 coins in the bag.
The value of the coins is $1.25. How many
nickels are in the bag?
5. Open-Ended Describe a real-world
problem that can be modeled by a linear
system. Then graph and solve the system
and interpret the solution in the context of
the problem.
34
Algebra 1
Chapter 7 Resource Book
6. Short Response At a grocery store,
a customer pays a total of $11.10 for
1.6 pounds of chicken and 2 pounds of fish.
Another customer pays a total of $12.15 for
2.4 pounds of chicken and 1.8 pounds of
fish. How much do 2 pounds of chicken and
2 pounds of fish cost? Explain.
7. Open-Ended Find values for m and b so
that the system y 5 2x 2 5 and
y 5 mx 1 b has (6, 7) as a solution.
8. Gridded Response During one day,
two cars are sold at a car dealership. The
two customers each arrange payment plans
with the salesperson. The graph shows the
amount y of money (in dollars) paid for the
car after x months. After how many months
will each customer have paid the same
amount?
y
6000
4000
2000
0
0 1 2 3 4 5 6 x
Months since purchase
9. Extended Response A chemist needs
900 milliliters of a 40% acid solution for
a chemistry experiment. The chemist
combines x milliliters of a 20% acid solution
and y milliliters of a 70% acid solution to
make the 40% acid solution.
a. Write a linear system that represents
the situation.
b. How many milliliters of the 20% acid
solution and the 70% acid solution are
combined to make the 40% acid
solution?
c. The chemist also needs 900 milliliters of a 45% acid solution. Does the
chemist need more of the 20% acid
solution than the 70% acid solution to
make this new mix? Explain.
Copyright © Holt McDougal. All rights reserved.
1. Multi-Step Problem You are selling
Amount paid
(dollars)
LESSON 7.4
For use with pages 4392469
Name ———————————————————————
Date ————————————
Challenge Practice
LESSON
7.4
For use with pages 463–469
Terry has a summer job mowing lawns in a neighborhood that has only two different lot
sizes. After the first day of work, Terry’s boss observed that Terry mowed 1 small lawn and
2 large lawns in 5 hours. After the second day of work, Terry’s boss observed that Terry
mowed 3 small lawns and 3 large lawns in 8 hours.
LESSON 7.4
In Exercises 1–3, use the following information.
1. Write a linear system to model this situation, where x represents the number of small
lawns mowed and y represents the number of large lawns mowed.
2. Solve the linear system written in Exercise 1. What does the solution represent?
3. If Terry mows 2 large lawns in a 9-hour day, how many small lawns will he be able
to mow?
In Exercises 4 – 6, use the following information.
Greyson has a paper delivery route which he completes by riding his bicycle. The drop-off
station where he picks up his papers for delivery is located in the neighborhood where he
delivers papers. When riding between his house and the drop-off station, Greyson averages
10 miles per hour. On Monday through Saturday the paper is a small daily and Greyson
1
averages }2 mile per hour while making his deliveries. When delivering the small daily it
takes Greyson 4 hours and 18 minutes, from the moment he leaves his house to the
moment he returns in order to complete his route. On Sundays, the paper is much larger
1
and he averages }3 mile per hour while making his deliveries, which adds an additional
2 hours to the time it takes to complete his route.
Copyright © Holt McDougal. All rights reserved.
4. Write a linear system to model this situation, where x represents the miles from
Greyson’s house to the drop-off station and y represents the length of the route
(in miles).
5. What is the distance from Greyson’s house to the drop-off station?
6. What is the length of the paper route?
Algebra 1
Chapter 7 Resource Book
35
Name ———————————————————————
Date ————————————
Practice A
LESSON
7.5
For use with pages 471– 477
Identify the slope of the linear equation.
1. y 2 3x 5 8
2. 4x 1 2y 5 6
3. 9x 2 3y 5 15
Match the linear system with its graph. Then use the graph to tell whether
the linear system has one solution, no solution, or infinitely many
solutions.
4. 23x 1 y 5 2
5. x 2 y 5 5
26x 1 2y 5 4
x1y55
y
A.
6. 4x 1 y 5 2
24x 2 y 5 1
y
B.
5
C.
1
LESSON 7.5
3
23
23
y
3
3 x
1
21
21
21
21
3 x
1
1
3
x
5
23
Graph the linear system. Then use the graph to tell whether the linear
system has one solution, no solution, or infinitely many solutions.
7. x 1 y 5 24
8. y 2 2x 5 3
y 5 2x 1 1
9. 2x 1 2y 5 4
x1y52
y 5 2x 1 2
y
y
y
3
3
3
1
1
23
21
21
1
3 x
23
21
21
1
3 x
21
21
23
1
3
x
23
23
10. 3x 2 y 5 1
23
11. 4x 1 2y 5 8
2x 1 y 5 22
12. 2x 2 4y 5 4
3x 2 y 5 3
y
x 1 2 5 2y
y
y
3
3
3
1
1
1
23
21
21
1
3 x
23
21
21
1
3
x
23
21
21
23
23
36
Algebra 1
Chapter 7 Resource Book
23
1
3 x
Copyright © Holt McDougal. All rights reserved.
1
Name ———————————————————————
LESSON
7.5
Practice A
Date ————————————
continued
For use with pages 471–477
Solve the linear system by using substitution or elimination.
13. 25x 1 5y 5 210
14. 4x 2 4y 5 218
3x 2 3y 5 5
7x 2 7y 5 24
16. 24x 1 3y 5 1
17. 4x 2 y 5 2
3x 2 4y 5 1
15. 2x 2 5y 5 0
5
}x 2 y 5 0
2
18. 2x 1 4y 5 1
212x 1 3y 5 0
6x 1 12y 5 3
Without solving the linear system, tell whether the linear system has one
solution, no solution, or infinitely many solutions.
1
19. y 5 } x 1 3
2
20. y 5 6x 1 4
y 5 26x 2 10
22. y 2 3x 5 8
23. 3y 1 6x 5 8
3x 1 y 5 8
2x 1 y 5 210
2
26. 2} x 1 y 5 2
3
25. 4x 2 6y 5 21
3
1
2}2 x 1 y 5 }4
26x 1 3y 5 6
24. 4x 1 3y 5 9
3
}x 1 y 5 3
4
LESSON 7.5
y 5 22x 1 3
21. y 5 3x 2 5
6
y 5 }2 x2 5
27. 9x 2 15y 5 15
3
x 1 }5 y 5 1
Copyright © Holt McDougal. All rights reserved.
28. Water Park A water park charges a fee for admission to the park and a fee to rent
a tube for the day. One admission to the water park costs x dollars and a tube rental
for the wave pool costs y dollars. A group pays $263.25 for admission for 15 people
and 8 tube rentals. Another group pays $358 for admission for 20 people and 13 tube
rentals. Is there enough information to determine the cost of one admission to the
water park? Explain.
29. Movie Tickets The table below shows the ticket sales at a small theater on a
Thursday night and a Friday night.
Number of
adult tickets
Number of
children’s tickets
Total sales
(dollars)
Thursday
45
10
425
Friday
225
50
2125
Day
a. Let x represent the cost (in dollars) of one adult ticket and let y represent the
cost (in dollars) of one children’s ticket. Write a linear system that models
the situation.
b. Solve the linear system.
c. Can you determine how much each kind of ticket costs? Why or why not?
Algebra 1
Chapter 7 Resource Book
37
Name ———————————————————————
Date ————————————
Practice B
LESSON
7.5
For use with pages 471– 477
Match the linear system with its graph. Then use the graph to tell whether
the linear system has one solution, no solution, or infinitely many
solutions.
1. y 1 3 5 4x
2. 2x 1 y 5 1
3y 5 12x 2 9
A.
3. 3x 1 y 5 1
2x 1 y 5 5
B.
y
22x 1 y 5 23
C.
y
y
1
3
21
21
3
x
23
21
21
1
3 x
LESSON 7.5
1
21
21
23
x
Graph the linear system. Then use the graph to tell whether the linear
system has one solution, no solution, or infinitely many solutions.
4. 26x 1 2y 5 22
5. 2y 2 x 5 24
23x 1 y 5 2
6. 4x 2 y 5 2
2x 1 y 5 3
y
2x 1 3y 5 9
y
y
3
3
1
3 x
1
1
5 x
1
23
21
21
1
3
x
3
x
23
8. 3x 1 y 5 4
1
x 1 }3 y 5 2
7. x 1 2y 5 3
2x 1 2y 5 22
9. 2x 2 y 5 4
22x 1 y 5 24
y
y
y
1
3
23
21
21
1
21
3
5
23
3
x
1
23
38
Algebra 1
Chapter 7 Resource Book
21
1
3
5 x
1
Copyright © Holt McDougal. All rights reserved.
23
21
21
Name ———————————————————————
LESSON
7.5
Practice B
Date ————————————
continued
For use with pages 471–477
Solve the linear system by using substitution or elimination.
10. 3x 2 2y 5 24
11. 3x 1 2y 5 4
x 1 2y 5 8
12. x 1 y 5 50
26x 2 4y 5 28
13. 2x 1 4y 5 23
23x 1 2y 5 0
14. 2x 1 3y 5 9
23x 1 2y 5 1
15. 2x 1 y 5 6
2x 1 y 5 10
2x 1 y 5 27
Without solving the linear system, tell whether the linear system has one
solution, no solution, or infinitely many solutions.
8
17. y 1 2x 5 }
3
16. 26x 1 6y 5 24
2x 2 2y 5 5
2x 1 y 5 210
2
20. 2} x 1 y 5 2
3
19. 4x 2 6y 5 21
3
1
2}2 x 1 y 5 }4
21. 9x 2 15y 5 15
3
x 1 }5 y 5 1
26x 1 3y 5 6
22. 23x 1 4y 5 2
3
2y 5 }2 x 1 1
23. 3x 1 y 5 4
1
x 1 }3 y 5 2
24. 24x 1 3y 5 2
LESSON 7.5
18. 4x 1 3y 5 9
3
}x 1 y 5 3
4
4 2 6y 5 28x
Copyright © Holt McDougal. All rights reserved.
25. Golf Clubs A sporting goods store stocks a “better” set of golf clubs in both left-
handed and right-handed sets. The set of left-handed golf clubs sells for x dollars
and the set of right-handed golf clubs sells for y dollars. In one month, the store
sells 2 sets of left-handed golf clubs and 12 sets of right-handed golf clubs for a
total of $1859.30. The next month, the store sells 2 sets of left-handed golf clubs
and 22 sets of right-handed golf clubs for a total of $3158.80. Is there enough
information to determine the cost of each kind of set? Explain.
26. Comedy Tickets The table below shows the ticket sales at an all-ages comedy
club on a Friday night and a Saturday night.
Number of
adult tickets
Number of
student tickets
Total sales
(dollars)
Friday
30
20
910
Saturday
45
30
1365
Day
a. Let x represent the cost (in dollars) of one adult ticket and let y represent the cost
(in dollars) of one student ticket. Write a linear system that models the situation.
b. Solve the linear system.
c. Can you determine how much each kind of ticket costs? Why or why not?
Algebra 1
Chapter 7 Resource Book
39
Name ———————————————————————
Date ————————————
Practice C
LESSON
7.5
For use with pages 471– 477
Match the linear system with its graph. Then use the graph to tell whether
the linear system has one solution, no solution, or infinitely many
solutions.
1. 6x 1 4y 5 25
5
3x 1 2y 5 2}2
2. 3x 1 4y 5 12
A.
B.
3
3. y 5 } x 1 5
5
24x 1 3y 5 29
y
23x 1 5y 5 210
C.
y
y
1
1
21
21
21
21
23
1
x
Graph the linear system. Then use the graph to tell whether the linear
system has one solution, no solution, or infinitely many solutions.
4. 4y 5 3x 1 20
5. 3x 1 2y 5 8
4y 1 12 5 5x
y
y
6
5
3
2
3
1
22
22
2
6
10 x
7. 3x 1 4y 5 224
1
1
}y 1 }x 5 1
3
4
23
1
21
21
23
26
1
26
210
Algebra 1
Chapter 7 Resource Book
x
21
21
6 x
23
3 x
y
3
1
1
23
1
4y 2 10x 5 8
3
21
21
3 x
23
y
2
1
9. 25x 1 2y 5 3
22x 1 3y 5 1
2
22
22
3
8. 2x 1 3y 5 21
y
26
6. 3y 2 4x 5 6
4
y 5 }3 x 1 2
22x 1 3y 5 6
y
40
1
23
x
1
21
x
3
1
3 x
23
21
21
23
Copyright © Holt McDougal. All rights reserved.
LESSON 7.5
1
3
Name ———————————————————————
LESSON
7.5
Practice C
Date ————————————
continued
For use with pages 471–477
Solve the linear system by using substitution or elimination.
10. 2x 1 2y 5 24
23x 1 4y 5 4
13. 22x 1 5y 5 210
5y 2 2x 5 5
11. 4x 1 3y 5 2
3
2x 1 }2 y 5 1
12. x 1 8y 5 16
1
14. 22x 1 3y 5 2}
2
15. 2y 2 10x 5 28
23x 1 8y 5 28
2y 2 x 5 4
3x 1 2y 5 4
Without solving the linear system, tell whether the linear system has one
solution, no solution, or infinitely many solutions.
16. 4y 5 12x 2 1
19. 5y 2 4x 5 3
10y 5 8x 1 6
18. 22x 1 3y 5 4
1
20. y 2 } x 5 22
4
21. 3y 1 5x 5 1
3x 2 2y 5 5
25x 2 3y 5 1
x 2 2y 5 8
22. 2y 2 x 5 3
23. 23x 1 4y 5 24
2x 1 y 5 6
4x 1 3y 5 2
24. 4y 5 25x 1 3
5
3
2y 1 }2 x 5 }2
LESSON 7.5
–12x 1 3y 5 21
17. x 1 4y 5 3
1
} x 1 2y 5 4
2
25. Restaurant Sales The table below shows the number of each of the specials that
has been sold on a Friday night and a Saturday night.
Number of
vegetarian specials
Number of
chicken specials
Total sales
(dollars)
Friday
28
44
964.40
Saturday
21
33
723.30
Copyright © Holt McDougal. All rights reserved.
Day
a. Let x represent the cost (in dollars) of the vegetarian special and let y represent
the cost (in dollars) of the chicken special. Write a linear system that models the
situation.
b. Solve the linear system.
c. Can you determine how much each kind of special costs? Why or why not?
26. Retail Prices Two employees at a store are given the task of putting price tags on items.
One person starts pricing items at a rate of 10 items per minute. The second person starts
10 minutes after the first person and prices items at a rate of 8 items per minute.
a. Let y be the number of items priced x minutes after the first person starts pricing.
Write a linear system that models the situation.
b. Solve the linear system.
c. Does the solution of the linear system make sense in the context of the problem?
Explain.
Algebra 1
Chapter 7 Resource Book
41
Name ———————————————————————
LESSON
7.5
Date ————————————
Review for Mastery
For use with pages 471– 477
GOAL
Identify the number of solutions of a linear system.
Vocabulary
A linear system with no solution is called an inconsistent system.
A linear system with infinitely many solutions is called a dependent
system.
EXAMPLE 1
A linear system with no solution
LESSON 7.5
Show that the linear system has no solution.
25x 1 4y 5 16
Equation 1
5x 2 4y 5 8
Equation 2
Solution
Graphing
y
Graph the linear system.
3
The lines are parallel because they have the same
slope but different y-intercepts. Parallel lines do not
intersect, so the system has no solution.
Method 2
1
21
21
3
x
Elimination
Add the equations.
25x 1 4y 5 16
5x 2 4y 5 8
0 5 24
This is a false statement.
Copyright © Holt McDougal. All rights reserved.
Method 1
The variables are eliminated and you are left with a false statement regardless of the
values of x and y. This tells you that the system has no solution.
EXAMPLE 2
A linear system with infinitely many solutions
Show that the linear system has infinitely many solutions.
2
y 5 }3 x 1 5
Equation 1
22x 1 3y 5 15
Equation 2
y
7
Solution
Method 1 Graphing
3
Graph the linear system.
The equations represent the same line, so any point
on the line is a solution. So, the linear system has
infinitely many solutions.
42
Algebra 1
Chapter 7 Resource Book
1
23
21
21
1
3
x
Name ———————————————————————
LESSON
7.5
Review for Mastery
For use with pages 471–477
Method 2
Date ————————————
continued
Substitution
2
Substitute }3 x 1 5 for y in Equation 2 and solve for x.
22x 1 3y 5 15
22x 1 31 }3 x 1 5 2 5 15
2
22x 1 2x 1 15 5 15
15 5 15
Write Equation 2.
2
Substitute }3 x 1 5 for y.
Distributive property
Simplify.
The variables are eliminated and you are left with a statement that is true regardless
of the values of x and y. This tells you the system has infinitely many solutions.
LESSON 7.5
Exercises for Examples 1 and 2
Tell whether the linear system has no solution or infinitely
many solutions.
1. 215x 1 3y 5 6
2. 24x 1 y 5 5
y 5 5x 1 2
EXAMPLE 3
y 5 4x 1 3
Identify the number of solutions
Without solving the linear system, tell whether the linear system has
one solution, no solution, or infinitely many solutions.
a. 7x 2 2y 5 9
Copyright © Holt McDougal. All rights reserved.
7x 2 2y 5 21
Solution
7
9
a. y 5 } x 2 }
2
2
7
1
y 5 }2 x 1 }2
Equation 1
b. 3x 1 y 5 210
Equation 2
Equation 1
26x 2 2y 5 20
Equation 2
Write Equation 1 in slope-intercept form.
Write Equation 2 in slope-intercept form.
Because the lines have the same slope but different y-intercepts, the system has no
solution.
b. y 5 23x 2 10
y 5 23x 2 10
Write Equation 1 in slope-intercept form.
Write Equation 2 in slope-intercept form.
The lines have the same slope and y-intercept, so the system has infinitely many
solutions.
Exercises for Example 3
Without solving the linear system, tell whether the linear system has
one solution, no solution, or infinitely many solutions.
3. x 2 3y 5 7
4x 5 12y 1 28
4.
2x 1 3y 5 17
3x 1 2y 5 14
5. 24x 1 y 5 5
28x 2 14y 5 228
Algebra 1
Chapter 7 Resource Book
43
Name ———————————————————————
LESSON
7.5
Date ————————————
Challenge Practice
For use with pages 471–477
In Exercises 1–3, use the linear system.
1
ax 1 }4 y 5 7
1
6
1
3
}x 1 }y 5 3
1. For what values of a does the system have no solution?
2. For what values of a does the system have infinitely many solutions?
3. For what values of a does the system have exactly one solution?
LESSON 7.5
In Exercises 4 and 5, suppose a, b, and c are non-zero constants.
Use the linear system.
ax 1 by 5 3
cax 1 cby 5 12
4. Does the number of solutions depend on the values of a, b, and c?
5. Describe the number of solutions in each possible case.
In Exercises 6 – 9, suppose a1, a2, b1, b2, c1, and c2 are non-zero constants.
Use the linear system.
a1x 1 b1 y 5 c1
a2x 1 b2 y 5 c2
Solve for x and y in terms of a1, a2, b1, b2, c1, and c2.
7. State the relationship between the values of a1, a2, b1, b2, c1, and c2 that will
guarantee there is exactly one solution.
8. State the relationship between the values of a1, a2, b1, b2, c1, and c2 that will
guarantee there is no solution.
9. State the relationship between the values of a1, a2, b1, b2, c1, and c2 that will
guarantee there are infinitely many solutions.
44
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
6.
Name ———————————————————————
Graphing Calculator Activity Keystrokes
LESSON
7.6
For use with pages 485–486
TI-83 Plus
Example 1
Y=
Casio CFX-9850GC Plus
Example 1
X,T,,n
ENTER
2
From the main menu, choose GRAPH.
3
ENTER
1
X,T,,n
ENTER
ENTER
ENTER
GRAPH
Press one of the arrow keys to place the cursor
on the screen. Use the arrow keys to move the
cursor to points in the graph of the system.
()
4
3
X,T,,n
1
ENTER
ENTER
F3
3
F6
X,T,,n
F1
X,,T
1
EXE
2
EXE
F3
F6
F4
F6
Press F1 [Trace] to place the cursor on the
screen. Use the arrow keys to move the cursor
to points in the graph of the system.
Example 2
From the main menu, choose GRAPH.
Example 2
Y=
Date ————————————
X,T,,n
ENTER
ENTER
5
ENTER
4
F3
F6
F1
()
F3
F6
F4
X,,T
ab c
3
X,,T
5
EXE
1
EXE
F6
To identify a solution, press F1 [Trace] and use
the cursor to locate a point in the graph of the
system, or simply identify a solution visually.
GRAPH
To identify a solution, use the cursor to locate
a point in the graph of the system, or simply
identify a solution visually.
Copyright © Holt McDougal. All rights reserved.
LESSON 7.6
Algebra 1
Chapter 7 Resource Book
45
Name ———————————————————————
Date ————————————
Practice A
LESSON
7.6
For use with pages 478–484
Tell whether the ordered pair is a solution of the system of inequalities.
1. (2, 1)
3. (0, 21)
2. (23, 2)
y
y
y
3
1
3
25
23
21
21
x
1
21
3
1
1 x
23
21
1
3 x
1
3 x
2
6 x
2
6 x
5
4. (22, 0)
5. (2, 4)
6. (22, 3)
y
y
y
3
3
3
1
25
23
1
1 x
21
1
3
21
21
x
23
23
7. x 1 y ≥ 4
8. x 1 y ≤ 4
x < 22
x<2
10. y 1 x ≤ 4
y>2
11. x 2 y ≤ 4
12. y 1 x ≥ 4
x > 22
y<2
A.
9. x 2 y ≥ 4
y < 22
B.
y
6
C.
y
10
y
6
6
22
22
2
6
10 x
26
D.
2
E.
y
26
2
6 x
22
22
26
F.
y
y
6
6
2
2
2
26
22
22
22
6 x
26
46
Algebra 1
Chapter 7 Resource Book
26
2
6
10 x
22
Copyright © Holt McDougal. All rights reserved.
LESSON 7.6
Match the system of inequalities with its graph.
Name ———————————————————————
Practice A
LESSON
7.6
Date ————————————
continued
For use with pages 478–484
Graph the system of inequalities.
13. x > 21
14. y > 23
15. x ≥ 2
y≤0
x<4
y>0
y
y
y
3
3
1
1
3
1
23
23
21
21
1
3
x
21
21
1
3
21
21
23
x
1
3
x
23
23
23
16. x < 1
18. y ≤ 3
17. x > 0
y ≤ 22
y≤x
y > 2x
y
y
y
3
3
3
1
1
1
23
21
21
1
3
x
23
21
21
1
3
23
x
21
21
1
3 x
23
23
23
and let y represent the number of boxes of 8-ounce cups.
Write a system of linear inequalities for the number of
cups that can be bought.
b. Graph the system of inequalities.
c. Identify two possible combinations of cups you can buy.
20. Studying You need at least 4 hours to do your science and
history homework. It is 1:00 P.M. on Sunday and your friend
wants you to go to the movies at 7:00 P.M.
a. How much time do you have between now and 7:00 P.M.
to do your homework?
b. Let x represent the number of hours spent on science
homework and let y represent the number of hours spent
on history homework. Write and graph a system of linear
inequalities that shows the number of hours you can work
on each subject if you go to the movies.
Boxes of 8-ounce cups
a. Let x represent the number of boxes of 5-ounce cups
Hours spent on history
Copyright © Holt McDougal. All rights reserved.
the summer. You need to order 5-ounce and 8-ounce cups.
The storage room will only hold 10 more boxes of cups.
A box of 5-ounce cups costs $15 and a box of 8-ounce cups
costs $18. A maximum of $90 is budgeted for cups.
y
9
8
7
6
5
4
3
2
1
0
LESSON 7.6
19. Ordering Cups You work at an Italian ice shop during
0 1 2 3 4 5 6 7 8 9 x
Boxes of 5-ounce cups
y
6
5
4
3
2
1
0
0 1 2 3 4 5 6 x
Hours spent on science
Algebra 1
Chapter 7 Resource Book
47
Name ———————————————————————
Date ————————————
Practice B
LESSON
7.6
For use with pages 478–484
Tell whether the ordered pair is a solution of the system of inequalities.
1. (3, 0)
2. (2, 2)
y
3. (22, 2)
y
y
5
1
3
21
3
1
x
1
23
21
21
23
1
5 x
3
21
21
3 x
Match the system of inequalities with its graph.
1
4. } x 1 y ≥ 3
2
1
5. y 2 } x ≤ 3
2
1
6. y ≤ } x 1 3
2
x > 21
x < 21
x > 21
A.
B.
y
C.
y
1
y
1
23
23
3 x
1
1
1
3 x
23
1
3 x
1
3 x
1
3 x
7. x > 21
8. y ≥ 2
x<1
9. x 1 y > 1
x≤y
y<3
y
y
y
3
3
1
23
1
3 x
21
23
21
21
23
1
3 x
23
21
23
10. x ≥ y 1 2
23
11. y ≥ 2
2x 1 y < 4
12. x ≤ 2y
x 1 y ≤ 23
2x 2 y < 4
y
y
y
1
3
3
1
1
21
21
48
1
Algebra 1
Chapter 7 Resource Book
3
x
25
23
x
21
21
23
21
21
23
Copyright © Holt McDougal. All rights reserved.
LESSON 7.6
Graph the system of inequalities.
Name ———————————————————————
LESSON
7.6
Practice B
Date ————————————
continued
For use with pages 478–484
Write a system of inequalities for the shaded region.
13.
14.
y
y
15.
y
3
1
25
23
21
21
1
23
x
21
21
3 x
23
21
1
3 x
1
3 x
23
23
25
16.
17.
y
18.
y
3
y
5
3
3
21
21
1
3 x
3 x
23
19. Cookout You are planning a cookout. You figure that you will
need at least 5 packages of hot dogs and hamburgers. A package
of hot dogs costs $1.90 and a package of hamburgers costs $5.20.
You can spend a maximum of $20 on the hot dogs and hamburgers.
represent the number of packages of hamburgers. Write a system
of linear inequalities for the number of packages of each that
can be bought.
b. Graph the system of inequalities.
c. Identify two possible combinations of packages of hot dogs and
hamburgers you can buy.
23
21
21
Y
Y
20. Chores You need at least 4 hours to do your chores, which are
cleaning out the garage and weeding the flower beds around
your house. It is 1:30 P.M. on Sunday and your friend wants you
to go to the movies at 7:00 P.M.
a. How much time do you have between now and 7:00 P.M. to
do your chores?
b. Let x represent the number of hours spent cleaning out the
garage and let y represent the number of hours spent on
weeding the flower beds. Write and graph a system of linear
inequalities that shows the number of hours you can work on
each chore if you go to the movies.
c. Identify two possible combinations of time you can spend on each chore.
X
0ACKAGES OF HOT DOGS
LESSON 7.6
Copyright © Holt McDougal. All rights reserved.
a. Let x represent the number of packages of hot dogs and let y
1
0ACKAGES OF HAMBURGERS
23
1
(OURS WEEDING
23
1
X
(OURS CLEANING
Algebra 1
Chapter 7 Resource Book
49
Name ———————————————————————
Date ————————————
Practice C
LESSON
7.6
For use with pages 478–484
Tell whether the ordered pair is a solution of the system of inequalities.
2. (0, 21)
1. (0, 1)
3. (1, 4)
y
y
y
3
23
3 x
1
3 x
1
21
1
21
21
21
x
1
3
5
Match the system of inequalities with its graph.
4. 3x 1 2y ≥ 4
5. 3x 1 2y ≥ 24
y>42x
6. 3x 2 2y ≤ 4
x1y<4
y
A.
y
B.
20
x1y<4
y
C.
10
6
12
6
2
22
x
220
212
24
24
2
6
x
x
26
4
22
22
7. x ≥ 22
9. 3x 1 y < 0
8. x < 0
y≤5
y > 21
4x 2 y ≤ 1
y
y
3
3
3
1
1
23
1
23
y
5
21
21
1
21
21
x
3 x
1
23
23
10. x ≥ 0, y ≥ 0
11. x > 4, x < 8
2x 1 y < 3
y ≥ 2x 1 1
y
21
21
1
3 x
23
12. y > 22, x ≥ 0
y ≥ 3x
y
y
20
3
12
1
3
1
21
21
50
1
Algebra 1
Chapter 7 Resource Book
3
x
23
4
24
24
4
12
x
21
21
23
1
3 x
Copyright © Holt McDougal. All rights reserved.
LESSON 7.6
Graph the system of inequalities.
Name ———————————————————————
LESSON
7.6
Practice C
Date ————————————
continued
For use with pages 478–484
Write a system of inequalities for the shaded region.
13.
14.
y
15.
y
y
3
3
1
3
23
21
1
1
3 x
1
23
1
16.
x
21
21
3
2
6
23
3 x
17.
y
1
18.
y
y
3
3
23
21
21
1
1
3 x
23
21
21
2
1
22
3 x
x
23
19. School Play The tickets for a school play cost $8 for adults and
Student tickets
$5 for students. The auditorium in which the play is being held can
hold at most 525 people. The organizers of the school play must
make at least $3000 to cover the costs of the set construction,
costumes, and programs.
a. Write a system consisting of an equation and an inequality
that describes the situation.
b. Draw a graph to show the possible combinations of hours
that you could exercise.
c. Interpret the graph in the context of the problem.
Swimming
type of ticket sold.
b. Graph the system of inequalities.
c. If the organizers sell out and sell twice as many student tickets
as adult tickets, can they reach their goal? Explain how you got
your answer.
20. Exercise You exercise 15 hours per week by swimming and
running. You want to spend at least twice the amount of time
swimming as running.
y
14
12
10
8
6
4
2
0
0
200 400 600 x
Adult tickets
LESSON 7.6
Copyright © Holt McDougal. All rights reserved.
a. Write a system of linear inequalities for the number of each
y
600
500
400
300
200
100
0
0 2 4 6 8 10 12 14 x
Running
Algebra 1
Chapter 7 Resource Book
51
Name ———————————————————————
LESSON
7.6
Date ————————————
Review for Mastery
For use with pages 478–484
GOAL
Solve systems of linear inequalities in two variables.
Vocabulary
A system of linear inequalities in two variables, or simply a system
of inequalities, consists of two or more linear inequalities in the same
variables.
A solution of a system of linear inequalities is an ordered pair that is
a solution of each inequality in the system.
The graph of a system of linear inequalities is the graph of all
solutions of the system.
EXAMPLE 1
Graph a system of two linear inequalities
y
Graph the system of inequalities.
1
y < }2 x 1 2
Inequality 1
y ≥ 22x 1 5
Inequality 2
7
5
3
Solution
Graph both inequalities in the same coordinate plane.
The graph of the system is the intersection of the two
half-planes, which is shown as the shaded region.
1
23
21
21
x
1
Inequality 1
1
EXAMPLE 2
Inequality 2
y < }2 x 1 2
y ≥ 22x 1 5
2?
< }2 (2) 1 2
2?
≥ 22(2) 1 5
2<3✓
2≥1✓
1
Copyright © Holt McDougal. All rights reserved.
LESSON 7.6
CHECK Choose a point in the shaded region, such as (2, 2). To check this solution,
substitute 2 for x and 2 for y into each inequality.
Graph a system of three linear inequalities
Graph the system of inequalities.
y≤5
x<4
y ≥ 22x 1 2
Inequality 1
Inequality 2
Inequality 3
y
6
Solution
Graph all three inequalities in the same coordinate plane.
The graph of the system is the triangular region shown.
2
26
22
22
26
52
Algebra 1
Chapter 7 Resource Book
6 x
Name ———————————————————————
LESSON
7.6
Review for Mastery
Date ————————————
continued
For use with pages 478–484
Exercises for Examples 1 and 2
Graph the system of linear inequalities.
1. y > 3x 2 2
2.
y ≤ }2 x 1 1
3
EXAMPLE 3
x > 22
3. y > 2
y > 23
y<8
3
4
y ≥ 4x 2 1
y ≤ }x 1 2
Write a system of linear inequalities
y
Write a system of inequalities for the shaded
region.
5
Solution
3
Inequality 1 One boundary for the shaded region has
a slope of 24 and a y-intercept of 5. So, its equation is
y 5 24x 1 5. Because the shaded region is below the
solid line, the inequality is y ≤ 24x 1 5.
1
21
21
1
5
x
3
Inequality 2 Another boundary line for the shaded region has a slope of }5 and a
3
y-intercept of 22. So, its equation is y 5 }5 x 2 2. Because the shaded region is
3
Copyright © Holt McDougal. All rights reserved.
The system of inequalities for the shaded region is: y ≤ 24x 1 5
Inequality 1
3
y > }5 x 2 2
Inequality 2
Exercises for Example 3
LESSON 7.6
above the dashed line, the inequality is y > }5 x 2 2.
Write a system of inequalities that defines the shaded region.
4.
5.
y
y
6
3
1
23
21
2
1
6
10
x
3 x
Algebra 1
Chapter 7 Resource Book
53
Name ———————————————————————
LESSONS
7.5–7.6
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 471–484
32 pages per hour. Anthony starts
15 minutes after Stacy and can read
28 pages per hour.
a. Let y be the number of pages read
x hours after Stacy began reading.
Write a linear system that models
the situation.
b. Solve the linear system.
c. Does the solution of the linear system
make sense in the context of the
problem? Explain.
2. Multi-Step Problem A restaurant offers
LESSON 7.6
two different meals each evening and has at
least 260 customers. For Friday night, the
restaurant offers salmon and lemon chicken.
The restaurant expects that more people
will order the chicken than the salmon. The
salmon costs $6 per serving and the chicken
costs $4 per serving. The restaurant has a
budget of at most $1600 for meat for
Friday night.
a. Let x be the number of customers
who ordered salmon and let y be the
number of customers who ordered
lemon chicken. Write a system of
linear inequalities that models the
situation.
b. Graph the system of inequalities.
c. Use the graph to determine whether
120 orders of salmon and 160 orders
of chicken can be ordered.
3. Open-Ended Write a linear system so that
it has no solution and one of the equations is
5x 2 4y 5 26.
4. Gridded Response What is the area (in
square units) of the garden defined by the
system of inequalities below?
y≥0
x≥0
y≤4
x1y≤8
54
Algebra 1
Chapter 7 Resource Book
5. Short Response During a sale at a
clothing store, all shirts are priced the same
and all shorts are priced the same. Lucy
buys 6 shirts and 3 shorts for $78. The next
day, while the sale is still in progress, Lucy
goes back and buys 2 shirts and 1 pair of
shorts for $26. Is there enough information
to determine the cost of 1 shirt? Explain.
6. Extended Response During the summer,
you want to earn at least $120 per week. You
earn $9 per hour babysitting and you earn
$6 per hour working at a grocery store. You
can work at most 25 hours per week.
a. Write and graph a system of linear
inequalities that models the situation.
b. You work 7 hours per week
babysitting and 8 hours per week at
the grocery store. Will you earn at
least $120 per week? Explain.
c. You are scheduled to work 12 hours
babysitting. What is the range of hours
you can work at the grocery store to
earn at least $120 per week?
7. Short Response Is it possible to find a
value for c so that the linear system below
has no solution? Explain.
3x 2 7y 5 14
3
y 5 }7 x 1 c
Equation 1
Equation 2
8. Extended Response Martin decides to
make a walkway in his backyard. He spends
$94 on 6 large bricks and 8 small bricks.
Then he decides to make another walkway
using the same kinds of bricks. He spends
$188 for 12 large bricks and 16 small bricks.
a. Write a system of linear equations that
models the situation.
b. Is there enough information given
to determine the cost of one brick of
each type? Explain.
c. A large brick costs $4 more than a
small brick. What is the cost of one
brick of each type?
Copyright © Holt McDougal. All rights reserved.
1. Multi-Step Problem Stacy can read
Name ———————————————————————
LESSON
7.6
Date ————————————
Challenge Practice
For use with pages 478–484
Graph the system of inequalities.
1. y ≥ ⏐x⏐
y ≤ 6 2 ⏐x⏐
2.
⏐x⏐ ≤ 2
⏐y⏐ ≤ 2
In Exercises 3–6, use the following information.
Your school club decides to hold a fundraiser by selling trail mix, and you are in charge of
making the mix. You plan to offer two mixes, Country Blend and Premium Mix, each
1
sold in one pound bags. Each pound of Country Blend consists of }2 pound of toasted
1
1
oats, }4 pound of peanuts, and }4 pound of raisins. Each pound of Premium Mix consists
1
1
1
of }4 pound of toasted oats, }4 pound of peanuts, and }2 pound of raisins. You have available
to use at most 40 pounds of oats, 22 pounds of peanuts, and 35 pounds of raisins.
3. Model the situation above by letting x represent the number of pounds of Country
Blend and y represent the number of pounds of Premium Mix. Your algebraic model
should be a system of five inequalities. (Remember that you cannot make a negative
number of pounds of trail mix.)
4. Graph the system of inequalities from Exercise 3.
Premium Mix. How many bags of each type of mix should you make in order to
maximize your income? (Hint: the maximum income must occur at one of the
vertices of the graph.)
6. Using the answer from Exercise 5, what will be your club’s income if all the bags of
mix are sold?
Algebra 1
Chapter 7 Resource Book
LESSON 7.6
Copyright © Holt McDougal. All rights reserved.
5. You sell the trail mix for $5 per pound for Country Blend and $7 per pound for
55
Name ———————————————————————
CHAPTER
7
Date ————————————
Chapter Review Game
For use after Chapter 7
Magic Square
Solve each linear system in the table using any method. Place the
indicated coordinate on the line given in the box. When the puzzle is
completed correctly, the sum of each row, column, and diagonal should
be the same. Place the sum of each row, column, and diagonal on the
given lines next to the square.
Diagonal: _______
3
2
1
}x 1 y 5 4
4
} x 1 2y 5 12
y-coordinate
______
7x 2 y 5 225
2x 1 5y 5 14
y-coordinate
______
CHAPTER REVIEW GAME
Column 1: _______
56
Algebra 1
Chapter 7 Resource Book
1
y 5 2}2 x 1 8
3x 1 y 5 4
22x 1 y 5 21
x-coordinate
______
y 5 2x 2 7
x-coordinate
______
4x 1 3y 5 8
x 2 2y 5 13
x-coordinate
______
y5x14
y 5 5x 2 8
y-coordinate
______
y 5 22x 1 21
3x 1 2y 5 8
3x 2 4y 5 2
x-coordinate
______
1
y 5 }x 1 7
3
y-coordinate
______
Column 2: _______
Column 3: _______
Row 1: _______
Row 2: _______
Row 3: _______
Diagonal: _______
Copyright © Holt McDougal. All rights reserved.
2x 2 y 5 0
2x 1 y 5 4
y-coordinate
______
Name ———————————————————————
CHAPTER
8
Date ————————————
Family Letter
For use with Chapter 8
Lesson Title
Lesson Goals
Key Applications
8.1: Apply Exponent Properties
Involving Products
Use properties of exponents
involving products.
• Bees
• Ice Cream Composition
• Coastal Landslide
8.2: Apply Exponent Properties
Involving Quotients
Use properties of exponents
involving quotients.
• Fractal Tree
• Astronomy
• Space Travel
8.3: Define and Use Zero and
Negative Exponents
Use zero and negative
exponents.
• Mass
• Botany
• Medicine
Copyright © Holt McDougal. All rights reserved.
Focus on Operations
Use fractional exponents.
8.4: Use Scientific Notation
Read and write numbers in
scientific notation.
• Blood Vessels
• Insect Lengths
• Agriculture
8.5: Write and Graph Exponential
Growth Functions
Write and graph exponential
growth models.
• Collector Car
• Compound Interest
• Investments
8.6: Write and Graph Exponential
Decay Functions
Write and graph exponential
decay functions.
• Forestry
• Cell Phones
• Guitars
Focus on Functions
Identify, graph, and write
geometric sequences.
CHAPTER SUPPORT
Chapter Overview One way you can help your student succeed in Chapter 8 is
by discussing the lesson goals in the chart below. When a lesson is completed, ask your
student the following questions. “What were the goals of the lesson? What new words and
formulas did you learn? How can you apply the ideas of the lesson to your life?”
Key Ideas for Chapter 8
In Chapter 8, you will apply the key ideas listed in the Chapter Opener (see page 503)
and reviewed in the Chapter Summary (see page 559).
1. Applying properties of exponents to simplify expressions
2. Working with numbers in scientific notation
3. Writing and graphing exponential functions
Algebra 1
Chapter 8 Resource Book
57
Name ———————————————————————
CHAPTER
Family Letter
continued
For use with Chapter 8
Key Ideas Your student can demonstrate understanding of key concepts by working
through the following exercises with you.
Lesson
Exercise
8.1
A farming corporation plants 103 seeds per acre of land. The corporation plants 105
acres. Use order of magnitude to find the number of seeds that were planted.
8.2
A city has 1000 gas pumps. During the past year, 94,750,000 gallons of gas were
sold in the city. Use order of magnitude to find the approximate number of gallons
sold per gas pump.
8.3
Simplify the expression. Write your answer using only positive exponents.
(b) (22x4y24z)23
(a) (23x)5 • (23)27
(5x)23 p y4
1
(4x)
(c) }
22
(d) }
6 26
2x y
Evaluate the expression 1002 • 10023/2.
Focus on
Operations
8.4
Evaluate the expression (4.3 3 106)(2.1 3 1022).
(a) Write the answer in scientific notation.
(b) Write the answer in standard form.
8.5
You inherited a stamp collection valued at $400 when you were 10 years old. The
value of the collection increases at a rate of 4.3% per year. How much will it be
worth when you turn 18? Round your answer to the nearest dollar.
8.6
Find the value of a $20,000 boat after 5 years if the boat depreciates
8% per year. Round your answer to the nearest dollar.
Focus on
Functions
Tell whether the sequence is arithmetic or geometric. Then write the next term of
the sequence. 7, 14, 21, 28, 35, 42, ...
Home Involvement Activity
Directions Investigate five different banks or credit unions to learn their interest
rates and how money is compounded (monthly, yearly) in their savings accounts. Then
find the balance for each account after one, five, and ten years with principal amounts
of $500, $2000, and $10,000. Analyze your findings to determine the best account for
short-term and long-term investments.
Focus on Operations: 10 8.4: (a) 9.03 3 104 (b) 90,300 8.5: $560 8.6: $13,182
Focus on Functions: arithmetic; 49
28x z
2125x
8.1: 108 seeds 8.2: about 105 gallons 8.3: (a) }
(b) }
(c) 16x2 (d) }9
12 3
9
Answers
y12
y 10
Algebra 1
Chapter 8 Resource Book
x5
58
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
8
Date ————————————
Nombre ——————————————————————
CAPÍTULO
8
Fecha ———————————
Carta para la familia
Usar con el Capítulo 8
Título de la lección
Objetivos de la lección
Aplicaciones clave
8.1: Aplicar propiedades de
exponentes con productos
Usar propiedades de
exponentes con productos
• Abejas
• Redacción del helado
• Derrumbamiento de
tierras costales
8.2: Aplicar propiedades de
exponentes con cocientes
Usar propiedades de
exponentes con cocientes
• Árbol de factores
• Astronomía
• Viaje espacial
8.3: Definir y usar cero y
exponentes negativos
Usar cero y exponentes
negativos
Usar exponentes fraccionales
• Masa
• Botánica
• Medicina
8.4: Usar la notación científica
Leer y escribir números
usando la notación científica
• Vasos sanguíneos
• Longitudes de insectos
• Agricultura
8.5: Escribir y graficar funciones
de crecimiento exponencial
Escribir y graficar modelos
de crecimiento exponencial
• Carro coleccionable
• Interés compuesto
• Inversiones
8.6: Escribir y graficar funciones
de decrecimiento exponencial
Escribir y graficar modelos
de decrecimiento exponencial
Identificar, graficar y escribir
secuencias geométricas
• Silvicultura
• Teléfonos celulares
• Guitarras
Copyright © Holt McDougal. All rights reserved.
Enfoque en las operaciones
Enfoque en las funciones
CHAPTER SUPPORT
Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener éxito en
el Capítulo 8 es hablar sobre los objetivos de la lección en la tabla a continuación. Cuando
se termina una lección, pregúntele a su hijo lo siguiente: “¿Cuáles fueron los objetivos de
la lección? ¿Qué palabras y fórmulas nuevas aprendiste? ¿Cómo puedes aplicar a tu vida
las ideas de la lección?”
Ideas clave para el Capítulo 8
En el Capítulo 8, aplicarás las ideas clave enumeradas en la Presentación del capítulo
(ver la página 503) y revisadas en el Resumen del capítulo (ver la página 559).
1. Aplicar propiedades de exponentes para simplificar expresiones
2. Trabajar con números en notación científica
3. Escribir y graficar funciones exponenciales
Algebra 1
Chapter 8 Resource Book
59
Nombre ——————————————————————
Carta para la familia
CAPÍTULO
continúa
Usar con el Capítulo 8
Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer los
siguientes ejercicios con usted.
Lección
Ejercicio
8.1
Una compañía agrícola siembra 103 semillas por acre de tierra. La compañía
siembra 105 acres. Usa el orden de magnitud para hallar el número de semillas
que se sembraron.
8.2
Una ciudad tiene 1000 bombas de gasolina. Durante el año pasado, 94,750,000
galones de gasolina se vendieron en la ciudad. Usa el orden de magnitud para
hallar el número aproximado de galones vendidos por bomba de gasolina.
8.3
Simplifica la expresión. Escribe tu respuesta usando solo exponentes positivos.
(a) (23x)5 • (23)27
(5x)23 p y4
1
(4x)
(c) }
22
Enfoque en las
operaciones
(b) (22x4y24z)23
(d) }
6 26
2x y
Evalúa la expresión 1002 • 10023/2.
8.4
Evalúa la expresión (4.3 3 106)(2.1 3 1022).
(a) Escribe la respuesta usando la notación científica.
(b) Escribe la respuesta en forma usual.
8.5
Heredaste una colección de sellos con un valor de $400 cuando tenías 10 años.
El valor de la colección aumenta a una tasa de 4.3% por año. ¿Qué será su valor
cuando cumples los 18 años? Redondea tu respuesta al dólar más próximo.
8.6
Halla el valor de un barco de $20,000 después de 5 años si se deprecia 8% por
año. Redondea tu respuesta al dólar más próximo.
Enfoque en las
funciones
Indica si la secuencia es aritmética o geométrica. Luego escribe el término que
sigue en la secuencia. 7, 14, 21, 28, 35, 42, ...
Actividad para la familia
Instrucciones Investiga cinco bancos o cooperativos de crédito diferentes para
saber sus tasas de interés y cómo se compone el dinero (mensualmente, anualmente)
en sus cuentas de ahorros. Luego halla el saldo de cada cuenta después de un, cinco y
diez años con cantidades principales de $500, $2000 y $10,000. Analiza tus hallazgos
para determinar la mejor cuenta para inversiones de corto plazo y de largo plazo.
Enfoque en las operaciones: 10 8.4: (a) 9.03 3 104 (b) 90,300 8.5: $560 8.6: $13,182
Enfoque en las funciones: aritmética; 49
2125x
(c) 16x2 (d) }9
y 10
28x z
8.1: 108 semillas 8.2: aproximadamente 105 galones 8.3: (a) }
(b) }
12 3
9
Respuestas
y12
Algebra 1
Chapter 8 Resource Book
x5
60
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
8
Fecha ———————————
Name ———————————————————————
LESSON
8.1
Date ————————————
Activity Support Master
For use with page 504
Expression
Expression as
repeated multiplication
Number
of factors
Simplified
expression
74 p 75
(7 p 7 p 7 p 7) p (7 p 7 p 7 p 7 p 7)
9
79
(24)2 p (24)3
[(24) p (24)] p [(24) p (24) p (24)]
x1 p x5
Expanded expression
Expression
as repeated
multiplication
Number
of factors
Simplified
expression
(53)2
(53) p (53)
(5 p 5 p 5) p (5 p 5 p 5)
6
56
F (26)2 G4
F (26)2 G p F (26)2 G p
F (26)2 G p F (26)2 G
LESSON 8.1
Expression
Copyright © Holt McDougal. All rights reserved.
(a 3)3
Algebra 1
Chapter 8 Resource Book
61
Name ———————————————————————
Date ————————————
Practice A
LESSON
8.1
For use with pages 504–510
Name the property that is demonstrated by the example.
1. (2x)3 5 23 p x3 5 8x3
2. x4 p x5 5 x 415 5 x9
( y3)2 5 y 3p2 5 y 6
3.
Fill in the blanks.
4.
(z3)5 5 z 3
5. (5x)4 5 5
5
5z
5
6. 33 p 31 5 33
1
7.
LESSON 8.1
5
(x 2y4)3 5 ( x ) ( y )
5x
x
(24y 2)3 5 (24) ( y )
53
8.
px
y
9. x2(x3y)2 5 x
y
(x ) y
5x
x
5x
y
y
Simplify the expression. Write your answer using exponents.
10. 82 p 85
13.
11. 52 p 54
(24)5
14.
16. (13 p 18)2
12. 7 p 78
(63)7
15.
(42)9
17. (21 p 25)5
18. (7 p 154)6
20. y2 p y 6
21. z10 p z3
19. x3 p x
22.
(m4)7
23.
25. (3n)3
(b 9)2
24.
26. (2x)5
( p5)3
27. (xy)6
28. State Populations The table below shows the populations of selected states in 1870.
Write the order of magnitude of each of the populations.
State
Wisconsin
Nebraska
New Jersey
Oregon
Population
1,054,670
122,993
906,096
90,923
29. U.S. National Parks Hot Springs National Park in Arkansas covers an area of about
101 square miles. Kenai Fjords National Park in Arkansas covers an area that is
about 102 times the area of Hot Springs National Park. Find the approximate area of
Kenai Fjords National Park. Write your answer using exponents.
30. Mining In 2000, Canada mined approximately 104 metric tons of uranium. The
amount of metric tons of zinc mined in Canada in 2000 was approximately 102 times
this amount. About how many metric tons of zinc were mined in Canada in 2000?
62
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
Simplify the expression.
Name ———————————————————————
LESSON
8.1
Date ————————————
Practice B
For use with pages 504–510
Simplify the expression. Write your answer using exponents.
1. 54 p 58
2. (24)7 p (24)3
3. (210)5 p (210)2
4. 82 p 84 p 8
5. 25 p 2 p 24
6.
(35)2
8.
(152)4
7.
(93)7
9.
[(24)5]9
10. (13 p 19)4
11. (48 p 27)6
12. (135 p 8)5
Simplify the expression.
14. y3 p y p y4
15. a10 p a2 p a6
16.
(z5)5
18.
[(b 1 1)2]3
17.
(b7)2
19. (23x)4
20. 2(3x)4
21. (2ab)5
22.
23.
(3m7)4 p m3
LESSON 8.1
13. x 5 p x 2
(2x 3y)6
24. 4p2 p (3p5)2
Find the missing exponent.
Copyright © Holt McDougal. All rights reserved.
25. x6 p x? 5 x12
26.
(x 4)? 5 x12
27.
(3z?)3 5 27z18
28. Newspaper Circulation In 1996, the newspaper circulation in the country of
Algeria was approximately 103 times the newspaper circulation in the country
of Mauritania. The newspaper circulation in Mauritania was 103. What was the
newspaper circulation in Algeria?
29. Metric System The metric system has names for very large weights.
a. One gigaton is 102 times the weight of a hectaton. One hectaton is 102 ton.
Write one gigaton in tons.
b. One teraton is 109 times the weight of a kiloton. One kiloton is 103 ton.
Write one teraton in tons.
c. One exaton is 106 times the weight of a teraton. Use your answer to
part (b) to write one exaton in tons.
30. Wall Mural You are designing a wall mural that will be composed of squares of
different sizes. One of the requirements of your design is that the side length of each
square is itself a perfect square.
a. If you represent the side length of a square as x 2, write an expression for the area
of a mural square.
b. Find the area of a mural square when x 5 5.
c. Find the area of a mural square when x 5 10.
Algebra 1
Chapter 8 Resource Book
63
Name ———————————————————————
Date ————————————
Practice C
LESSON
8.1
For use with pages 504–510
Simplify the expression. Write your answer using exponents.
1. (29)10 p (29)4
4.
(48)7
2. 103 p 105 p 10
5.
7. (20 p 31)5
3. (27) p (27)3 p (27)4
(113)9
6.
8. (125 p 8)8
[(26)6]3
9. [(216) p 26]6
Simplify the expression.
LESSON 8.1
10. x4 p x p x7
11.
[(c 1 5)3]6
12.
(24c7)3
13. 2(4c7)3
14.
(5x8y5)4
15.
16.
(5p3)3 p 2p4
17. 10m4 p (2m5)6
18.
(6x3)2(24x5)3
19. 2(4n4)3(212n5)
20.
1 }13 z4 2 (3z2)4
3
(210a7b)5
21. (210c)3(22c2)5
Find the missing exponent.
22.
(5d 4)? 5 625d16
23.
(2a4)? p 3a5 5 96a25
24. 5a6 p (10a5)? 5 5000a21
25. Write three expressions that involve products of powers, powers of powers, or
powers of products and are equivalent to 24x12.
The number of personal computers in use in Bahrain in 2001 was 10 times the
number used in Samoa. The number of personal computers in use in Australia in
2001 was 10 times the number used in Bahrain. How many personal computers were
in use in Australia in 2001? Explain how you got your answer.
27. Bananas In 1999, Venezuela produced approximately 106 metric tons of bananas.
This is 102 times the number of bananas produced in Samoa in 1999. How many
metric tons of bananas were produced in Samoa in 1999? Explain how you got your
answer.
28. Storage Cubes You are designing open storage cubes that will hang on the
walls of your room. These cubes will be artistic as well as functional. One of the
requirements of your design is that the side length of the cube be a perfect square.
a. If you represent the side length of a cube as x2, write an expression for the
volume of a wall cube.
b. Find the volume of a wall cube when x 5 5.
c. Find the volume of a wall cube when x 5 10.
64
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
26. Personal Computers In 2001, there were 103 personal computers in use in Samoa.
Name ———————————————————————
LESSON
8.1
Date ————————————
Review for Mastery
For use with pages 504–510
GOAL
Use properties of exponents involving products.
Vocabulary
The order of magnitude of a quantity can be defined as the power of
10 nearest the quantity.
EXAMPLE 1
Use the product of powers property
Simplify the expression.
a. 26 p 28 5 26 1 8
LESSON 8.1
5 214
b. (23)7 p (23) 5 (23)7 p (23)1
5 (23)711
5 (23)8
c. (27)3 p (27) p (27)4 5 (27)3 p (27)1 p (27)4
5 (27)3 1 1 1 4
5 (27)8
d. m p m5 p m6 5 m1 1 5 1 6
5 m12
Copyright © Holt McDougal. All rights reserved.
Exercises for Example 1
Simplify the expression.
EXAMPLE 2
1. 83 p 811
2. 6 p 63
3. y3 p y6 p y 2
4. (210)2 p (210) p (210)5
Use the power of a power property
Simplify the expression.
a.
(33)6 5 33 p 6
5 318
b.
[(212)7]6 5 (212)7 p 6
5 (212)42
c.
(d 5)2 5 d 5 p 2
5 d 10
d.
[(x 2 3)3]4 5 (x 2 3)3 p 4
5 (x 2 3)12
Algebra 1
Chapter 8 Resource Book
65
Name ———————————————————————
LESSON
8.1
Review for Mastery
Date ————————————
continued
For use with pages 504–510
Exercises for Example 2
Simplify the expression.
EXAMPLE 3
5.
(133)10
6.
[(28)7]3
7.
( f 8)2
8.
[(w 1 8)9]2
Use the power of a product property
Simplify the expression.
a. (16 p 21)4 5 164 p 214
LESSON 8.1
b. (6mn)3 5 (6 p m p n)3
5 63m3n3
5 216m3n3
c. (25p)3 5 (25 p p)3
5 (25)3 p p3
5 2125p3
d. 2(2q)4 5 2(2 p q)4
5 2(24 p q4)
5 216q4
Use all three properties
Simplify (23y 5)3 • 2y 2.
Solution
(23y5)3 p 2y2 5 (23)3 p ( y 5)3 p 2y 2
Power of a product property
5 227 p y15 p 2y2
Power of a power property
5 254y17
Product of powers property
Exercises for Examples 3 and 4
Simplify the expression.
9. (5 p 18)6
11.
66
Algebra 1
Chapter 8 Resource Book
(23x 2 y 5)2
10. 2(11p)3
12. (2m)3 p (m4)5
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 4
Name ———————————————————————
LESSON
8.1
Date ————————————
Challenge Practice
For use with pages 504–510
In Exercises 1–5, simplify the expression, if possible. Write your answer as
a power.
1. a x/3a 3
2.
(a2b)5y p (ab2)2y
3.
(x1/2 p y1/4)2
4.
[(xy)(x 3y 5)]2
5. (x 1 2)2a 1 1 p (x 1 2)3a 2 5
In Exercises 6–9, use the following information.
6. Write an expression that gives the volume of the storage bin in terms of a.
7. Suppose the length and width of the storage bin are doubled. By what factor would
the height of the bin have to change so that the volume of the bin remains the same?
LESSON 8.1
You are constructing a storage bin to hold bird seed. You decide the length, width, and
height of the bin will each have a length of a feet.
8. Suppose the length of the original storage bin is tripled and the width of the storage
bin is halved. By what factor would the height of the bin have to change so that the
volume of the bin is doubled?
9. Suppose the length, width, and height of the bin each have 1 foot added to them.
Write an expression for the volume of the storage bin.
Copyright © Holt McDougal. All rights reserved.
10. An exam has 10 true-false questions and 10 multiple choice questions. Each
multiple choice question has 6 possible answers. Assuming a student guesses at each
question on the exam, write an exponential expression for the number of different
ways it is possible to answer the 20 questions.
11. Using the fact that 6 5 2 p 3, write the expression from Exercise 10 as powers of
2 and 3.
Algebra 1
Chapter 8 Resource Book
67
Name ———————————————————————
Date ————————————
Practice A
LESSON
8.2
For use with pages 511–517
Name the property demonstrated by the example.
x5
1. }3 5 x523 5 x2
x
a 4
2.
1 }b 2
5.
1 }34 2
a4
b
2m8
3. }
5 2m826 5 m2
m6
3
86
8
6. }
5}
4
8 p 82
8
5 }4
Fill in the blanks.
38
4. }5 5 38
3
5
4
5}
4
53
58
Simplify the expression. Write your answer using exponents.
910
8. }
97
47
7. }3
4
(25)4
10. }3
(25)
(27)5
11. }1
(27)
7
13.
36
9. }1
3
5
12.
9
1 }53 2
14.
1 }27 2
1 }14 2
1
15. 45 p }2
4
1
16. }5 p y11
y
1
17. z3 p }2
z
1
18. }4 p m8
m
19.
1 }y 2
21.
1 }1z 2
20.
a 13
1 }b 2
x 3
9
22. Internet Users The table shows the numbers of Internet users in selected countries
in 2001.
Country
Internet Users
Albania
Jamaica
Marshall Islands
Romania
104
105
103
106
a. How many times greater is the number of users from Romania than the number
of users from the Marshall Islands?
b. How many times greater is the number of users from Albania than the number of
users from the Marshall Islands?
c. How many times greater is the number of users from Jamaica than the number of
users from the Marshall Islands?
d. How many times greater is the number of users from Romania than the number
of users from Albania?
68
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 8.2
Simplify the expression.
Name ———————————————————————
Date ————————————
Practice B
LESSON
8.2
For use with pages 511–517
Simplify the expression. Write your answer using exponents.
614
1. }
68
145
2. }4
14
(25)7
3. }2
(25)
125 p 123
4. }
124
817
5. }
3
8 p 87
6.
1 }34 2
1 2}15 2
1
8. 38 p }1
3
9.
1 }14 2
1
11. z16 p }7
z
12.
1 }b 2
15.
1 2
18.
2m
1}
3n 2
7.
6
5
5
p 413
Simplify the expression.
1
10. }9 p y15
y
3 4
3
13.
1 2}6z 2
16.
m
1}
5n 2
14.
a
1}
2b 2
17.
1 2
4 3
9
5
3x7 4
2y
a 8
3x4 5
y
}
6
55
}
12
9
19. Area The area of New Zealand is 104,454 square miles and the area of Saint
Kitts and Nevis, islands in the Caribbean Sea, is 104 square miles. Use order of
magnitude to estimate how many times greater New Zealand’s area is than Saint
Kitts and Nevis’ area.
20. Cell Phone Subscribers The table below shows the approximate number of cell
phone subscribers in selected countries in 2001.
Copyright © Holt McDougal. All rights reserved.
Number of subscribers
Algeria
Dominican Republic
Poland
Solomon Islands
105
106
107
103
a. How many times greater is the number of cell phone subscribers in Poland than
LESSON 8.2
Country
in the Solomon Islands?
b. How many times greater is the number of cell phone subscribers in the
Dominican Republic than in the Solomon Islands?
21. Glass Vase You are taking a glass-blowing class and have created a vase in the
shape of a sphere. The vase will have a hole in the top so you can put flowers in it
21
and it will sit on a stand. The radius of your vase is }
inches. Use the formula
2
4
V 5 }3 πr 3 to write an expression for the volume of your vase.
Algebra 1
Chapter 8 Resource Book
69
Name ———————————————————————
Date ————————————
Practice C
LESSON
8.2
For use with pages 511–517
Simplify the expression. Write your answer using exponents.
152 p 159
1. }
156
613
2. }
4
6 p 65
1
4. 813 p }6
8
5.
1 }15 2
7
p 517
3.
7
1 2}89 2
1 3
6. 108 p 2}
10
1
2
Simplify the expression.
6 4
a 7
7.
1 2}b 2
10.
4a
1}
5b 2
8.
2 3
11.
3
3x
1}
y 2
7x
1}
8y 2
9
3 2
7
3y 3
10x7 2
14. } p }
5
9y8
2x2 5
1
13. }5 p }
4x
y3
1 2
1 2
7
6
9.
m
1}
2n 2
12.
3x
1}
10y 2
15.
1 2}6x 2
10
5 3
2
3
5
x
p }4
x4 5
3y
1 2
p }7
bx
b x 12
16. Find the values of x and y if you know that }y 5 b5 and }
5 b4. Explain how you
b
b 2y
found your answer.
17. U.S. Postal Service In 2004, the U.S. Postal Service handled 97,926,396 pieces
of first class mail and 848,633 pieces of priority mail. Use order of magnitude to
estimate how many times greater a volume of first class mail the U.S. Postal Service
handled than the volume of priority mail.
and British systems. In the American system, one quintillion is the name for the
number 1018. In the British system, one quintillion is the name for the number
1030. How many times larger is one quintillion in the British system than in the
American system?
19. Lawn Ornaments You have learned how to make lightweight plant containers using
a mixture of peat, sand, and cement. You are going to take these skills and make
4
lawn ornaments in the shapes of spheres. Use the formula for volume V 5 }3 πr 3
to write an expression for the volume of each sphere shown.
1
2
ft
3
4
3
2
70
Algebra 1
Chapter 8 Resource Book
ft
ft
Copyright © Holt McDougal. All rights reserved.
LESSON 8.2
18. Large Numbers Very large numbers are named differently in the American
Name ———————————————————————
LESSON
8.2
Date ————————————
Review for Mastery
For use with pages 511–517
GOAL
EXAMPLE 1
Use properties of exponents involving quotients.
Use the quotient of powers property
Simplify the expression.
713
a. }
5 71328
78
(21)6
b. }2 5 (21)622
(21)
5 75
23 p 29
212
c. }
5}
4
2
24
5 (21)4
y18
1
d. }7 p y18 5 }
y
y7
5 212 2 4
5 28
5 y18 2 7
5 y11
Exercises for Example 1
Simplify the expression.
EXAMPLE 2
1215
1. }
126
(28)20
2. }
(28)16
136 p 138
3. }
139
1
4. }
p w 21
w16
Use the power of a quotient property
LESSON 8.2
Copyright © Holt McDougal. All rights reserved.
Simplify the expression.
m5
n
3 3
33
27
} 5 }3 5 }3
p
p
p
m 5
a.
1 }n 2
b.
12
5 }5
Exercises for Example 2
Simplify the expression.
5.
7
1 }bc 2
6.
4
1 2}w3 2
Algebra 1
Chapter 8 Resource Book
71
Name ———————————————————————
LESSON
8.2
Review for Mastery
Date ————————————
continued
For use with pages 511–517
EXAMPLE 3
Use the properties of exponents
Simplify the expression.
a.
(2x3)2
2x3 2
5}
5y
(5y2)2
22(x3)2
1 2
}2
5}
2 2 2
5 (y
)
4x6
25y
5 }4
b.
32(k3)2 l3
3k3 2 l2
p }2 5 }
p}
4l
6k
42(l5)2 6k 2
9k 6
l3
}
5}
p
16l10 6k2
9k6l3
5}
96l10k2
3k 4
5 }7
32l
1 2
}5
Power of a quotient property
Power of a product property
Power of a power property
Power of a quotient property
Power of a power property
Multiply fractions.
Quotient of a powers property
Exercises for Example 3
Simplify the expression.
EXAMPLE 4
3s5 3
t
1 2
}
4
3m2n 3
1
8. }4 p }
3m
n2
1
2
Solve a real world problem
Distances The distance from Earth to the nearest galaxy is about 1022 meters. The
distance from Earth to the North Star is about 1019 meters. How many times farther
from Earth is the nearest galaxy than the North Star?
Solution
Distance to the nearest galaxy
1022
5 1022 219 5 103
}}} 5 }
Distance to the North Star
1019
The nearest galaxy is about 103 times farther than the North Star.
Exercise for Example 4
9. The distance from the sun to Saturn is 1012 meters. The distance from the sun
to Venus is 1011 meters. How much further is Saturn than Venus from the sun?
72
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 8.2
7.
Name ———————————————————————
LESSON
8.2
Date ————————————
Challenge Practice
For use with pages 511–517
ax
1. Solve for the value of a if }
5 a 3 and x 5 a 5y.
a 2y
4(b 2 1)2
(b 1 1)2
}
2. Solve for the value of b if }
5
.
b2
b2
c xc y
3. Solve for the values of x and y if }
5 c and c y 2 1 5 c 3.
c xy
4. Solve for the value of c if 2c 1 4 5 3b 2 and b 6 5 c 3.
d 3x
5. Solve for the value of y if }
5 d 3x 2 y.
d 3y
In Exercises 6–8, use the following information.
A common formula used to compute annual salary raises is
Salary 5 Starting Salary p (1 1 r)n
where r is the rate of annual raise and n is the number of years of employment.
Example:
Find the salary of an employee who has worked for 2 years and whose starting salary was
$25,000 at a company that gives annual raises at a rate of r 5 0.1.
Solution:
New Salary 5 $25,000(1 1 0.1)2
5 $25,000(1.21)
Copyright © Holt McDougal. All rights reserved.
Suppose a company gives annual raises at a rate of r 5 0.05.
6. What is the salary of an employee whose starting salary was $40,000 per year and
has worked at the company for 10 years?
LESSON 8.2
5 $30,250
7. What is the salary of an employee whose starting salary was $50,000 per year and
has worked at the company for 5 years?
8. What is the salary of an employee whose starting salary was $100,000 per year and
has worked at the company for 20 years?
Algebra 1
Chapter 8 Resource Book
73
Name ———————————————————————
Date ————————————
Practice A
LESSON
8.3
For use with pages 519–524
Match the equivalent expressions.
22
1 }23 2
22
1 }32 2
2. 222 p 322
3.
4
B. }
9
9
C. }
4
4. 523
5. 822
6. 225
7. (23)24
8. (29)21
9. 60
1.
1
A. }
36
Evaluate the expression.
10. (25)0
13.
21
1 }34 2
0
11.
1 }12 2
14.
1 }25 2
12.
23
22
1 }16 2
15. 022
Simplify the expression. Write your answer using only positive exponents.
16. x25
17. m29
18. 6y23
19. 8a210
20. (3b)24
21. x3y22
22. x24y 3
23. a21b22
24. 2x23y1
4 21
25. Finger Thickness Your friend tells you that her finger is }
inch thick. Evaluate
3
1 2
26. Floor Tile The minimum recommended width of the space between 6-inch by
6-inch tiles is 222 inch and the maximum recommended width is 221 inch. Simplify
the expressions for the minimum and maximum widths of the space between the
6-inch by 6-inch floor tiles.
27. Hole Punch Your hole punch makes holes in your paper that have a diameter of
421 inch.
a. Write an expression for the area of one punched hole. Use the formula for the
LESSON 8.3
area of a circle A 5 πr2.
b. Your hole punch makes three holes in a page. Write an expression for the total
area punched out of one sheet of paper.
74
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
the expression that represents the thickness of your friend’s finger.
Name ———————————————————————
Date ————————————
Practice B
LESSON
8.3
For use with pages 519–524
Evaluate the expression.
1. 325
2. 1023
3. (22)26
4. 50
5. (26)0
6.
7.
22
1 }58 2
10. 1022 p 1023
8.
3
1 }74 2
11. 426 p 43
0
1 }43 2
9. 025
1
12. }
524
Simplify the expression. Write your answer using only positive exponents.
13. x27
14. 6y24
15. (2b)25
16. (23m)24
17. a2b24
18. 3x22y25
19.
(4x24y2)23
x2
22. }
y24
20.
(8mn3)0
x26
23. }5
4y
c23
21. }
d25
1
24. }
23 27
3x y
25. Paper A sheet of 67-pound paper has a thickness of 10021 inch.
a. Write and evaluate an expression for the total thickness of 5 sheets of
26. Frogs A frog egg currently has a radius of 521 centimeter. Write an expression
for the volume of the frog egg. Use the formula for the volume of a sphere
4
V 5 }3 πr3.
27. Metric System The metric system has names for very small lengths.
a. One micrometer is 103 times the length of one nanometer. One nanometer is
1029 meter. Write one micrometer in meters.
b. One femtometer is 103 times the length of one attometer. One attometer is
10218 meter. Write one femtometer in meters.
c. One centimeter is 1010 times the length of one picometer. One picometer is
10212 meter. Write one centimeter in meters.
LESSON 8.3
Copyright © Holt McDougal. All rights reserved.
67-pound paper.
b. Write and evaluate an expression for the total thickness of 23 sheets of
67-pound paper.
Algebra 1
Chapter 8 Resource Book
75
Name ———————————————————————
LESSON
8.3
Date ————————————
Practice C
For use with pages 519–524
Evaluate the expression.
1. 324 p 321
2. 924 p 98
3.
1
4. }
1025
526
5. }
529
8210
6. }
828
3 21
7. 15 }
5
224
8. 32 }
23
7
9. 4 2 2 p }0
12
1 2
1 2
(521)4
1 2
Simplify the expression. Write your answer using only positive exponents.
1
11. }
9x24y28
1
12. }
6x4y210
1
13. }
25
(4x )22
8
14. }
(22d 2)24
(2x)24y8
15. }
2x5y23
x26y4
16. }
(23x2)24y21
20x3y24
17. }
(2x24y21)2
(4x24y7)2
18. }
24x26y2
10.
(4x23y4)22
Tell whether the statement is true or false for all nonzero values of a and
b. If it is false, give a counterexample.
a25
1
19. }
5 }a
26
a
b21
a
20. }
5}
21
b
a
1
21. }
5a1b
21
a 1 b21
22. Guitar The world’s smallest guitar is only 1026 meter tall. An average guitar is
23. Knitting Needles A size 1 knitting needle has a diameter of about 421 centimeter
and a size 8 knitting needle has a diameter of about 221 centimeter.
LESSON 8.3
a. How many times larger is the diameter of a size 8 needle than the diameter of a
76
size 1 needle?
b. Suppose that each needle is 14 inches long. Write expressions for the
approximate volume of each size of knitting needle. Use the formula for the
volume of a cylinder V 5 πr2h.
c. How many times larger is the approximate volume of a size 8 needle than the
approximate volume of a size 1 needle?
d. Are your approximations in part (b) overestimates or underestimates? Explain
your reasoning.
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
about 100 meter tall. How many times taller is an average guitar than the world’s
smallest guitar?
Name ———————————————————————
LESSON
8.3
Date ————————————
Review for Mastery
For use with pages 519–524
GOAL
EXAMPLE 1
Use zero and negative exponents.
Use definition of zero and negative exponents
Evaluate the expression.
1
a. 423 5 }3
4
Definition of negative exponents
1
5}
64
Evaluate exponent.
b. 150 5 1
23
c.
1 }32 2
Definition of zero exponent
1
5 }3
Definition of negative exponents
3
}
2
1 2
1
5}
Evaluate exponents.
27
}
8
1 2
8
5}
27
Simplify.
Exercises for Example 1
Evaluate the expression.
0
1 2}12 2
2. (25)24
23
1
3. }
622
EXAMPLE 2
4.
1 }52 2
Evaluate exponential expressions
Evaluate the expression.
a. 1316 p 13214 5 1316 2 14
b.
Product of powers property
5 132
Subtract exponents.
5 169
Evaluate power.
[(22)24]2 5 (22)24 p 2
5 (22)28
1
(22)
1
5}
256
5 }8
Power of a power property
Multiply exponents.
Definition of negative exponents
Evaluate power.
Algebra 1
Chapter 8 Resource Book
LESSON 8.3
Copyright © Holt McDougal. All rights reserved.
1.
77
Name ———————————————————————
LESSON
8.3
Review for Mastery
Date ————————————
continued
For use with pages 519–524
Exercises for Example 2
Evaluate the expression.
EXAMPLE 3
825
5. }
825
1
6. }
922
7. (24)7 p (24)29
102
8. }
1023
Use properties of exponents
Simplify the expression. Write your answer using only
positive exponents.
(3m22n3)3 5 33 p (m22)3 p (n3)3
5 27 p
27n
m
m26
pn
9
Power of a product property
Power of a power property
9
5}
6
(25st)2t24
(25st)2t 8
}
b. }
5
210s3t28
210s3t 4
(25s2t 2)t8
5}
3 4
210s t
25s2t10
210s t
5t 6
5}
22s
5}
3 4
Definition of negative exponents
Definition of negative exponents
Power of a product property
Product of powers property
Quotient of powers property
Exercises for Example 3
Simplify the expression. Write your answer using only
positive exponents.
9.
(5x2y23z)4
4m22np3
10. }
12m2n25p
LESSON 8.3
(2r2t)23rst 4
11. }
6r 6s23
78
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
a.
Name ———————————————————————
LESSONS
8.1–8.3
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 505–524
1. Multi-Step Problem A department store
sells plastic cubical containers that can be
used to store food.
a. One of the containers has a side length
3
of 3 }4 inches. Find the container’s
volume by writing the side length as an
improper fraction and substituting the
length into the formula for the volume
of a cube.
b. Identify the property of exponents you
used to find the volume in part (a).
2. Multi-Step Problem There are about
3
10 white corpuscles in 1 cubic millimeter
of blood.
a. Copy and complete the table by
Blood (cubic
millimeters)
Number of white
corpuscles
10
?
100
?
1000
?
10,000
?
100,000
?
b. A particular sample of blood is
95,000 cubic millimeters. Use order
of magnitude to write an expression
you can use to find the approximate
number of white corpuscles in the
sample of blood. Simplify the
expression. Verify your answer using
the table.
3. Short Response A carrot seed has a mass
of about 1024 gram and is 103 times less
massive than a sweet corn seed. A student
says that a sweet corn seed has a mass of
about 1 gram. Is the student correct?
Explain.
measurement of length and their equivalents
in meters.
Name of unit
Length (meters)
Terameter
1012
Kilometer
103
Centimeter
1022
Micrometer
1026
a. Use the table to write a conversion
problem that can be solved by
applying a property of exponents
involving products.
b. Use the table to write a conversion
problem that can be solved by
applying a property of exponents
involving quotients.
5. Gridded Response The mass of a grain
of sand is about 1023 gram. About how
many grains of sand are in a bag of sand that
weighs 2.8 grams?
6. Extended Response For an experiment,
a scientist dropped about 1024 cubic inch of
olive oil into a container of water to see how
the oil would spread out over the surface of
the water. The scientist found that the oil
spread until it covered an area of about
1022 square inch.
a. About how thick was the layer of oil
that spread out across the water? Check
your answer using unit analysis.
b. The water has a surface area of
102 square inches. If the oil spreads to
the same thickness as in part (a), how
many cubic inches of olive oil would
be needed to cover the entire surface
of the water?
c. Explain how you could find the
amount of oil needed to cover a
container of water with a surface area
of 10 x square inches.
Algebra 1
Chapter 8 Resource Book
LESSON 8.3
Copyright © Holt McDougal. All rights reserved.
finding the number of white
corpuscles for the given amounts of
blood (in cubic millimeters).
4. Open-Ended The table shows units of
79
Name ———————————————————————
LESSON
8.3
Date ————————————
Challenge Practice
For use with pages 519–524
In Exercises 1–5, a and b are real numbers such that a > 0 and b > 0. Tell
whether the statement is always true, sometimes true, or never true.
If it is sometimes true, give a pair of values for which it is true and a pair
of values for which it is false.
b4
a23
1. }
5 }3
24
a
b
2. (a 1 b)22 5 a 22 1 b22
3.
(a 2 1 b 2)1/2 5 a 1 b
4.
(a 2 1 b 2)2 5 a 4 1 2a2b2 1 b 4
5.
a 1 b 5 a2 1 b2
6. Determine which positive values of a make a 23 > a 24 a true statement.
In Exercises 7–10, evaluate the given expression for the given values of a.
[(a 1 1)22]3
7. }; a 5 0
[(a 2 1)23]2
[(a2 1 3) a 2 2]3
8. }}; a 5 2
[(a 2 1)2]4
9.
[(a 1 2)a]a 2 1 2 2a 1 1; a 5 0
LESSON 8.3
Copyright © Holt McDougal. All rights reserved.
(a a)22a
10. }
;a52
(a 1 1)2a
80
Algebra 1
Chapter 8 Resource Book
Name ———————————————————————
FOCUS ON
8.3
Date ————————————
Practice
For use with pages 525–526
Evaluate the expression.
1. 6251/2
2. 16921/2
3. 45/2
4. 923/2
5. 19621/2
6. 493/2
7. 1251/3
8. 34321/3
9. (227)2/3
10. 6424/3
11. (264)1/3
1
12. }
824/3
13. 253/2 • 251/2
21621/3
14. }
2162/3
1
15. }
• 3623/2
3622
16. (264)2/3 • (264)21/3
811/2
17. 813/2 4 }
8121/2
(28)1/3
18. (28)5/3 • }
(28)2/3
19. Reasoning Show that the product of the cube root of a and a can be written as a4/3
using an argument similar to the one given for square roots on page 525.
FOCUS ON 8.3
Copyright © Holt McDougal. All rights reserved.
23
20. Challenge Evaluate the expression 644/3 • }
÷ 256.
163/2
Algebra 1
Chapter 8 Resource Book
81
Name ———————————————————————
FOCUS ON
8.3
Date ————————————
Review for Mastery
For use with pages 525–526
GOAL
Use fractional exponents.
In lesson 2.7, you learned to write the square root of a number using
a radical sign. You can also write a square root of a number using
}
exponents. For a nonnegative number a, Ï a 5 a1/2. You can work
1
1
with exponents of }2 and multiples of }2 just as you work with integer
exponents.
Evaluate expressions involving square roots
}
a. 811/2 5 Ï 81
59
c. 363/2 5 36(1/2)?3
1
b. 10021/2 5 }
1001/2
1
5}
}
Ï100
1
5}
10
d. 925/2 5 9(1/2) • (25)
5 (91/2)25
}
5 1 Ï9 225
5 325
5 (361/2)3
}
5 1 Ï 36 23
5 63
5 216
1
3
1
5}
243
5 }5
Exercises for Example 1
Evaluate the expression.
1.
14421/2
FOCUS ON 8.3
3. 6423/2
82
Algebra 1
Chapter 8 Resource Book
2. 93/2
4. 251/2
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 1
Name ———————————————————————
FOCUS ON
8.3
Review for Mastery
Date ————————————
continued
For use with pages 525–526
EXAMPLE 2
Evaluate expressions involving cube roots
3}
1
b. 6421/3 5 }
641/3
a. 81/3 5 Ï 8
3
}
5 Ï23
1
Ï64
1
5 }4
5}
3}
52
d. 2722/3 5 27(1/3) • (22)
c. 1254/3 5 125(1/3) • 4
5 (271/3)22
5 (1251/3)4
3}
3 } 22
5 1 Ï 125 24
= 1 Ï27 2
5 54
5 625
5 322
1
3
1
5 }9
5 }2
Exercises for Example 2
Evaluate the expression.
EXAMPLE 3
6. 274/3
7. 642/3
8. 12524/3
Use properties of exponents
a. 723/2 • 77/2 5 7(23/2) 1 (7/2)
= 74/2
= 72
= 49
57/3 • 5
5(7/3) 1 1
b. }
5}
1/3
1/3
5
5
57/3 • 5
510/3
}
}
5
1/3
51/3
5
7/3
5 •5
}
5 5(10/3) 2 (1/3)
1/3
7/3
55 • 5
}
5 53
1/3
7/3
55 • 5
}
5 125
1/3
5
Exercises for Example 3
Evaluate the expression.
9. 1621/2 • 162
10. 273 • 2728/3
84/3
11. 87/3 • }
82
41/2
12. 423/2 • }
423/2
Algebra 1
Chapter 8 Resource Book
FOCUS ON 8.3
Copyright © Holt McDougal. All rights reserved.
5. 21621/3
83
Name ———————————————————————
Graphing Calculator Activity Keystrokes
LESSON
For use with page 535
Casio CFX-9850GC Plus
TI-83 Plus
10
1.1
^
ⴛ
21
10
)
^
(ⴚ)
ENTER
8
)
(
1.4
ⴛ
From the main menu, choose RUN.
(
10
ⴛ
1.1
^
21
10
)
^
(ⴚ)
8
)
(
1.4
ⴛ
EXE
Copyright © Holt McDougal. All rights reserved.
LESSON 8.4
8.4
(
Date ————————————
84
Algebra 1
Chapter 8 Resource Book
Name ———————————————————————
LESSON
8.4
Date ————————————
Practice A
For use with pages 528–534
1. 0.004
2. 0.04
3. 4000
A. 40 3 1023
B. 4 3 103
C. 4 3 1023
Write the number in scientific notation.
4. 6.4
5. 85.2
6. 0.25
7. 0.104
8. 540
9. 9124.5
10. 0.0095
11. 630,000
12. 0.03
13. 23,960
14. 0.0457
15. 0.000045
LESSON 8.4
Match the equivalent numbers.
Write the number in standard form.
16. 5.2 3 104
17. 9.1 3 108
18. 6.25 3 105
19. 6.05 3 102
20. 8.125 3 106
21. 1.113 3 1010
22. 4.7 3 1023
23. 1.6 3 1028
24. 4.45 3 1026
25. 9.24 3 1024
26. 7.1123 3 1023
27. 2.0123 3 1025
Order the numbers from least to greatest.
28. 21,000; 4.5 3 103; 15,625; 3 3 104
29. 0.0006; 7.8 3 1026; 0.0012; 2.15 3 102
Copyright © Holt McDougal. All rights reserved.
30. 1.765; 1.3 3 1022; 0.0125; 6.15 3 1021
31. Body Makeup The table below shows the amounts (in pounds) of some elements
that are in the body of a 150-pound person. Complete the table.
Element
Oxygen
Chlorine
Cobalt
Magnesium
Sodium
Hydrogen
Weight in
decimal
form
97.5
?
0.00024
?
0.165
?
Weight in
scientific
notation
?
3 3 1021
?
6 3 1022
?
1.5 3 101
32. Internet Users In 2003, there were about 5.8078 3 108 people using the Internet in
the world and about 1.6575 3 108 of these people were in the United States. What
percent of Internet users in 2003 were in the United States? Round your answer to
the nearest tenth of a percent.
Algebra 1
Chapter 8 Resource Book
85
Name ———————————————————————
Practice B
LESSON
8.4
LESSON 8.4
Date ————————————
For use with pages 528–534
Write the number in scientific notation.
1. 10.4
2. 6751
3. 0.54
4. 0.000103
5. 415,620
6. 0.08104
7. 3,412,000
8. 525.5
9. 104.25
10. 0.0000456
11. 0.000000207
12. 23,551
Write the number in standard form.
13. 15.8 3 104
14. 3.21 3 108
15. 450.21 3 107
16. 8.1045 3 105
17. 17.22 3 106
18. 1.012 3 102
19. 8.12 3 1024
20. 4.014 3 1027
21. 8.1025 3 1023
22. 3.12056 3 1029
23. 1.211 3 1022
24. 7.00135 3 1025
Order the numbers from least to greatest.
25. 1.3759 3 104; 14,205; 9.287 3 103; 3.0214 3 104
26. 0.16; 2.5 3 1023; 1.04 3 1023; 0.0985
27. 8.79 3 102; 1146; 1.0085 3 103; 1023
28. 1.2 3 1025; 0.001023; 1.045 3 1023; 0.01036
29.
(6 3 108)(5 3 1022)
4.5 3 1025
30. }
9 3 1022
31.
(2 3 1025) 5
32. Pixels The images on a computer screen are made up of more than 5000 pixels,
or dots, per square inch. How many pixels are on a computer screen that measures
108 square inches? Write your answer in scientific notation.
33. Oregon Oregon has an area of approximately 2.52 3 105 square kilometers. In
2000, the population of Oregon was approximately 3.42 3 106 people. How many
people were there per square kilometer in Oregon in 2000?
34. Uranus’ Moons The table below shows the masses in kilograms of some of Uranus’
moons.
Moon
Mass (kg)
Miranda
Titania
Ariel
Oberon
Umbriel
6.6 3 1019
3.52 3 1021
13.5 3 1020
30.1 3 1020
11.7 3 1020
a. Write the moons in order of largest mass to smallest mass.
b. How many times larger is the moon of largest mass than the moon of
smallest mass?
86
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
Evaluate the expression. Write your answer in scientific notation.
Name ———————————————————————
LESSON
8.4
Date ————————————
Practice C
For use with pages 528–534
1. 0.0015
2. 30,400
3. 0.0000046
4. 9,120,006
5. 24.5
6. 0.1256
7. 705
8. 100,456
9. 0.000000501
LESSON 8.4
Write the number in scientific notation.
Write the number in standard form.
10. 1.325 3 105
11. 7.05123 3 108
12. 8.15 3 1028
13. 9.044 3 1022
14. 5.1 3 103
15. 3.1112 3 1010
16. 8.1101 3 1025
17. 7.7 3 1027
18. 6.25 3 107
Order the numbers from least to greatest.
19. 758.4; 7.208 3 103; 72,165; 7.914 3 103
20. 1.305 3 1023; 0.000526; 2.018 3 1023; 0.00205
21. 0.000316; 3.28 3 1024; 3.016 3 1024; 0.003028
Evaluate the expression. Write your answer in scientific notation.
22.
(5.7 3 103)(2.2 3 1026)
6.5 3 1027
23. }
1.3 3 1023
24.
(3 3 1029) 5
25. California California has an area of approximately 4.11 3 105 square kilometers.
Copyright © Holt McDougal. All rights reserved.
In 2000, the population of California was approximately 3.39 3 107 people. How
many people were there per square kilometer in California in 2000?
26. Helium Atom A proton and a neutron each weigh 1.67 3 10224 gram. An electron
weighs 9.11 3 10228 gram. One helium atom contains 2 protons, 2 neutrons, and
2 electrons. Find the mass of one helium atom.
27. Saturn’s Moons The table below shows the masses in kilograms of some of
Saturn’s moons.
Moon
Mass (kg)
Mimas
Calypso
Tethys
Dione
Phoebe
3.75 3 1019
4 3 1015
6.27 3 1020
11 3 1020
4 3 1017
a. Write the moons in order of largest mass to smallest mass.
b. How many times larger is the moon of largest mass than the moon of
smallest mass?
c. There are approximately 2.2 pounds in one kilogram. Write each mass
in pounds.
Algebra 1
Chapter 8 Resource Book
87
Name ———————————————————————
LESSON
LESSON 8.4
8.4
Date ————————————
Review for Mastery
For use with pages 528–534
GOAL
Read and write numbers in scientific notation.
Vocabulary
A number is written in scientific notation when it is of the form
c 3 10n where 1 ≤ c < 10 and n is an integer.
EXAMPLE 1
Write numbers in scientific notation
Write the number in scientific notation.
a. 397,000,000
b. 0.000712
Solution
a. 397,000,000 5 3.97 3 108
b. 0.000712 5 7.12 3 1024
EXAMPLE 2
Move decimal point 8 places to the left.
Exponent is 8.
Move decimal point 4 places to the right.
Exponent is 24.
Write numbers in standard form
Write the number in standard form.
b. 9.131 3 1023
a. 3.02 3 104
a. 3.02 3 104 5 30,200
b. 9.131 3 1023 5 0.009131
Exponent is 4.
Move decimal point 4 places to the right.
Exponent is 23.
Move decimal point 3 places to the left.
Exercises for Examples 1 and 2
Write the number in scientific notation.
1. 0.0000079
2. 1,356,000
Write the number in standard form.
3. 1.012 3 103
EXAMPLE 3
4. 3.7 3 1025
Order numbers in scientific notation
Order 5.2 3 107, 910,000, and 13,200,000 from least to greatest.
Solution
STEP 1
Write each number in scientific notation, if necessary.
9,100,000 5 9.1 3 106
88
Algebra 1
Chapter 8 Resource Book
13,200,000 5 1.32 3 107
Copyright © Holt McDougal. All rights reserved.
Solution
Name ———————————————————————
LESSON
8.4
Review for Mastery
Date ————————————
continued
For use with pages 528–534
Order the numbers. First order the numbers with different powers of 10.
Then order the numbers with the same power of 10.
Because 106 < 107, you know that 9.1 3 106 is less than both 1.32 3 107 and
5.2 3 107. Because 1.32 < 5.2, you know that 1.32 3 107 is less than 5.2 3 107.
So, 9.1 3 106 < 1.32 3 107 < 5.2 3 107.
STEP 3
LESSON 8.4
STEP 2
Write the original numbers in order from least to greatest.
9,100,000, 13,200,000, 5.2 3 107
EXAMPLE 4
Compute with numbers in scientific notation
Evaluate the expression. Write your answer in scientific notation.
a.
(3.2 3 103)(4.7 3 104)
5 (3.2 p 4.7) 3 (103 p 104)
5 15.04 3 10
Commutative and associative properties
7
Product of powers property
5 (1.504 3 101) 3 107
Write 15.04 in scientific notation.
5 1.504 3 (101 3 107)
Associative property
5 1.504 3 108
Copyright © Holt McDougal. All rights reserved.
b.
Product of powers property
(3.8 3 10 ) 5 3.8 3 (10 )
24 2
2
24 2
Power of a product property
5 14.44 3 1028
Power of a power property
5 (1.444 3 101) 3 1028
Write 14.44 in scientific notation.
5 1.444 3 1027
Associative property and product of
powers property
2.6 3 106
2.6
106
}
c. }
5}
3
22
6.5
6.5 3 10
1022
Product rule for fractions
5 0.4 3 108
Quotient of powers property
5 (4.0 3 1021) 3 108
Write 0.4 in scientific notation.
5 4.0 3 107
Associative property and product of
powers property
Exercises for Examples 3 and 4
5. Order 361,000, 2.1 3 106, and 2.8 3 105 from least to greatest.
Evaluate the expression. Write your answer in scientific notation.
7.2 3 1023
6. }
1.8 3 106
8.
7.
(9.1 3 107)(2.3 3 1025)
(2.9 3 106)2
Algebra 1
Chapter 8 Resource Book
89
Name ———————————————————————
LESSON
LESSON 8.4
8.4
Date ————————————
Challenge Practice
For use with pages 528–534
In Exercises 1–5, evaluate the expression without using a calculator.
Write your answer in scientific notation.
1.
(1.2 3 1023) p (1.2 3 105)
(2.5 3 106) p (1 3 108)
2. }}
5 3 103
3.
(3 3 106) 1 (5 3 105)
4. 6(4 3 1022) 1 4
2.2(2 3 104) 1 1.2(2 3 105)
5. }}}
(7.1 3 1022) 1 (2.13 3 1021)
6. The population of Earth in the year 2000 was estimated to be 6 3 109 people. The
population of the U.S. in the year 2000 was estimated to be 3 3 108 people. What
proportion of the world’s population in the year 2000 resided in the U.S.?
7. The population of the People’s Republic of China in the year 2000 was estimated to
be 1.3 3 109 people. The population of the Republic of China (Taiwan) in the year
2000 was estimated to be 2.6 3 107 people. What was the proportion of the population of the Republic of China to the People’s Republic of China?
8. In the year 2002 there were approximately 9.6 3 105 dogs registered with the
In Exercises 9 and 10, convert the decimal expressions to scientific notation
and then simplify the expression. Write your answer in decimal form.
(0.0000032) p (2000000)
9. }}
(8 3 103) p (8 3 1024)
(0.0000012) 2 (0.000002)
10. }}
16
90
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
American Kennel Club (AKC) and 2.3 3 104 of those dogs were Rottweilers.
What proportion of the dogs registered to the AKC in the year 2002 were
Rottweilers?
Name ———————————————————————
CHAPTER
8.5
Date ————————————
Spreadsheet Activity Keystrokes
For use with pages 544 and 545
EXCEL
Select cell A1.
Years since 1984, t
TAB
Value, C (dollars)
ENTER
Select cell A2.
0
TAB
11000
ENTER
Select cell A3.
5A2 + 1
TAB
5B2*1.069
ENTER
Copyright © Holt McDougal. All rights reserved.
LESSON 8.5
Select cells A3–A22. From the Edit menu, choose Fill. From the Fill submenu, choose Down. Select
cells B2 and B3. From the Format menu, choose Cells. Select the Number tab. In the Category list,
choose Number. For Decimal places, use the up and down arrows to set the number of decimal places
to 2. Click OK. Select cells B3–B22. From the Edit menu, choose Fill. From the Fill submenu, choose
Down.
Algebra 1
Chapter 8 Resource Book
91
Name ———————————————————————
Date ————————————
Practice A
LESSON
8.5
For use with pages 536–543
Write a rule for the function.
1.
x
21
0
1
2
3
y
}
1
3
1
3
9
27
2.
x
21
0
1
2
3
y
}
1
5
1
5
25
125
Match the function with its graph.
3. y 5 5x
4. y 5 (2.5)x
LESSON 8.5
A.
B.
y
23
5. y 5 (1.5)x
C.
y
y
5
5
5
3
3
3
21
21
1
23
3 x
21
21
1
23
3 x
21
21
1
3 x
1
3
x
1
3
x
Graph the function and identify its domain and range.
7. y 5 10 x
8. y 5 6 x
y
23
y
5
10
5
3
6
3
1
2
1
21
21
1
3
x
9. y 5 (3.5) x
23
21
22
92
1
3
x
10. y 5 (1.4) x
y
23
y
23
21
21
11. y 5 (2.2) x
y
y
5
5
5
3
3
3
1
1
1
21
21
1
Algebra 1
Chapter 8 Resource Book
3
x
23
21
21
1
3
x
23
21
21
Copyright © Holt McDougal. All rights reserved.
6. y 5 4 x
Name ———————————————————————
LESSON
8.5
Practice A
continued
For use with pages 536–543
5 x
13. y 5 }
2
7 x
12. y 5 }
3
7 x
14. y 5 }
4
1 2
1 2
1 2
y
y
23
Date ————————————
y
5
5
5
3
3
3
1
1
1
21
21
1
3
21
21
23
x
1
x
3
23
21
21
1
3
x
1
3
x
Graph the function. Compare the graph with the graph of y 5 4x.
1
17. y 5 } p 4x
4
16. y 5 3 p 4x
y
LESSON 8.5
15. y 5 24x
y
y
3
3
3
1
1
1
23
21
21
1
3
x
21
21
23
1
x
3
23
21
21
23
23
23
Copyright © Holt McDougal. All rights reserved.
In the growth model, identify the growth rate, the growth factor, and the
initial amount.
18. y 5 3(1 1 0.05)t
19. y 5 2(1 1 0.25)t
20. y 5 0.1(1.75)t
21. Investments You deposit $200 in a savings account that earns 3% interest
compounded yearly. Find the balance in the account after the given amounts of time.
a. 1 year
b. 2 years
c. 5 years
22. Grade Point Average From Chad’s freshman year to his senior year, his grade point
average has increased by approximately the same percentage each year. Chad’s grade
point average in year t can be modeled by
5 t
y 5 21 }4 2
where t 5 0 corresponds to Chad’s freshman year. Complete the table showing
Chad’s grade point average throughout his high school career.
Year, t
0
1
2
3
Grade point average
?
?
?
?
Algebra 1
Chapter 8 Resource Book
93
Name ———————————————————————
Date ————————————
Practice B
LESSON
8.5
For use with pages 536–543
Write a rule for the function.
1.
x
22
21
0
1
2
y
}
1
121
}
1
11
1
11
121
2.
x
21
0
1
2
3
y
}
1
8
}
1
4
}
1
2
1
2
Graph the function and identify its domain and range.
3. y 5 12x
4. y 5 (1.75)x
y
y
10
LESSON 8.5
5. y 5 (3.1)x
y
3
3
6
23
2
21
22
23
1
3
x
9 x
6. y 5 }
2
1 2
21
21
1
3
x
23
23
21
21
1
3
x
1
3
x
1
3
x
23
3 x
8. y 5 2 }
2
1 2
7. y 5 25x
y
y
y
5
1
3
23
1
1
21
3
x
21
21
23
1
3
x
23
23
25
25
4 x
10. y 5 2 p }
3
1 2
9. y 5 5 p 2x
y
11. y 5 23 p 2x
y
y
1
10
5
6
3
2
1
21
21
23
23
94
21
22
Algebra 1
Chapter 8 Resource Book
1
3 x
23
21
21
1
3
x
25
Copyright © Holt McDougal. All rights reserved.
1
Name ———————————————————————
LESSON
8.5
Practice B
Date ————————————
continued
For use with pages 536–543
Graph the function. Compare the graph with the graph of y 5 6x.
12. y 5 2 p 6 x
1
14. y 5 } p 6 x
2
13. y 5 26 x
y
y
y
1
10
23
1
21
3
5
x
3
6
23
1
2
23
21
22
1
3
25
x
23
1
16. y 5 2} p 6 x
4
y
y
21
23
3
x
1
3
x
3
17. y 5 2} p 6 x
2
y
2
3
23
1
1
3
1
x
23
21
21
23
1
3
x
21
22
LESSON 8.5
15. y 5 23 p 6 x
21
21
26
29
23
210
215
25
18. Investments You deposit $500 in a savings account that earns 2.5% interest
a. 1 year
b. 5 years
c. 20 years
19. College Tuition From 1995 to 2005, the tuition at a
college increased by about 7% per year. Use the graph
to write an exponential growth function that models
the tuition over time.
Tuition (dollars)
Copyright © Holt McDougal. All rights reserved.
compounded yearly. Find the balance in the account after the given amounts of time.
y
16,000
12,000
8,000
(0, 8000)
4,000
0
0 1 2 3 4 5 6 7 8 9 10 t
Years since 1995
20. Profit A business had $10,000 profit in 2000. Then the profit increased by 8%
each year for the next 10 years.
a. Write a function that models the profit in dollars over time.
b. Use the function to predict the profit in 2009.
Algebra 1
Chapter 8 Resource Book
95
Name ———————————————————————
Date ————————————
Practice C
LESSON
8.5
For use with pages 536–543
Write a rule for the function.
1.
x
22
21
y
2}
16
1
2}4
1
2.
0
1
2
21
24
216
x
21
0
1
2
3
y
}
5
2
5
10
20
40
Graph the function and identify its domain and range.
3. y 5 15x
4. y 5 (2.25)x
5. y 5 (5.2)x
LESSON 8.5
y
15
3
5
9
1
3
23
3
23
y
y
21
23
1
3
21
21
x
9 x
6. y 5 }
8
1 2
3 x
1
1
23
23
21
21
x
1
3
x
1 2
y
y
3
3
5 x
8. y 5 2 }
2
7. y 5 27x
23
1
21
21
y
1
3
3
x
1
23
1
21
21
1
3
x
23
21
21
25
23
23
27
3 x
10. y 5 4 p }
2
1 2
9. y 5 3 p 6 x
y
11. y 5 22 p 4x
y
y
10
2
15
23
6
21
22
9
26
2
3
23
96
21
23
Algebra 1
Chapter 8 Resource Book
23
1
3 x
21
22
1
3
x
210
1
3 x
Copyright © Holt McDougal. All rights reserved.
23
Name ———————————————————————
Practice C
LESSON
8.5
Date ————————————
continued
For use with pages 536–543
Graph the function. Compare the graph with the graph of y 5 5x.
12. y 5 2 p 5x
1
14. y 5 } p 5x
2
13. y 5 25x
y
y
y
10
3
1
6
23
21
21
3 x
1
2
23
21
22
23
23
3 x
1
1
21
21
23
21
23
3
x
29
215
23
3
3
1
1
21
21
1
3
x
y
y
1
LESSON 8.5
y
x
3
17. y 5 2} p 5x
4
1
16. y 5 2} p 5x
2
3
3
23
25
15. y 5 23 p 5x
1
1
3
x
23
21
21
23
23
Copyright © Holt McDougal. All rights reserved.
18. Investments You deposit $375 in a savings account that earns 2.75% interest
compounded yearly. Find the interest earned by the account after the given amounts
of time. Explain how you got your answers.
a. 1 year
b. 5 years
c. 20 years
19. Population A town had a population of 65,000 in 2000. Then the population
increased by 2.5% each year for the next 5 years.
a. Write a function that models the population over time.
b. Use the function to predict the population in 2004.
20. Internet Users The number of students who have applied for Internet privileges at
school has doubled each month.
a. What is the percent of increase each month?
b. Ten students had applied for Internet privileges initially. Write a function that
models the number of students applying for Internet privileges over time.
c. How many students will have applied for Internet privileges in 4 months?
Algebra 1
Chapter 8 Resource Book
97
Name ———————————————————————
LESSON
8.5
Date ————————————
Review for Mastery
For use with pages 536–543
GOAL
Write and graph exponential growth models.
Vocabulary
An exponential function is a function of the form y 5 abx where
a Þ 0, b > 0, and b Þ 1.
When a > 0 and b > 1, the function y 5 abx represents exponential
growth.
Compound interest is interest earned on both an initial investment and
on previously earned interest.
LESSON 8.5
EXAMPLE 1
Write a function rule
Write a rule for the function.
11
Solution
EXAMPLE 2
Tell whether the function is
exponential. Here, the y-values are
multiplied by 5 for each increase
of 1 in x, so the table represents an
exponential function of the form
y 5 a p bx where b 5 5.
11
11
x
22
21
0
1
2
y
}
2
5
2
10
50
250
35
35 35 35
STEP 2
Find the value of a by finding the value of y when x 5 0. When x 5 0,
y 5 ab0 5 a p 1 5 a. The value of y when x 5 0 is 10, so a 5 10.
STEP 3
Write the function rule. A rule for the function is y 5 10 p 5 x.
Copyright © Holt McDougal. All rights reserved.
STEP 1
11
Graph an exponential function
Graph the function y 5 5 • 3x. Identify its domain and range.
Solution
STEP 1
Make a table by choosing a few values
for x and finding the values of y. The
domain is all real numbers.
x
22
21
0
1
2
y
}
5
9
}
5
3
5
15
45
y
STEP 2
Plot the points.
45
STEP 3
Draw a smooth curve through the points.
From either the table or the graph, you can
see that the range is all positive real numbers.
35
25
15
23
98
Algebra 1
Chapter 8 Resource Book
21
1
3
x
Name ———————————————————————
LESSON
8.5
Review for Mastery
Date ————————————
continued
For use with pages 536–543
EXAMPLE 3
Compare graphs of exponential functions
1
Graph y 5 2} p 4x and y 5 2 • 4x. Compare each graph with the graph of
2
y 5 4x.
Solution
To graph each function, make a table of values, plot the points, and draw a smooth
curve through the points.
y
x
y54
x
1
y 5 2} p 4x
2
y52p4
x
1
}
1
16
2}
32
21
}
1
4
2}8
y5
1
8
}
1
23
1
2
}
y 5 4x
2(4)x
21
1
y5
3
x
1
22 (4)x
23
1
0
1
2}2
2
1
4
22
8
2
16
28
32
1
LESSON 8.5
22
3
1
Because the y-values for y 5 2}2 p 4x are 2}2 times the corresponding y-values for
1
Copyright © Holt McDougal. All rights reserved.
y 5 4x, the graph of y 5 2}2 p 4x is a vertical shrink and a reflection in the x-axis of
the graph of y 5 4x.
Because the y-values for y 5 2 p 4x are 2 times the corresponding y-values for y 5 4x,
the graph of y 5 2 p 4x is a vertical stretch of the graph of y 5 4x.
Exercises for Examples 1, 2, and 3
1. Write a rule for the function.
x
22
21
0
1
2
y
1
3
9
27
81
2. Graph y 5 4 p 3 x and identify its domain and range.
3. Graph y 5 25 p 6 x. Compare the graph with the graph of y 5 6 x.
Algebra 1
Chapter 8 Resource Book
99
Name ———————————————————————
LESSON
8.5
Date ————————————
Problem Solving Workshop:
Worked Out Example
For use with pages 536–543
PROBLEM
Savings You put $125 in a savings account that earns 3% annual interest
compounded yearly. You do not make any deposits or withdrawals. How much will
your investment be worth in 4 years?
STEP 1
Read and Understand
What do you know? The amount deposited, the annual interest, and the years
What do you want to find out? How much is in the account after 4 years?
STEP 2
Make a Plan Use what you know to write and solve an exponential growth
model.
STEP 3
Solve the Problem Write and solve an exponential growth model.
LESSON 8.5
y 5 a(1 1 r)t
5 125(1 1 0.03)4
5 125(1.03)4
ø 140.69
Write exponential growth model.
Substitute 125 for a, 0.03 for r, and 4 for t.
Simplify.
Use a calculator.
You will have $140.69 in 4 years.
STEP 4
Look Back Use the simple interest formula to estimate the amount of
interest earned.
I 5 Prt
5 (125)(0.03)(4) 5 15
Write simple interest formula.
Substitute 125 for P, 0.03 for r, and 4 for t.
PRACTICE
1. Internet In 1996, consumer spending
per person per year for the Internet
was $13.24. The spending increased
by about 36% per person per year from
1996 to 2007. Predict the spending per
person per year on the Internet in 2007.
2. Error Analysis Describe and correct
the error made in solving Exercise 1.
x
y 5 13.24(0.36)
5 13.24(0.36)11
ø 0.10
The consumer spending per person per
year for the Internet increased by $.10
from 1996 to 2007. The spending in
2007 was $13.34.
100
Algebra 1
Chapter 8 Resource Book
3. Population In 1960, the population of
the United States was 179,323,175. By
2000, the population was 281,423,231.
Write an exponential model for the U.S.
population from 1960 to 2000. Use the
model to predict the U.S. population in
2010.
4. Pond When a stone is dropped into a
pond, the initial 1-foot radius of the ripple
increases at a rate of about 50% per
second. Find the radius of the initial ripple
5 seconds after the stone is dropped.
5. What If? Suppose a larger stone is
dropped into the pond and the initial
1-foot radius of the ripple increases at
a rate of about 75% per second. Find
the radius of the initial ripple 5 seconds
after the stone is dropped.
Copyright © Holt McDougal. All rights reserved.
The compounded interest is slightly more than $15. So, the answer is correct.
Name ———————————————————————
LESSON
8.5
Date ————————————
Challenge Practice
For use with pages 536–543
In Exercises 1–5, find an exponential function of the form f(x) 5 ab x that
passes through the given points.
1. (0, 1), (2, 9), (4, 81)
3
2. (0, 3), (1, 6), 21, }
2
1
2
3.
125
1
, 3, }
1 0, }12 2, 1 21, }
10 2 1
2 2
4.
1 0, }19 2, 1 1, }13 2, (2, 1)
5.
1 0, }32 2, (1, 3), (3, 12)
6. f (x) 5 3 p 28x
LESSON 8.5
In Exercises 6–10, use the properties of exponents to write both functions
so that each has the same constant raised to a power, then determine
which function has the greater value when x 5 1.
g(x) 5 3 p 46x
7. f (x) 5 2 p 42x 2 1
g(x) 5 5 p 16x 1 2
8. f (x) 5 25x 1 1
1 22x
Copyright © Holt McDougal. All rights reserved.
g(x) 5 1 }5 2
9. f (x) 5 6 p 16x
1
g(x) 5 }2 p 64x
10. f (x) 5 1000 p (2.25)5x
g(x) 5 2000 p (1.5)3x
Algebra 1
Chapter 8 Resource Book
101
Name ———————————————————————
LESSON
8.6
Date ————————————
Practice A
For use with pages 547–554
Tell whether the table represents an exponential function. If so, write a
rule for the function.
1.
x
22
21
0
1
2
y
100
10
1
}
1
10
}
2.
1
100
x
21
0
1
2
3
y
25
23
21
1
3
Match the function with its graph.
1 x
3. y 5 }
2
1 2
1 x
5. y 5 2 }
2
1 2
4. y 5 2x
A.
B.
y
C.
y
3
1
23
y
3
3
21
21
1
23
3 x
x
21
23
3
21
21
23
23
1
3 x
1
3
x
1
3
x
23
Graph the function and identify its domain and range.
2 x
7. y 5 }
5
3 x
8. y 5 }
8
1 2
1 2
1 2
y
LESSON 8.6
y
23
5
5
5
3
3
3
1
1
1
21
21
1
3
x
9. y 5 (0.4)x
23
21
21
102
1
3
x
10. y 5 (0.7)x
y
23
y
23
21
21
11. y 5 (0.2)x
y
y
5
5
5
3
3
3
1
1
1
21
21
1
Algebra 1
Chapter 8 Resource Book
3
x
23
21
21
1
3
x
23
21
21
Copyright © Holt McDougal. All rights reserved.
1 x
6. y 5 }
6
Name ———————————————————————
Practice A
LESSON
8.6
Date ————————————
continued
For use with pages 547–554
x
1
Graph the function. Compare the graph with the graph of y 5 1 }
2.
3
1 x
12. y 5 2 p }
3
1 x
13. y 5 2 }
3
1 2
1
1 x
14. y 5 } p }
3
3
1 2
y
1 2
y
y
3
3
5
1
3
23
21
21
1
1
x
3
23
1
23
21
21
1
3
21
21
3 x
1
23
x
23
Tell whether the graph represents exponential growth or exponential
decay.
15.
16.
y
17.
y
y
3
3
23
21
21
1
1
3 x
23
1
21
21
1
21
21
x
1
3
x
23
18.
19.
y
Copyright © Holt McDougal. All rights reserved.
225
21
y
3
25
75 x
23
21
21
3
1
3 x
275
225
21
25
75 x
LESSON 8.6
3
275
20.
y
21. Car Value You buy a used car for $12,000. It depreciates at the rate of 15% per
year. Find the value of the car after the given number of years.
a. 1 year
b. 3 years
c. 5 years
22. Declining Employment A business had 4000 employees in 2000. Each year for the
next 5 years, the number of employees decreased by 2%.
a. Write a function that models the number of employees over time.
b. Use the function to predict the number of employees in 2004. Round to the
nearest whole number.
Algebra 1
Chapter 8 Resource Book
103
Name ———————————————————————
LESSON
8.6
Date ————————————
Practice B
For use with pages 547–554
Tell whether the table represents an exponential function. If so, write a
rule for the function.
1.
x
22
21
0
1
2
y
25
5
1
}
1
5
}
2.
1
25
x
21
0
1
2
3
y
1
4
7
10
13
Graph the function and identify its domain and range.
1 x
3. y 5 }
12
7 x
4. y 5 }
8
1 2
1 x
5. y 5 2 }
8
1 2
1 2
y
y
y
5
2
10
23
3
21
1
3
x
3
x
3
x
6
22
21
26
23
21
22
1
3
x
1 x
6. y 5 2 p }
5
1 2
6 x
2
7. y 5 2 p (0.25)x
y
8. y 5 20.5 p (0.3)x
y
y
10
7
6
5
23
21
20.5
2
23
21
22
1
3
Copyright © Holt McDougal. All rights reserved.
LESSON 8.6
0.5
x
1
23
21
1
3
x
x
1
Graph the function. Compare the graph with the graph of y 5 1 }
2.
8
1 x
9. y 5 2 p }
8
1
1 x
11. y 5 } p }
4
8
1 x
10. y 5 2 }
8
1 2
1 2
y
1 2
y
y
20
5
2
23
23
104
21
24
1
Algebra 1
Chapter 8 Resource Book
3
x
21
1
3
3
x
23
21
21
1
Name ———————————————————————
LESSON
8.6
Practice B
Date ————————————
continued
For use with pages 547–554
Decide whether the given statement is always, sometimes, or never true.
Justify your answer.
12. For a positive real number b other than 1, the graphs of y 5 bx and y 5 2bx
are reflections in the y-axis.
1 x
13. For a positive real number b other than 1, the graphs of y 5 bx and y 5 }
b
1 2
intersect.
14. For a nonzero number a and a positive real number b, the graphs of y 5 abx and
1
x
y5}
a • b are not identical.
Tell whether the graph represents exponential growth or exponential
decay. Then write a rule for the function.
15.
16.
y
(21, 4)
(0, 3)
21
21
y
5
(26, 17)
(0, 4)
12
(22, 1)
4 (0, 2)
1
23
17.
y
5
1
3 x
26
22
24
2
23
6 x
1
21
21
1
x
18. Computer Value You buy a computer for $3000. It depreciates at the rate of
20% per year. Find the value of the computer after the given number of years.
a. 1 year
19. Unemployment Rate In 2000, the unemployment rate
of a city decreased by approximately 2.1% each month.
In January, the unemployment rate was 7%.
a. Use the graph at the right to write a function that
models the unemployment rate of the city over time.
b. What was the unemployment rate in December?
Unemployment
rate (percent)
Copyright © Holt McDougal. All rights reserved.
c. 5 years
y (0, 7)
7
6
5
4
0
0 1 2 3 4 5 6 7 8 9 10 11 t
Months since January
LESSON 8.6
b. 3 years
20. Indoor Water Park An indoor water park had a declining attendance from 2000
to 2005. The attendance in 2000 was 18,000. Each year for the next 5 years, the
attendance decreased by 5.5%.
a. Write a function that models the attendance since 2000.
b. What was the attendance in 2005?
Algebra 1
Chapter 8 Resource Book
105
Name ———————————————————————
Date ————————————
Practice C
LESSON
8.6
For use with pages 547–554
Tell whether the table represents an exponential function. If so, write a
rule for the function.
1.
x
–2
–1
0
1
2
y
}
100
81
}
10
9
1
}
9
10
}
2.
81
100
x
22
21
y
2}
2
17
2}
4
33
0
1
28
2}
4
31
2
15
2}
2
Graph the function and identify its domain and range.
1 x
3. y 5 }
15
4 x
4. y 5 }
9
1 2
1 x
5. y 5 2 }
4
1 2
1 2
y
y
y
15
3
3
9
1
1
3
23
23
21
23
3 x
1
23
21
21
3 x
1
21
21
1
3
x
3
x
3
x
23
23
1 x
6. y 5 4 p }
9
1 2
7. y 5 3 p (0.25)x
y
y
y
0.1
30
10
18
6
6
2
23
21
20.1
1
20.3
23
21
26
1
3
x
23
21
22
1
3
Copyright © Holt McDougal. All rights reserved.
LESSON 8.6
8. y 5 20.2 p (0.3)x
20.5
x
1 x
Graph the function. Compare the graph with the graph of y 5 1 }
2.
5
1
9. y 5 5 p }
5
x
1
10. y 5 2 }
5
1 2
x
1
1 x
11. y 5 2} p }
5
5
1 2
y
1 2
y
y
3
25
1
15
23
21
21
5
23
21
25
1
3
x
Algebra 1
Chapter 8 Resource Book
3
1
x
23
21
21
23
23
25
106
1
1
Name ———————————————————————
Practice C
LESSON
8.6
Date ————————————
continued
For use with pages 547–554
Decide whether the given statement is always, sometimes, or never true.
Justify your answer.
12. For a positive real number b other than 1, the graphs of y 5 bx and y 5 2bx
are reflections in the x-axis.
1 x
13. For a positive real number b other than 1, the graphs of y 5 bx and y 5 }
b
1 2
have the same range.
14. For a positive real number b, the function y 5 2bx is an exponential growth
function.
Tell whether the graph represents exponential growth or exponential
decay. Then write a rule for the function.
15.
y
16.
y
5 (0, 5)
(2, 0.8)
1
1
3
y
10
(0, 3)
(21, 2.4)
3
21
21
17.
5
(21, 7.5)
1
5 x
25
23
21
21
(1, 4.8)
2
1 x
23
21
22
1
3 x
18. Truck Value You buy a used truck for $15,000. It depreciates at a rate of 18% per
year. Find how much the value of the truck depreciated after the given number of
years have passed.
a. 1 year
19. Sleeping Behavior On average, as people grow older,
they sleep fewer hours during the night. The amount of
sleep that your great-aunt gets has decreased by 1.8%
since 2000.
a. Use the graph at the right to write a function that models
the number of hours your great-aunt sleeps each night
over time.
b. How many hours of sleep did your aunt average a night
in 2003?
Hours of sleep
Copyright © Holt McDougal. All rights reserved.
c. 5 years
y (0, 8)
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 t
Years since 2000
LESSON 8.6
b. 3 years
20. Investment You invested $2000 into the stock market in 2000. Your investment
increased 6% each year for five years. Over the next five years your investment
decreased in value of 6% each year. Did you have the $2000 again at the end of ten
years? Explain your reasoning.
Algebra 1
Chapter 8 Resource Book
107
Name ———————————————————————
LESSON
8.6
Date ————————————
Review for Mastery
For use with pages 547–554
GOAL
Write and graph exponential decay functions.
Vocabulary
When a > 0 and 0 < b < 1, the function y 5 a p b x represents
exponential decay.
EXAMPLE 1
Write a function rule
Tell whether the table represents an exponential function. If so, write a
rule for the function.
11
x
21
y
2}3
1
11
11
0
1
2
22
212
272
36
36
The y-values are multiplied by 6 for each increase
of 1 in x, so the table represents an exponential
function of the form y 5 abx with b 5 6.
36
The value of y when x 5 0 is 22, so a 5 22.
The table represents the exponential function y 5 22 p 6 x.
LESSON 8.6
1. Tell whether the table represents an
exponential function. If so, write a
rule for the function.
EXAMPLE 2
x
22
21
0
1
2
y
1
3
9
27
81
Copyright © Holt McDougal. All rights reserved.
Exercise for Example 1
Graph an exponential function
x
1
Graph the function y 5 1 }
2 . Identify its domain and range.
10
Solution
STEP 1
Make a table of values. The domain is
all real numbers.
y
9
x
21
0
1
2
7
y
10
1
}
1
10
}
1
100
5
STEP 2
Plot the points.
STEP 3
Draw a smooth curve through the points.
From either the table or the graph, you can
see the range is all positive real numbers.
y5
(101 )
x
1
108
Algebra 1
Chapter 8 Resource Book
23
21
1
3
x
Name ———————————————————————
LESSON
8.6
Review for Mastery
Date ————————————
continued
For use with pages 547–554
EXAMPLE 3
Classify and write rules for functions
Tell whether the graph represents exponential growth or exponential
decay. Then write a rule for the function.
a.
b.
y
y
(1, 8)
7
15
(0, 15)
5
(0, 4)
9
(1, 5)
1
3
23
21
1
3
23
x
21
1
3
x
Solution
a. The graph represents exponential decay ( y 5 abx where 0 < b < 1). The
y-intercept is 15, so a 5 15. Find the value of b by using the point (1, 5)
and a 5 15.
y 5 abx
Write function.
5 5 15 p b1
Substitute.
1
3
Solve.
}5b
1 x
Copyright © Holt McDougal. All rights reserved.
b. The graph represents exponential growth ( y 5 ab x where b > 1). The y-intercept
is 4, so a 5 4. Find the value of b by using the point (1, 8) and a 5 4.
y 5 ab x
Write function.
8 5 4 p b1
Substitute.
25b
Solve.
LESSON 8.6
A function rule is y 5 15 p 1 }3 2 .
A function rule is y 5 4 p 2x.
Exercises for Examples 2 and 3
2. Graph y 5 (0.7) x and identify its domain and range.
1
3. The graph of an exponential function passes through the points (0, 4) and 1, } .
2
1
2
Graph the function. Tell whether the graph represents exponential growth or
exponential decay. Then write a rule for the function.
Algebra 1
Chapter 8 Resource Book
109
Name ———————————————————————
LESSONS
8.4–8.6
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 528–554
a. Write each radius in scientific
notation.
b. The surface area S of a sphere with
radius r is given by S 5 4πr 2. Assume
Jupiter and Callisto are spheres. Find
their surface areas. Write your answers
in scientific notation.
c. What is the ratio of the surface area of
Jupiter to the surface area of Callisto?
What does the ratio tell you?
2. Multi-Step Problem The half-life of a
pesticide is the time it takes for the pesticide
to reduce to half of its original amount in
soil. A certain pesticide has a half-life of
about 45 days.
pesticide. Write a function that models
the amount of the pesticide in the soil
over time.
b. How much of the 20 ounces sprayed
will be in the soil after 180 days?
3. Multi-Step Problem The graph shows
the number of mobile phone subscribers in
the world over time.
Number of
subscribers
(millions)
LESSON 8.6
a. A yard is sprayed with 20 ounces of
y
1600
1200
(0, 91)
800
(1, 145)
400
0
0 1 2 3 4 5 6 7 x
Years since 1995
a. Does the graph represent exponential
growth or exponential decay?
b. Write a function that models the
number of mobile phone subscribers
over time.
c. How many mobile phone subscribers
were there in 1998?
110
Algebra 1
Chapter 8 Resource Book
4. Short Response In 2004, a family bought
a boat for $7000. The boat depreciates (loses
value) at a rate of 15% annually. In 2006,
a person offers to buy the boat for $5500.
Should the family sell the boat? Explain.
5. Gridded Response A new television
costs $400. The value of the television
decreases over time. The value V in dollars
of the television after t years is given by the
function V 5 400(0.86)t. What is the decay
rate, written as a decimal, of the value of
the television?
6. Open-Ended Write two numbers in
scientific notation whose product is
5.4 3 107. Write two numbers in scientific
notation whose quotient is 5.4 3 107.
7. Short Response The graph shows the
value of a car over time.
y
(0, 20,000)
20,000
16,000
12,000
(1, 18,800)
8,000
4,000
0
0 2 4 6 8 10 12 14 16 18 x
Time (years)
a. Write an equation for the function
whose graph is shown.
b. At what rate is the car losing value?
Explain.
8. Extended Response A skier is saving
money to buy a new pair of ski boots. The
skier puts $200 in a saving account that pays
4% annual interest compounded yearly.
a. Write a function that models the amount
of money in the account over time.
b. Graph the function.
c. The skier wants a pair of ski boots that
cost $234.99. Will there be enough in
the account after 3 years to buy the ski
boots? Explain.
Copyright © Holt McDougal. All rights reserved.
Jupiter is about 71,492 kilometers. The
radius of Callisto, one of Jupiter’s moons,
is about 2400 kilometers.
Value (dollars)
1. Multi-Step Problem The radius of
Name ———————————————————————
LESSON
8.6
Date ————————————
Challenge Practice
For use with pages 547–554
In Exercises 1–5, find an exponential function of the form f(x) 5 ab x that
passes through the given points.
1.
3
1 1, }32 2, 1 2, }34 2, 1 4, }
16 2
2.
2
2
, 5, }
1 1, }23 2, 1 3, }
27 2 1 243 2
36
108
3. (0, 4), 2, } , 3, }
25
125
1
21
2
4
4. (1, 1), 2, } , 3, }
5
25
1
5.
21
2
2
1 0, }73 2, (1, 1), 1 2, }37 2
In Exercises 6–9, use the properties of exponents to write both functions
so that each has the same constant raised to a power, then determine
which function has the greater value when x 5 1.
1 5x
6. f(x) 5 3 p }
9
1 2
1 6x
g(x) 5 4 p 1 }3 2
1 2x 2 1
7. f (x) 5 2 p }
4
1 2
1 x12
Copyright © Holt McDougal. All rights reserved.
LESSON 8.6
g(x) 5 5 p 1 }
16 2
1 x11
8. f (x) 5 }
5
1 2
1 2x
g(x) 5 1 }
25 2
3 2x
9. f (x) 5 6 p }
4
1 2
1
18 x
g(x) 5 }2 p 1 }
32 2
Algebra 1
Chapter 8 Resource Book
111
Name ———————————————————————
FOCUS ON
8.6
Date ————————————
Practice
For use with pages 555–556
Tell whether the sequence is arithmetic or geometric. Then graph
the sequence.
1. 2, 4, 6, 8, ...
2. 64, 232, 16, 28, ...
Y
3. 21, 23, 25, 27, ...
Y
Y
/
X
/
X
/
X
Write a rule for the nth term of the geometric sequence. Then graph the
sequence, and identify the domain and the range.
5. 1, 26, 36, 2216, ...
Y
Y
Y
/
X
X
/
FOCUS ON 8.6
6. 3, 6, 12, 24, ...
1 1
1
8. 21, }, 2}, }, ...
4 8
2
1 1 1
7. 1, }, }, }, ...
4 16 64
X
/
Y
9. 281, 227, 29, 23, ...
Y
Y
/
X
X
/
X
/
112
Number of generations, n
1
2
3
Number of new pea-plants, an
1
6
36 216
Algebra 1
Chapter 8 Resource Book
4
.UMBER OF .EW 0EA 0LANTS A
generation. Write a rule for the nth term of the sequence in the
table. Then graph the first six terms of the sequence.
N
10. Challenge A certain type of pea-plant germinates 6 seeds per
Y
X
.UMBER OF 'ENERATIONS N
Copyright © Holt McDougal. All rights reserved.
4. 64, 16, 4, 1, ...
Name ———————————————————————
FOCUS ON
8.6
Date ————————————
Review for Mastery
For use with pages 555–556
GOAL
Identify, graph, and write geometric sequences.
Vocabulary
In a geometric sequence, the ratio of any term to the previous term is
constant.
This constant ratio is called the common ratio and is denoted by r.
The General Rule for a Geometric Sequence is given by an 5 a1r n 2 1.
EXAMPLE 1
Identify a geometric sequence
Tell whether the sequence is arithmetic or geometric. Then write the
next term of the sequence.
a. 4, 8, 12, 16, 20, ...
b. 486, 162, 54, 18, 6, ...
Solution
a. The first term is a1 5 4. Find the ratios of consecutive terms:
a2
a3
8
4
}
a 5}=2
1
12
8
a4
1
2
}
a 5 } 5 1}
2
16
12
a5
1
3
}
a 5 } 5 1}
3
20
16
1
4
}
a 5 } 5 1}
4
a2 2 a1 5 8 2 4 = 4
a3 2 a2 5 12 2 8 = 4
a4 2 a3 5 16 2 12 = 4
a5 2 a4 5 20 2 16 = 4
The common difference is 4, so the sequence is arithmetic. The next term of
the sequence is a6 5 a5 1 4 5 24.
b. The first term is a1 5 486. Find the ratios of consecutive terms:
a2
162
486
1
3
}
a 5}5}
1
a3
54
162
1
3
}
a 5}5}
2
a4
18
54
1
3
}
a 5}5}
3
a5
6
18
FOCUS ON 8.6
Copyright © Holt McDougal. All rights reserved.
Because ratios are not constant, the sequence is not geometric. To see if the
sequence is arithmetic, find the differences of consecutive terms.
1
3
}
a 5}5}
4
Because the ratios are constant, the sequence is geometric. The common ratio
1
1
is }3 . The next term of the sequence is a6 5 a5 • }3 5 2.
Exercises for Example 1
Tell whether the sequence is arithmetic or geometric. Then write the
next term of the sequence.
1. 4, 20, 100, 500, ...
2. 0.5, 1.25, 2, 2.75, ...
3. 32, 16, 8, 4, ...
Algebra 1
Chapter 8 Resource Book
113
Name ———————————————————————
FOCUS ON
8.6
Review for Mastery
Date ————————————
continued
For use with pages 555–556
EXAMPLE 2
Graph a geometric sequence
To graph the sequence from part (b) of Example 1, let each term’s position number in
the sequence be the x-value. The term is the corresponding y-value. Then make and
plot the points.
y
Position, x
Term, y
1
2
3
4
5
486
162
54
18
6
480
400
320
240
160
80
O
1
2
3
4
5 x
Exercises for Example 2
Graph the sequence.
4. 4, 20, 100, 500, ...
5. 0.5, 1.25, 2, 2.75, ...
6. 32, 16, 8, 4, ...
EXAMPLE 3
Write a rule for a geometric sequence
Solution
To write a rule for the nth term of the sequence, substitute the values for a1 and r in the
1
1 n21
general rule an 5 a1r n 2 1. Because a1 5 486 and r 5 }3, an 5 486 • 1 }3 2
. The 10th
1 10 2 1 2
term of the sequence is a10 5 486 • }3
=}
.
81
1 2
Exercises for Example 3
Write a rule for the nth term of the geometric sequence. Then find a10.
1 1 1
7. 1, 22, 4, 28, ...
8. 1, }, }, }, ...
3 9 27
9. 10, 20, 40, 80, ...
114
Algebra 1
Chapter 8 Resource Book
Copyright © Holt McDougal. All rights reserved.
FOCUS ON 8.6
Write a rule for the nth term of the geometric sequence in Example 1.
Then find a10.
Name ———————————————————————
CHAPTER
8
Date ————————————
Chapter Review Game
For use after Chapter 8
Math History
Solve the following exercises. Find the answer at the right of the page.
Place the letter associated with the correct answer on the line with the
exercise number to answer the following question.
Who was the first mathematician to use exponential notation the way we use it today?
Exercises
Answers
1. Simplify: x 3 p x5
(S) 1.495 3 1011
(B) 0
2. Write in scientific notation: 31,009,100
(R) x 8
(N) 1
3. Simplify: (8x 4y 3)0
(E) 0.055
(D) 0.891
x2 3
4. Simplify: }
y
(L) 1.055
(E) }3
5. What is the decay factor in the model y 5 35(0.891)t?
(A) 3.0 3 10 25
(P) x 15
6. Simplify: 2(3x 2)4(2x)2
(T) 2}
4 6
7. Write in standard form: 9.87 3 1025
(K) 987,000
16x22y 4
8. Simplify: }
(2x23y)3
(R) }6
7.5 3 1023
9. Evaluate: }
2.5 3 102
(U) }
y
x6
y
500
x y
(C) 2x 7y
y14
16x
8x 7
10. Simplify: (2x23y 4)2 p (4y22)23
(E) 3.10091 3 107
5 3 2x 2
11. Simplify: 2}2 p }3
x
y
(F) }
y
12. What is the growth rate in the model y 5 17(1.055)t ?
(S) 0.0000987
13. Evaluate: (6.5 3 106)(2.3 3 104)
(E) 2324x10
1
x6
2 1 2
1
5
6
7
2
8
3
9
4
10
11
12
13
Algebra 1
Chapter 8 Resource Book
CHAPTER REVIEW GAME
Copyright © Holt McDougal. All rights reserved.
1 2
115
Name ———————————————————————
CHAPTER
9
Date ————————————
Family Letter
For use with Chapter 9
Copyright © Holt McDougal. All rights reserved.
Lesson Title
Lesson Goals
Key Applications
9.1: Add and Subtract
Polynomials
Add and subtract
polynomials.
• Baseball Attendance
• Backpacking and Camping
• Car Costs
9.2: Multiply Polynomials
Multiply polynomials.
• Skateboarding
• Swimming Pool
• Sound Recordings
9.3: Find Special Products of
Polynomials
Use special product
patterns to multiply
polynomials.
• Border Collies
• Pea Plants
• Football Statistics
9.4: Solve Polynomial Equations
in Factored Form
Solve polynomial
equations.
• Armadillo
• Spittlebug
9.5: Factor x 2 1 bx 1 c
Factor trinomials of the
form
x2 1 bx 1 c.
• Banner Dimensions
• Card Design
• Construction
9.6: Factor ax 2 1 bx 1 c
Factor trinomials of the
form ax2 1 bx 1 c.
• Discus
• Diving
• Scrapbook Design
9.7: Factor Special Products
Factor special products.
• Falling Object
• Falling Brush
• Grasshopper
9.8: Factor Polynomials
Completely
Factor polynomials
completely.
• Terrarium
• Carpentry
• Jumping Robot
CHAPTER SUPPORT
Chapter Overview One way you can help your student succeed in Chapter 9 is by
discussing the lesson goals in the chart below. When a lesson is completed, ask your student the
following questions. ÒWhat were the goals of the lesson? What new words and formulas did you
learn? How can you apply the ideas of the lesson to your life?Ó
• Soccer
Key Ideas for Chapter 9
In Chapter 9, you will apply the key ideas listed in the Chapter Opener (see page 571)
and reviewed in the Chapter Summary (see page 634).
1. Adding, subtracting, and multiplying polynomials
2. Factoring polynomials
3. Writing and solving polynomial equations to solve problems
Algebra 1
Chapter 9 Resource Book
117
Name ———————————————————————
CHAPTER
Family Letter
For use with Chapter 9
continued
Key Ideas Your student can demonstrate understanding of key concepts by working through
the following exercises with you.
Lesson
Exercise
9.1
Find the sum or difference.
(a) (2x3 1 4x2 2 6x 2 8) 1 (x3 2 5x 1 4)
(b) (3x2 2 4x 2 5) 2 (22x2 2 8x 1 7)
9.2
You frame a picture that has a length of 10 inches and a width of 8 inches with a
border that is the same width on every side.
(a) Write a polynomial that represents the total area of the picture and border.
(b) Find the total area when the width of the border is 3 inches.
9.3
Find the product.
9.4
While lying on the ground, you throw a paper airplane straight up in the air with an
initial vertical velocity of 20 feet per second. The airplane’s height h, t seconds after
you throw it, can be modeled by h 5 216t 2 1 20t. After how many seconds does it
land on the ground?
9.5
Factor the trinomial.
(a) x2 1 2x 2 35
(b) y2 2 11y 1 24
9.6
Solve the equation.
(a) 2x2 1 9x 1 7 5 0
(b) 9y2 1 12y 2 12 5 0
9.7
A clothesline runs between two apartment buildings 144 feet in the air. A wet sock
is dropped while being placed on the line. Use the vertical motion model to write an
equation for the height h (in feet) of the sock as a function of the time t (in seconds)
after it is dropped. After how many seconds does the sock land on the ground?
9.8
Factor the expression completely: 6x2y 1 45xy2 1 75y3.
(a) (3x 2 4)2
(b) (x 1 5y)(x 2 5y)
Home Involvement Activity
Directions Measure the length and width of a rectangular-sized yard, to the
nearest foot. Suppose you were going to put a rectangular shaped pool in the yard
with a space x feet wide on all four sides. Find a model for the area of the pool. Write
it as a quadratic trinomial. If x 5 7, what is the area of the pool the yard could hold?
3 seconds 9.8: 3y(2x 1 5y)(x 1 5y)
(b) ( y 2 8)( y 2 3) 9.6: (a) x 5 2}2 , 21 (b) y 5 }3, 22 9.7: h 5 216t 2 1 144;
7
2
9.1: (a) 3x3 1 4x2 2 11x 2 4 (b) 5x 2 1 4x 2 12 9.2: (a) x 2 1 18x 1 80 (b) 143 in.2
9.3: (a) 9x2 2 24x 1 16 (b) x2 2 25y2 9.4: 1.25 seconds 9.5: (a) (x 1 7)(x 2 5)
Answers
118
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
9
Date ————————————
Nombre ——————————————————————
CAPÍTULO
9
Fecha ———————————
Carta para la familia
Usar con el Capítulo 9
Copyright © Holt McDougal. All rights reserved.
Cap’tulo 9 es hablar sobre los objetivos de la lecci—n en la tabla a continuaci—n. Cuando se termina
una lecci—n, pregœntele a su hijo lo siguiente: ÒÀCu‡les fueron los objetivos de la lecci—n? ÀQuŽ
palabras y f—rmulas nuevas aprendiste? ÀC—mo puedes aplicar a tu vida las ideas de la lecci—n?Ó
Título de la lección
Objetivos de la lección
Aplicaciones clave
9.1: Sumar y restar polinomios
Sumar y restar polinomios
• Asistencia de béisbol
• Ir de excursión y de
camping
• Gastos de carro
9.2: Multiplicar polinomios
Multiplicar polinomios
• Patinaje
• Piscina
• Grabaciones de sonidos
9.3: Hallar productos especiales
de polinomios
Usar patrones de productos
especiales para multiplicar
polinomios
• Pastor fronterizo
• Plantas de guisantes
• Estadística de fútbol
americano
9.4: Resolver ecuaciones de
polinomios en forma de
factores
Resolver ecuaciones de
polinomios
• Armadillo
• Insecto
9.5: Hallar factores de
x 2 1 bx 1 c
Hallar factores de trinomios
en forma de x2 1 bx 1 c.
• Dimensiones de
estandarte
• Diseño de tarjeta
• Construcción
9.6: Hallar factores de
ax 2 1 bx 1 c
Hallar factores de trinomios
en forma de ax2 1 bx 1 c.
• Disco
• Zambullidas
• Diseño de álbum de
recortes
9.7: Hallar factores de
productos especiales
Hallar factores de productos
especiales
• Objeto en caída
• Cepillo en caída
• Saltamontes
9.8: Hallar factores de
polinomios completamente
Hallar factores de
polinomios completamente
• Terrario • Carpintería
• Robot de saltos
CHAPTER SUPPORT
Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener Žxito en el
• Fútbol
Ideas clave para el Capítulo 9
En el Capítulo 9, aplicarás las ideas clave enumeradas en la Presentación del capítulo
(ver la página 571) y revisadas en el Resumen del capítulo (ver la página 634).
1. Sumar, restar y multiplicar polinomios
2. Hallar factores de polinomios
3. Escribir y resolver ecuaciones de polinomios para resolver problemas
Algebra 1
Chapter 9 Resource Book
119
Nombre ——————————————————————
CAPÍTULO
Carta para la familia
continúa
Usar con el Capítulo 9
Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer los
siguientes ejercicios con usted.
Lección
Ejercicio
9.1
Halla la suma o la diferencia.
(a) (2x3 1 4x2 2 6x 2 8) 1 (x3 2 5x 1 4)
(b) (3x2 2 4x 2 5) 2 (22x2 2 8x 1 7)
9.2
Enmarcas un cuadro que tiene un largo de 10 pulgadas y un ancho de 8 pulgadas
con un borde que tiene el mismo ancho en cada lado.
(a) Escribe un polinomio que represente el área total del cuadro y del borde.
(b) Halla el área total cuando el ancho del borde es 3 pulgadas.
9.3
Halla el producto.
9.4
Recostado en el suelo, tiras un avión de papel hacia arriba con una velocidad
vertical inicial de 20 pies por segundo. La altura del avión h, t segundos después
de tirarlo, se puede modelar por h 5 216t 2 1 20t. ¿Después de cuántos segundos
aterriza el avión?
9.5
Halla los factores del trinomio. (a) x2 1 2x 2 35
9.6
Resuelve la ecuación.
9.7
Un tendedero se extiende 114 pies en el aire entre dos edificios de apartamentos.
Un calcetín mojado se cae del tendedero. Usa el modelo de moción vertical para
escribir una ecuación para la altura h (en pies) del calcetín como una función del
tiempo t (en segundos) después de caerse. ¿Después de cuántos segundos aterriza el
calcetín?
9.8
Halla los factores de la expresión completamente: 6x2y 1 45xy2 1 75y3.
(a) (3x 2 4)2
(b) (x 1 5y)(x 2 5y)
(a) 2x2 1 9x 1 7 5 0
(b) y2 2 11y 1 24
(b) 9y2 1 12y 2 12 5 0
Actividad para la familia
Instrucciones Mide el largo y el ancho de un patio trasero rectangular al pie
más próximo. Supón que deseas poner una piscina rectangular en el patio con un espacio x pies de ancho en los cuatro lados. Halla un modelo para el área de la piscina.
Escríbelo como un trinomio cuadrático. Si x 5 7, ¿qué es el área de la piscina que se
puede poner en el patio?
3 segundos 9.8: 3y(2x 1 5y)(x 1 5y)
(b) ( y 2 8)( y 2 3) 9.6: (a) x 5 2}2 , 21 (b) y 5 }3, 22 9.7: h 5 216t 2 1 144;
7
2
9.1: (a) 3x3 1 4x2 2 11x 2 4 (b) 5x 2 1 4x 2 12 9.2: (a) x 2 1 18x 1 80 (b) 143 pulg2
9.3: (a) 9x2 2 24x 1 16 (b) x2 2 25y2 9.4: 1.25 segundos 9.5: (a) (x 1 7)(x 2 5)
Respuestas
120
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
9
Fecha ———————————
Name ———————————————————————
Date ————————————
Graphing Calculator Activity Keystrokes
LESSON
9.1
For use with page 578
TI-83 Plus
Casio CFX-9850GC Plus
Part a.
Part a.
From the main menu, choose GRAPH.
Y=
3
(
)
X,T,,n
x
2
2
(
2
5
x2
X,T,,n
2
x
X,T,,n
)
X,T,,n
3
ENTER
2
X,T,,n
2
4
EXE
Use the arrow keys to move the cursor to the
graph style icon in the first column before y2.
Press ENTER until you see the graph style thick.
WINDOW
()
ENTER
()
ENTER
GRAPH
ENTER
ENTER
5
5
1
1
ENTER
ENTER
2
3
EXE
1
5
EXE
Part b.
(
1
2
X,T,,n
MATH
3
ENTER
MATH
(
CLEAR
3
CLEAR
MATH
X,T,,n
)
()
4
X,T,,n
5
(
X,T,,n
3
7
X,T,,n
)
6
x
2
F4
F2
1
EXE
EXE
EXIT
X,,T
5
EXE
6
5
4
2
EXIT
()
5
3
X,,T
X,,T
SHIFT
F3
5
EXE
EXE
)
5
2
)
()
5
F6
()
4
EXE
1
5
EXE
EXE
3
^
X,,T
EXE
EXE
X,T,,n
X,,T
x
X,,T
2
2
Part b.
From the main menu, choose GRAPH.
(
Y=
X,,T
EXE
x2
^
^
F4
F2
1
EXE
EXIT
3
X,,T
X,,T
EXIT
()
)
7
10
X,,T
SHIFT
EXE
1
X,,T
2
3
F3
)
LESSON 9.1
5
5
(
X,T,,n
ENTER
X,,T
(
()
10
F6
ENTER
Copyright © Holt McDougal. All rights reserved.
Use the arrow keys to move the cursor to the
graph style icon in the first column before y2.
Press ENTER until you see the graph style thick.
WINDOW
()
5
10
ENTER
()
ENTER
GRAPH
ENTER
ENTER
5 ENTER 1
10 ENTER 1
Algebra 1
Chapter 9 Resource Book
121
Name ———————————————————————
LESSON
9.1
Date ————————————
Practice A
For use with pages 572–578
Write the polynomial so that the exponents decrease from left to right.
Identify the degree and leading coefficient of the polynomial.
1. 8n6
2. 29z 1 1
3. 4 1 2x 5
4. 18x 2 x 2 1 2
5. 3y 3 1 4y 2 1 8
6. m 2 20m3 1 5
7. 28 1 10a 4 2 3a7
8. 4z 1 z 3 2 5z 2 1 6z 4
9. 8h 3 2 6h 4 1 h7
LESSON 9.1
Tell whether the expression is a polynomial. If it is a polynomial, find its
degree and classify it by the number of its terms. Otherwise, tell why it is
not a polynomial.
10. 6m2
11. 3x
12. y 22 1 4
13. 3b2 2 2
1
14. } x 2 2 2x 1 1
2
15. 6x 3 2 1.4x
Find the sum or difference.
16. (6x 1 4) 1 (x 1 5)
17.
(4m2 2 5) 1 (3m2 2 2)
18.
(2y 2 1 y 2 1) 1 (7y 2 1 4y 2 3)
19.
(3x 2 1 5) 2 (x 2 1 2)
20.
(10a 2 1 4a 2 5) 2 (3a 2 1 2a 1 1)
21.
(m 2 2 3m 1 4) 2 (2m 2 1 5m 1 1)
Write a polynomial that represents the perimeter of the figure.
23.
x12
x11
2x 1 1
x14
x21
x15
x11
Copyright © Holt McDougal. All rights reserved.
22.
24. Library Books For 1995 through 2005, the number F of fiction books (in ten
thousands) and the number N of nonfiction books (in ten thousands) borrowed
from a library can be modeled by
F 5 0.01t 2 1 0.08t 1 7 and N 5 0.004t 2 1 0.05t 1 5
where t is the number of years since 1995. Find the total number B of books
borrowed from the library in a year from 1995 to 2005.
25. Photograph Mat A mat in a frame has an opening
for a photograph as shown in the figure. Find the
area of the mat if the area of the opening is given by
A 5 πab. Leave your answer in terms of π.
b ⫽ 4x ⫺ 2
4x
a⫽3
x
Not drawn to scale
122
Algebra 1
Chapter 9 Resource Book
Name ———————————————————————
Date ————————————
Practice B
LESSON
9.1
For use with pages 572–578
Write the polynomial so that the exponents decrease from left to right.
Identify the degree and leading coefficient of the polynomial.
1. 4n 5
2. 4x 2 2x 2 1 3
3. 6y 3 2 2y 2 1 4y 4 2 5
Tell whether the expression is a polynomial. If it is a polynomial, find its
degree and classify it by the number of its terms. Otherwise, tell why it is
not a polynomial.
4. 10 x
5. 26n 2 2 n 3 1 4
6. w 23 1 5
Find the sum or difference.
(3z 2 1 z 2 4) 1 (2z 2 1 2z 2 3)
8.
9.
(2x 2 1 5x 2 1) 1 (x 2 2 5x 1 7)
10.
(24m 2 1 3m 2 1) 2 (m 1 2)
12. (3m 1 4) 2 (2m 2 2 6m 1 5)
11.
(8c 2 2 4c 1 1) 1 (23c 2 1 c 1 5)
(10b 2 2 3b 1 2) 2 (4b 2 1 5b 1 1)
Write a polynomial that represents the perimeter of the figure.
13.
14.
3x
2x 1 1
4x 2 3
2x 2 1
2x 1 1
x12
2x 1 1
3x
Copyright © Holt McDougal. All rights reserved.
LESSON 9.1
7.
15. Floor Plan The first floor of a home has the floor plan shown. Find the area of the
first floor.
4x
x
x
2
8
x24
x
2
16. Profit For 1995 through 2005, the revenue R (in dollars) and the cost C (in dollars)
of producing a product can be modeled by
1
21
R 5 }4 t 2 1 }
t 1 400
4
and
1
13
C5}
t2 1 }
t 1 200
12
4
where t is the number of years since 1995. Write an equation for the profit earned
from 1995 to 2005. (Hint: Profit 5 Revenue 2 Cost)
Algebra 1
Chapter 9 Resource Book
123
Name ———————————————————————
LESSON
9.1
Date ————————————
Practice C
For use with pages 572–578
Tell whether the expression is a polynomial. If it is a polynomial, find its
degree and classify it by the number of its terms. Otherwise, tell why it is
not a polynomial.
1. 28
2. x 2 2 5x 1 x 21
1
3. 23b2 2 5 1 } b
2
LESSON 9.1
Find the sum or difference.
4.
(3m 3 1 2m 1 1) 1 (4m 2 2 3m 1 1)
5.
(24y 2 1 y 1 5) 1 (4 2 3y 2 y 2)
6.
(24c 1 c 3 1 8) 1 (c 2 2 5c 2 3)
7. (23z 1 6) 2 (4z 2 2 7z 2 8)
8.
(14x 4 2 3x 2 1 2) 2 (3x 3 1 4x 2 1 5)
9.
(5 2 x 4 2 2x 3) 2 (26x 2 1 5x 1 5)
10. Find the sum f (x) 1 g(x) and the difference f(x) 2 g(x) for the functions
f(x) 5 25x 2 1 2x 2 1 and g(x) 5 6x 3 1 2x 2 2 5.
Find the sum or difference.
11.
(10a 2b 2 2 7a 2b) 1 (24a 3b 2 1 5a 2b 2 2 3a 2b 1 5)
12.
(6m2n 2 5mn2 2 8n 1 2m) 2 (6n2m 1 3m2n)
(in thousand metric tons) and the amount L of perlite produced (in thousand
metric tons) in the United States can be modeled by
P 5 3.09t 4 2 36.74t 3 + 121.38t 2 2 77.65t 1 663.57
and
L 5 1.84t 4 2 20.04t 3 1 56.27t 2 2 48.77t 1 703.94
where t is the number of years since 1997.
a. Write an equation that gives the total number T of thousand metric tons of peat
and perlite produced as a function of the number of years since 1997.
b. Was more peat and perlite produced in 1997 or in 2003? Explain your answer.
14. Home Sales In 1997, the median sale price for a one-family home in the
Northeast was about $187,443 and the median sale price for a one-family home
in the Midwest was about $151,629. From 1997 through 2003, the median sale
price for a one-family home in the Northeast increased by about $13,857 per
year and the median sale price for a one-family home in the Midwest increased
by about $5457 per year.
a. Write two equations that model the median sale prices of a one-family home in
the Northeast and Midwest as functions of the number of years since 1997.
b. How much more did a home in the Northeast cost than a home in the Midwest
in 1997 and 2003? What was the change in the sale price of each area from
1997 to 2003?
124
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
13. Mineral Production For 1997 through 2003, the amount P of peat produced
Name ———————————————————————
LESSON
9.1
Date ————————————
Review for Mastery
For use with pages 572–578
GOAL
Add and subtract polynomials.
Vocabulary
A monomial is a number, a variable, or the product of a number and
one or more variables with whole number exponents.
The degree of a monomial is the sum of the exponents of the variables
in the monomial.
A polynomial is a monomial or a sum of monomials, each called a
term of the polynomial.
The degree of a polynomial is the greatest degree of its terms.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.
EXAMPLE 1
LESSON 9.1
When a polynomial is written so that the exponents of a variable
decrease from left to right, the coefficient of the first term is called
the leading coefficient.
Rewrite a polynomial
Write 12x 3 2 15x 1 13x 5 so that the exponents decrease from left to
right. Identify the degree and the leading coefficient of the polynomial.
Copyright © Holt McDougal. All rights reserved.
Solution
Consider the degree of each of the polynomial’s terms.
Degree is 3.
Degree is 1.
Degree is 5.
12x 3 2 15x 1 13x5
The polynomial can be rewritten as 13x5 1 12x 3 2 15x. The greatest degree is 5,
so the degree of the polynomial is 5, and the leading coefficient is 13.
Exercises for Example 1
Write the polynomial so that the exponents decrease from left to right.
Identify the degree and the leading coefficient of the polynomial.
1. 9 2 2x 2
2.
16 1 3y 3 1 2y
3. 6z 3 1 7z 2 2 3z 5
Algebra 1
Chapter 9 Resource Book
125
Name ———————————————————————
LESSON
9.1
Review for Mastery
Date ————————————
continued
For use with pages 572–578
EXAMPLE 2
Add polynomials
Find the sum.
a.
b.
(3x4 2 2x3 1 5x2) 1 (7x2 1 9x32 2x)
(7x22 3x 1 6) 1 (9x2 1 6x2 11)
Solution
a. Vertical format: Align like terms in vertical columns.
3x4 2 2x3 1 5x2
1
9x3 1 7x2 2 2x
______________________
LESSON 9.1
3x4 1 7x3 1 12x2 2 2x
b. Horizontal format: Use the associative and commutative
properties to group like terms and simplify.
(7x2 2 3x 1 6) 1 (9x2 1 6x 2 11) 5 (7x2 1 9x2) 1 (23x 1 6x) 1 (6 2 11)
5 16x2 1 3x 2 5
EXAMPLE 3
Subtract polynomials
Find the difference.
a.
(3x2 2 9x) 2 (2x2 2 5x 1 6)
b.
(11x2 1 6x 2 1) 2 (2x2 2 7x 1 5)
Solution
3x2 2 9x
2 (2x2 2 5x 1 6)
_______________
3x2 2 9x
2 2x2 1 5x 2 6
_____________
x2 2 4x 2 6
b. Horizontal format: Group like terms and simplify.
(11x2 1 6x 2 1) 2 (2x2 2 7x 1 5) 5 11x2 1 6x 2 1 2 2x2 1 7x 2 5
5 (11x2 2 2x2) 1 (6x 1 7x) 1 (21 2 5)
5 9x2 1 13x 2 6
Exercises for Examples 2 and 3
Find the sum or difference.
126
4.
(2a2 1 7) 1 (7a2 1 4a 2 3)
5.
(9b2 2 b 1 8) 1 (4b2 2 b 2 3)
6.
(7c3 2 6c 1 4) 2 (9c3 2 5c 2 2 c)
7.
(d 2 2 15d 1 10) 2 (212d 2 1 8d 2 1)
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
a. Vertical format: Align like terms in vertical columns.
Name ———————————————————————
LESSON
9.1
Date ————————————
Problem Solving Workshop:
Worked Out Example
For use with pages 5722578
PROBLEM
College Basketball Attendance During the period 1999 –2003, the attendance M and
W (in thousands) at men’s and women’s NCAA basketball games, respectively, can be
modeled by
M 5 73.3t3 2 372.4t2 1 722.2t 1 28,524.4 and W 5 40.3t3 2 208.6t2 1 727.7t 1 8035.7
where t is the number of years since 1999. About how many people attended NCAA
basketball games in 2003?
STEP 1
Read and Understand
What do you know? The equations that model the attendance for men’s and
women’s NCAA basketball games from 1999–2003.
STEP 2
Make a Plan Use what you know to add the two equations.
STEP 3
Solve the Problem Add the models for the attendance to men’s and
women’s games to find a model for A, the total attendance (in thousands).
A 5 (73.3t 3 2 372.4t 2 1 722.2t 1 28,524.4) 1 (40.3t 3 2 208.6t 2 1 727.7t 1 8035.7)
LESSON 9.1
What do you want to find out? The attendance of NCAA basketball games in 2003.
5 (73.3t 3 1 40.3t 3) 1 (2372.4t 2 2 208.6t 2) 1 (722.2t 1 727.7t) 1 (28,524.4 1 8035.7)
5 113.6t 3 2 581t 2 1 1449.9t 1 36,560.1
Substitute 4 for t in the model, because 2003 is 4 years after 1999.
A 5 113.6(4)3 2 581(4)2 1 1449.9(4) 1 36,560.1 5 40,334.1
Copyright © Holt McDougal. All rights reserved.
About 40,334,100 people attended NCAA basketball games in 2003.
STEP 4
PRACTICE
Look Back Substitute 4 into each attendance equation and then add to find the
total attendance in 2003.When you substitute 4 into the men’s attendance, you obtain
30,146. When you substitute 4 into the women’s attendance, you obtain 10,188.1.
When you add the men’s and women’s attendance, you get 40,334,100 people.
1. Hockey Attendance During the
2. Salaries During the period
period 199922003, the attendance S
and P (in thousands) at National
Hockey League regular season and
playoff games, respectively, can be
modeled by
1999–2003, the average salaries B and
F (in thousands of dollars) for Major
League Baseball and National Football
League players, respectively, can be
modeled by
S 5 2359.93t 2 1 2272.61t 1
17,084.14
B 5 236.57t 2 1 339.29t 1 1602.86
P 5 214t3 1 72t2 2 23t 1 1475.6
F 5 219.58t3 1 117.14t 2 2 17.49t 1
707.8
where t is the number of years since
1999. About how many people attended
National Hockey League games in
2003?
where t is the number of years since
1999. About how much more was the
average baseball salary than the average
football salary in 2003?
Algebra 1
Chapter 9 Resource Book
127
Name ———————————————————————
LESSON
9.1
Date ————————————
Challenge Practice
For use with pages 572–578
In Exercises 1–5, use the following information.
Suppose you have x number of quarters, x 1 4 number of dimes, 2x 1 1 number of
nickels, and 3x 1 5 number of pennies. For each combination of coins, determine whether
the number of coins is even, odd, or can’t be determined from the given information.
1. The total number of quarters and dimes
2. The total number of quarters and nickels
3. The total number of quarters and pennies
4. The total number of dimes and pennies
LESSON 9.1
5. The total number of dimes, nickels, and pennies
In Exercises 6–12, simplify the given expression. Assume x is positive.
6. (2x 1 1)[(3x 2 2 2x 1 5) 1 (2x 2 1 4x 2 3) 2 (5x 2 1 2x 1 2)]
[( 2
) ( 2
) ( 2
)]
7. (2x 1 1) 3x 2 2x 1 5 1 2x 1 4x 2 3 2 5x 1 2x 1 2
8. x3x 1 5 p x22x 2 2 p x2x 2 2
x
22x p x x
9. 3x p x
2
2
32x 2 5x 1 1 p 322x 1 5x 1 3
10. }}
26x 2 1 p 226x 1 3
2
52x 2 3x 2 4
11. }
2
52x 2 3x 2 6
2
128
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
4
2x 1 3
3x 2 2 2 }
12. }
4
2
2x
3x 2 5
Name ———————————————————————
LESSON
9.2
Date ————————————
Practice A
For use with pages 579–586
Find the product.
1. x(3x 2 2 2x 1 1)
2. 2y(3y 3 1 y 2 2 4)
3. 23m(m2 1 4m 2 1)
4. d 2(4d 2 2 3d 1 1)
5. 2w 3(w 2 1 3w)
6. 2a 2(a 2 1 3a 2 1)
Use a table to find the product.
7. (x 1 1)(x 2 4)
8. (y 1 6)(y 1 2)
9. (a 2 5)(a 2 3)
10. (2m 1 1)(m 1 3)
11. (3z 1 4)(z 2 5)
12. (d 1 6)(3d 2 1)
Use a vertical or a horizontal format to find the product.
13. (y 1 8)(y 2 3)
14. (n 1 5)(n 1 6)
15. (3x 2 2)(x 1 5)
16. (4a 1 1)(2a 2 1)
17. (w 1 1)(w 2 1 2w 1 1)
18. (m 2 2)(m2 2 2m 1 3)
Use the FOIL pattern to find the product.
19. (y 2 3)(8y 1 1)
20. (5b 2 1)(3b 1 2)
21. (2d 2 4)(3d 2 1)
22. (3x 1 1)(2x 1 2)
23. (6x 2 2)(x 1 4)
24. (2s 2 5)(s 1 3)
25. (8c 1 2)(5c 2 7)
26. (8p 2 3)(2p 2 5)
27. (14t 2 2)(t 1 2)
a wooden box to hold all of your sports equipment
as shown.
a. Write a polynomial that represents the volume
of the box.
b. Find the volume of the box when x 5 10.
24 in.
(3x 1 6) in.
(4x 1 8) in.
LESSON 9.2
Copyright © Holt McDougal. All rights reserved.
28. Volume You have come up with a plan for building
29. National Park System During the period 1990–2002, the number A of acres (in
thousands) making up the national park system in the United States and the percent
P (in decimal form) of this amount that is parks can be modeled by
A 5 211t 1 76,226
and
P 5 20.0008t 2 1 0.009t 1 0.6
where t is the number of years since 1990.
a. Find the values of A and P for t 5 0. What does the product A p P mean for
t 5 0 in the context of this problem?
b. Write an equation that models the number of acres (in thousands) that are just
parks as a function of the number of years since 1990.
Algebra 1
Chapter 9 Resource Book
129
Name ———————————————————————
LESSON
9.2
Date ————————————
Practice B
For use with pages 579–586
Find the product.
1. x 2(6x 2 2 3x 2 1)
2. 25a 3(4a 4 2 3a 1 1)
3. 4d 2(22d 3 1 5d 2 2 6d 1 2)
4. (3x 1 1)(2x 2 5)
5. (2y 1 3)(y 2 5)
6. (6a 2 3)(4a 2 1)
7. (b 2 8)(5b 2 2)
8. (8m 1 7)(2m 1 3)
9. (2p 1 2)(3p 2 1 1)
10. (2z 2 7)(–z 1 3)
11. (23d 1 10)(2d 2 1)
12. (n 1 1)(n 2 1 4n 1 5)
13. (w 2 3)(w 2 1 8w 1 1)
14. (2s 1 5)(s 2 1 3s 2 1)
15.
(x 2 2 4xy 1 y 2)(5xy)
Simplify the expression.
16. a(3a 1 1) 1 (a 1 1)(a 2 1)
17. (x 1 2)(x 1 5) 2 x(4x 2 1)
18. (m 1 7)(m 2 3) 1 (m 2 4)(m 1 5)
Write a polynomial for the area of the shaded region.
19.
20.
x
3x
2
5
x
4
LESSON 9.2
21. Flower Bed You are designing a rectangular flower bed
that you will border using brick pavers. The width of the
border around the bed will be the same on every side,
as shown.
x ft
6 ft
a. Write a polynomial that represents the total area of
the flower bed and the border.
b. Find the total area of the flower bed and border when
the width of the border is 1.5 feet.
22. School Enrollment During the period 1995–2002, the number S of students (in
thousands) enrolled in school in the U.S. and the percent P (in decimal form) of this
amount that are between 7 and 13 years old can be modeled by
S 5 32.6t 3 2 376.45t 2 1 1624.2t 1 66,939
and
P 5 0.000005t 4 2 0.0003t 3 1 0.003t 2 2 0.007t 1 0.4
where t is the number of years since 1995.
a. Find the values of S and P for t 5 0. What does the product S p P mean for
t 5 0 in the context of this problem?
b. Write an equation that models the number of students (in thousands) that are
between 7 and 13 years old as a function of the number of years since 1995.
c. How many students between 7 and 13 years old were enrolled in 1995?
130
Algebra 1
Chapter 9 Resource Book
x ft
5 ft
Copyright © Holt McDougal. All rights reserved.
x
Name ———————————————————————
LESSON
9.2
Date ————————————
Practice C
For use with pages 579–586
Find the product.
1. 28y 3(2y 4 2 5y 2 1 3)
2. (b 1 3)(3b 2 2 2b 1 1)
3. (6w 2 3)(4 2 3w)
4.
(9m3 1 1)(4m2 2 1)
5.
(2x 2 1 5x 2 2)(x 1 3)
6.
(8n 2 2 1)(3n 2 2 4n 1 5)
7.
(3p4 2 5)(2p 2 1 4)
8.
(28r 3 1 2)(6r 2 2 1)
9.
(25z 2 2 3)(22z 2 1 9)
10. xy(x 2 1 2y)
11. 23x(2xy 1 5y)
13. (x 2 y)(5x 1 6y)
14.
12. y 2(x 2 y 1 y 2 x )
(xy 2 1 70)(3x 1 2y)
15.
(x 2 2 4xy 1 y 2)(5xy)
Simplify the expression.
16. (7n 1 1)(3n 1 5) 1 (4n 2 2)(3n 1 1)
17. 5w 2(3w 3 2 2w 1 1) 1 w 4 (w 2 2 2w 1 3)
Write a polynomial for the area of the shaded region.
18.
19.
x13
x11
8
2x
x14
12
produced in the U.S. and the average price P (in dollars) spent on one of these cars
can be modeled by
C 5 2198.02t 1 6320.49 and P 5 1.67t 4 2 22.28t 3 1 44.84t 2 1 531.16t 1 16,860
where t is the number of years since 1995.
a. Write an equation that models the total amount spent (in thousands of dollars)
on new cars in the U.S. by consumers as a function of the number of years since
1995.
b. How much money was spent in the U.S. on new cars by consumers in 1995?
LESSON 9.2
Copyright © Holt McDougal. All rights reserved.
20. Car Production During the period 1995–2002, the number of cars C (in thousands)
21. Sporting Goods Equipment During the period 1990–2002, the amount of money
E (in millions of dollars) spent on sporting goods equipment in the U.S. and the
percent P (in decimal form) of this amount that is spent on exercise equipment can
be modeled by
E 5 25.56t 4 1 149.93t 3 2 1314.65t 2 1 4396.75t 1 14,439.09
and P 5 20.00002t 4 2 0.0005t 3 1 0.0028t 2 1 0.001t 1 0.126
where t is the number of years since 1990.
a. Find the values of E and P for t 5 0. What does the product E p P mean for
t = 0 in the context of this problem?
b. Write an equation that models the amount spent (in millions of dollars) on
exercise equipment as a function of the number of years since 1990.
c. How much money was spent in the U.S. on exercise equipment in 1990?
Algebra 1
Chapter 9 Resource Book
131
Name ———————————————————————
LESSON
9.2
Date ————————————
Review for Mastery
For use with pages 579–586
GOAL
EXAMPLE 1
Multiply polynomials.
Multiply a monomial and a polynomial
Find the product 5x 4(2x 3 2 3x 2 1 x 2 6).
Solution
5x4(2x3 2 3x2 1 x 2 6)
Write product.
5 5x4(2x3) 2 5x4(3x2) 1 5x4(x) 2 5x4(6)
7
6
5
Distributive property
4
5 10x 2 15x 1 5x 2 30x
Product of powers property
Exercises for Example 1
Find the product.
1. 3x2(7x2 2 2x 1 3)
EXAMPLE 2
2. 4x5(3x3 2 2x2 2 8x 1 9)
Multiply polynomials vertically
Find the product (5m 2 2 2m 1 3)(2m 1 7).
Solution
Multiply by 7.
STEP 2
LESSON 9.2
2
5m 2 2m 1 3
3
2m 1 7
2
35m 2 14m 1 21
Multiply by 2m.
STEP 3
2
3
5m 2 2m 1 3
2m 1 7
2
35m 2 14m 1 21
3
10m 2 4m2 1 6m
3
Add products.
5m2 2 2m 1 3
2m 1 7
35m2 2 14m 1 21
10m3 2 4m2 1 6m
10m3 1 31m2 2 8m 1 21
EXAMPLE 3
Multiply polynomials horizontally
Find the product (9x 2 2 x 1 6)(5x 2 2).
Solution
(9x2 2 x 1 6)(5x 2 2)
132
Algebra 1
Chapter 9 Resource Book
Write product.
5 9x2(5x 2 2) 2 x(5x 2 2) 1 6(5x 2 2)
Distributive property
5 45x3 2 18x2 2 5x2 1 2x 1 30x 2 12
Distributive property
5 45x3 2 23x2 1 32x 2 12
Combine like terms.
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
LESSON
9.2
Review for Mastery
Date ————————————
continued
For use with pages 579–586
EXAMPLE 4
Multiply binomials using FOIL pattern
Find the product (2x 2 1)(7x 1 6).
Solution
(2x 2 1)(7x 1 6)
Write product.
5 (2x)(7x) 1 (2x)(6) 1 (21)(7x) 1 (21)(6)
Write product of terms.
5 14x2 1 12x 1 (27x) 1 (26)
Multiply.
5 14x2 1 5x 2 6
Combine like terms.
Exercises for Examples 2, 3, and 4
Find the product.
3.
(m2 1 6m 1 4)(3m 2 1)
4. (2n 1 7)(3n 1 4)
5.
(2p2 2 p 1 6)( p 1 7)
6.
7. (5t 1 9)(3t 2 8)
EXAMPLE 5
(6q2 2 5q 2 4)(2q 2 3)
8. (8s 2 7)(9s 2 7)
Standardized Test Practice
The dimensions of a rectangle are 3x 2 1 and x 1 5. Which expression
represents the area of the rectangle?
Copyright © Holt McDougal. All rights reserved.
Solution
Area 5 length p width
Formula for area of a rectangle
5 (3x 2 1)(x 1 5)
Substitute for length and width.
5 (3x)(x) 1 (3x)(5) 1 (21)(x) 1 (21)(5)
Use FOIL pattern.
5
Multiply.
3x2
1 15x 1 (2x) 1 (25)
5 3x2 1 14x 2 5
LESSON 9.2
A 3x2 1 16x 2 5 B 3x2 1 14x 2 4 C 3x2 1 14x 2 5 D 4x 1 4
Combine like terms.
The correct answer is C.
Exercise for Example 5
9. The dimensions of a rectangle are y 1 9 and 2y 2 3. Write an expression for
the area of the rectangle.
Algebra 1
Chapter 9 Resource Book
133
Name ———————————————————————
LESSON
9.2
Date ————————————
Challenge Practice
For use with pages 579–586
In Exercises 1–5, find the product and simplify.
1.
(x3 1 2x)(x4 1 x2)
2.
(3y 2 y3)( y4 1 y)
3.
(2x3 1 2y)(x4 1 2y3)
4. x3(x5 1 4x3)(2x4 1 3x2)
5.
(x2 1 1)(x 1 2)(x2 1 2)
In Exercises 6–10, simplify the expression and write the result as a
polynomial in standard form.
6. x(x2 1 2x) 2 x2(x 1 2)
7. (x 1 1)(x 1 1) 2 (x 2 1)(x 2 1)
8.
(x2 1 1)(x2 1 1) 2 (x2 2 1)(x2 2 1)
9.
(2x2 1 3x 2 1)(x 2 1) 2 2x(x 1 1)
10. (x 1 3)(2x2 1 2) 1 2(x 1 1)(x 2 2) 1 3
In Exercises 11–13, use the following information.
134
11. Write an expression for the volume of the storage compartment in terms of x.
12. Simplify the expression found in Exercise 11 and write it as a polynomial
in standard form.
13. If x is 4, how many trailers will fit inside the storage compartment?
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 9.2
A ship storage compartment is being designed to carry trailers, each of which has
dimensions 50 feet long by 9 feet tall by 8 feet wide. It is decided that the storage
container will have dimensions 50x 1 150 feet long by 9x tall by 8x 1 16 feet wide.
Name ———————————————————————
LESSON
9.3
Date ————————————
Practice A
For use with pages 5872592
Find the missing term.
1. (a 2 b)2 5 a 2 2
?
1 b2
2. (m 1 n)2 5 m 2 1
3. (x 2 1)2 5 x 2 2
?
11
4. (x 1 5)2 5 x 2 1
5. (x 2 y)(x 1 y) 5 x 2 2
1 n2
?
1 25
?
6. (x 2 3)(x 1 3) 5 x 2 2
?
?
Match the product with its polynomial.
7. (2x 1 3)(2x 2 3)
8. (2x 1 3)2
9. (2x 2 3)2
A. 4x 2 1 12x 1 9
B. 4x 2 2 12x 1 9
C. 4x 2 2 9
Find the product of the square of the binomial.
10. (x 1 4)2
11. (m 2 8)2
12. (a 1 10)2
13. (p 2 12)2
14. (2y 1 1)2
15. (3y 2 1)2
16. (10r 2 1)2
17. (4n 1 2)2
18. (3c 2 2)2
Find the product of the sum and difference.
19. (z 1 5)(z 2 5)
20. (b 2 2)(b 1 2)
21. (n 2 8)(n 1 8)
22. (a 1 10)(a 2 10)
23. (2x 1 1)(2x 2 1)
24. (5m 2 1)(5m 1 1)
25. (4d 1 1)(4d 2 1)
26. (3p 1 2)(3p 2 2)
27. (2r 2 3)(2r 1 3)
28. 13 p 7
29. 24 p 36
30. 51 p 69
31. Total Profit For 1995 through 2005, the number N of units (in thousands)
produced by a manufacturing plant can be modeled by N 5 3t 1 2 and the
profit per unit P (in dollars) can be modeled by P 5 3t 2 2 where t is the
number of years since 1995. Write a polynomial that models the total profit T
(in thousands of dollars).
32. Eye Color In humans, the brown eye gene B is dominant
Mother
B
b
B
BB
Bb
b
bB
bb
Father
and the blue eye gene b is recessive. This means that humans
whose eye genes are BB, Bb, or bB have brown eyes and
those with bb have blue eyes. The Punnett square at the right
shows the results of eye colors for children of parents who
each have one B gene and one b gene.
a. Write a polynomial that models the possible gene
combinations of a child.
b. What percent of the possible gene combinations results
in a child with blue eyes?
Algebra 1
Chapter 9 Resource Book
LESSON 9.3
Copyright © Holt McDougal. All rights reserved.
Describe how you can use mental math to find the product.
135
Name ———————————————————————
LESSON
9.3
Date ————————————
Practice B
For use with pages 5872592
Find the product of the square of the binomial.
1. (x 2 9)2
2. (m 1 11)2
3. (5s 1 2)2
4. (3m 1 7)2
5. (4p 2 5)2
6. (7a 2 6)2
7. (10z 2 3)2
8. (2x 1 y)2
9. (3y 2 x)2
Find the product of the sum and difference.
10. (a 2 9)(a 1 9)
11. (z 2 20)(z 1 20)
12. (5r 1 1)(5r 2 1)
13. (6m 1 10)(6m 2 10)
14. (7p 2 2)(7p 1 2)
15. (9c 2 1)(9c 1 1)
16. (4x 1 3)(4x 2 3)
17. (4 2 w)(4 1 w)
18. (5 2 2y)(5 1 2y)
Describe how you can use mental math to find the product.
19. 15 p 25
20. 43 p 57
21. 182
Perform the indicated operation using the functions f(x) 5 4x 1 0.5 and
g(x) 5 4x 2 0.5.
22. f(x) p g(x)
23. (f(x))2
24. (g(x))2
for wrinkled seed shape. Any gene combination with an S results in a spherical seed
shape. Suppose two pea plants have the same gene combination Ss.
a. Make a Punnett square that shows the possible gene combinations of an offspring
pea plant and the resulting seed shape.
b. Write a polynomial that models the possible gene combinations of an offspring
pea plant.
c. What percent of the possible gene combinations of the offspring results in a
wrinkled seed shape?
Made
LESSON 9.3
26. Basketball Statistics You are on the basketball team
136
and you want to figure out some statistics about foul
shots. The area model shows the possible outcomes of
two attempted foul shots.
a. What percent of the two possible outcomes of two
attempted foul shots results in you making at least
one foul shot? Explain how you found your answer
using the table.
b. Show how you could use a polynomial to model the
possible results of two attempted foul shots.
Algebra 1
Chapter 9 Resource Book
Made
Missed
Missed
Copyright © Holt McDougal. All rights reserved.
25. Pea Plants In pea plants, the gene S is for spherical seed shape, and the gene s is
Name ———————————————————————
LESSON
9.3
Date ————————————
Practice C
For use with pages 5872592
Find the product.
1. (8x 2 5)2
2. (4p 1 4)2
3. (10m 2 11)2
4. (11s 2 10)2
5. (20b 2 15)2
6. (m 1 4n)2
7. (r 2 8s)2
8. (10a 1 3b)2
9. (2x 2 4y)2
10. (8p 2 3)(8p 1 3)
11. (11t 1 4)(11t 2 4)
12. (7n 2 5)(7n 1 5)
13. (9z 1 12)(9z 2 12)
14. (15 2 w)(15 1 w)
15. (6 2 5p)(6 1 5p)
16. (20 2 3m)(20 1 3m)
17. (10a 2 5b)(10a 1 5b)
18. (4x 2 3y)(4x 1 3y)
Describe how you can use mental math to find the product.
19. 36 p 44
20. 232
21. 492
Perform the indicated operation using the functions f(x) 5 9x 2 0.5 and
g(x) 5 9x 1 0.5.
22. f(x) p g(x)
23. (f(x) 1 g(x))2
24. (f(x) 2 g(x))2
25. Write two binomials that have the product x 2 2 144. Explain how you found
your answer.
26. Write a pattern for the cube of a binomial (a 2 b)3. Justify.
Made
to figure out some statistics about attempted goals. The area
model shows the possible outcomes of two attempted goals.
a. What percent of the two possible outcomes of two
attempted goals results in you making at least one
goal? Explain how you found your answer using
the table.
b. Show how you could use a polynomial to model the
possible results of two attempted goals.
Missed
Made
Missed
28. Greenhouse You are drawing up a plan to build a greenhouse
in the shape of a rectangular prism. The height of the greenhouse
is constant at 8 feet tall. You have 144 feet of material to form the
base of the greenhouse into a square with a side length of 12 feet.
You want to change the dimensions of the enclosed region. For
every 1 foot you increase the width, you must decrease the
length by 1 foot. Write a polynomial that gives the volume of
the prism after you increase the width by x feet and decrease the
length by x feet. Explain why any change in dimensions results
in a volume less than that of the original prism.
8 ft
LESSON 9.3
Copyright © Holt McDougal. All rights reserved.
27. Soccer Statistics You are on the soccer team and you want
Algebra 1
Chapter 9 Resource Book
137
Name ———————————————————————
LESSON
9.3
Date ————————————
Review for Mastery
For use with pages 587– 592
GOAL
Use special product patterns to multiply polynomials.
Square of a Binomial Pattern
EXAMPLE 1
Algebra
Example
(a 1 b)2 5 a2 1 2ab 1 b2
(x 1 3)2 5 x2 1 6x 1 9
(a 2 b)2 5 a2 2 2ab 1 b2
(3x 2 2)2 5 9x2 2 12x 1 4
Use the square of a binomial pattern
Find the product.
a. (7x 1 2)2
b. (6x 2 5y)2
Solution
a. (7x 1 2)2 5 (7x)2 1 2(7x)(2) 1 22
Square of a binomial pattern
5 49x2 1 28x 1 4
Simplify.
b. (6x 2 5y)2 5 (6x)2 2 2(6x)(5y) 1 (5y)2
5 36x 2 60xy 1 25y
2
2
Square of a binomial pattern
Simplify.
Exercises for Example 1
1. ( y 1 9)2
2. (3z 1 7)2
3. (2w 2 3)2
4. (10r 2 3s)2
Sum and Difference Pattern
Algebra
(a 1 b)(a 2 b) 5 a 2 b
LESSON 9.3
138
Example
2
Algebra 1
Chapter 9 Resource Book
2
(x 1 5)(x 2 5) 5 x2 2 25
Copyright © Holt McDougal. All rights reserved.
Find the product.
Name ———————————————————————
LESSON
9.3
Review for Mastery
Date ————————————
continued
For use with pages 587–592
EXAMPLE 2
Use the sum and difference pattern
Find the product.
a. (m 1 9)(m 2 9)
b. (4n 2 3)(4n 1 3)
Solution
a. (m 1 9)(m 2 9) 5 m2 2 92
Sum and difference pattern
5 m2 2 81
b. (4n 2 3)(4n 1 3) 5 (4n)2 2 32
5 16n2 2 9
Simplify.
Sum and difference pattern
Simplify.
Exercises for Example 2
Find the product.
5. (g 1 11)(g 2 11)
6. (7f 2 1)(7f 1 1)
7. (2h 1 9)(2h 2 9)
8. (6k 2 8)(6k 1 8)
EXAMPLE 3
Use special products and mental math
Solution
Notice that 37 is 3 less than 40 while 43 is 3 more than 40.
37 p 43 5 (40 2 3)(40 1 3)
Write as a product of difference and sum.
5 402 2 32
Sum and difference pattern
5 1600 2 9
Evaluate powers.
5 1591
Simplify.
Exercises for Example 3
Describe how you can use special products to find the product.
9. 552
10. 31 p 49
Algebra 1
Chapter 9 Resource Book
LESSON 9.3
Copyright © Holt McDougal. All rights reserved.
Use special products to find the product of 37 p 43.
139
Name ———————————————————————
LESSON
9.3
Date ————————————
Challenge Practice
For use with pages 587–592
In Exercises 1–5, simplify by multiplying and then adding and subtracting.
Write the result as a polynomial in standard form.
1.
(2x 1 3)2 1 (2x 2 3)2
2.
(2x2 1 1)2 1 (x2 1 2)2
3. (ax 1 by)2 1 (ax 2 by)2
4.
(ax2 1 by2)2 1 (ax2 2 by2)2
5. (x 1 5)2 2 (x 2 25)(x 1 1)
6. Show that (a 2 b 1 c)2 5 a2 1 b2 1 c2 2 2ab 1 2ac 2 2bc.
In Exercises 7 and 8, use the result from Exercise 6 to find the product.
7. (3x 2 2y 1 5z)2
8. (ax 2 by 1 cz)2
In Exercises 9–12, assume x is a positive integer.
9. Find an expression for the product of three consecutive even integers, with 2x as the
smallest of the three integers. Write the result as a polynomial in standard form.
10. Explain why the result from Exercise 9 is an even number.
11. Find an expression for the product of three consecutive odd integers, with 2x 1 1 as
the smallest of the three integers. Write the result as a polynomial in standard form.
LESSON 9.3
Copyright © Holt McDougal. All rights reserved.
12. Explain why the result from Exercise 11 is an odd number.
140
Algebra 1
Chapter 9 Resource Book
Name ———————————————————————
LESSON
9.4
Date ————————————
Practice A
For use with pages 5932598
1. (x 1 4)(x 1 5) 5 0
A. 25 and 4
2. (x 2 4)(x 1 5) 5 0
B. 25 and 24
3. (x 2 5)(x 2 4) 5 0
C. 4 and 5
LESSON 9.4
Match the equation with its solutions.
Solve the equation.
4. (x 1 6)(x 1 2) 5 0
5. (p 2 5)(p 1 3) 5 0
6. (b 2 7)(b 2 10) 5 0
7. (m 2 8)(m 1 1) 5 0
8. (a 2 9)(a 1 9) 5 0
9. (y 1 15)(y 1 12) 5 0
10. (c 2 25)(c 1 50) 5 0
11. (2z 2 2)(z 1 3) 5 0
12. (2n 2 6)(n 2 2) 5 0
Factor out the greatest common monomial factor.
13. 4m 2 2
14. 5x 2 10
15. 6y 1 15
16. 8x 1 8y
17. 7a 2 7b
18. 2a 1 10b
19. 9m 2 18n
20. 15p 2 3q
21. 12x 1 4y
22. 2c 2 1 4c
23. 9m3 1 m2
24. 2w2 1 4w
Copyright © Holt McDougal. All rights reserved.
Match the equation with its solutions.
25. 4a 2 1 a 5 0
A. 0 and 4
26. a 2 1 4a 5 0
B. 0 and 24
27. a 2 2 4a 5 0
1
C. 0 and 2}
4
Solve the equation.
28. a 2 1 8a 5 0
29. n2 2 7n 5 0
30. 2w 2 1 2w 5 0
31. 3p2 2 3p 5 0
32. 4c 2 2 8c 5 0
33. 5x 2 1 10x 5 0
34. Hot Air Balloon An object is dropped from a hot-air balloon 1296 feet above the
ground. The height of the object is given by
h 5 216(t 2 9)(t 1 9)
where the height h is measured in feet, and the time t is measured in seconds.
After how many seconds will the object hit the ground?
35. Kickball A kickball is kicked upward with an initial vertical velocity of 3.2 meters
per second. The height of the ball is given by
h 5 29.8t 2 1 3.2t
where the height h is measured in meters, and the time t is measured in seconds.
After how many seconds does the ball land?
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141
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LESSON
LESSON 9.4
9.4
Date ————————————
Practice B
For use with pages 5932598
Solve the equation.
1. (x 1 14)(x 2 3) 5 0
2. (m 2 12)(m 1 5) 5 0
3. (p 1 15)(p 1 24) 5 0
4. (n 2 8)(n 2 9) 5 0
1
5. (d 1 8) d 2 } 5 0
2
6.
7. (2z 2 8)(z 1 5) 5 0
8. (y 2 3)(5y 1 10) 5 0
9. (6b 2 4)(b 2 8) 5 0
10. (8x 1 4)(6x 2 3) 5 0
1
2
11. (3x 1 9)(6x 2 3) 5 0
1 c 1 }34 2(c 2 6) 5 0
12. (4x 1 5)(4x 2 5) 5 0
Factor out the greatest common monomial factor.
13. 10x 2 10y
14. 8x 2 1 20y
15. 18a2 2 6b
16. 4x 2 2 4x
17. r 2 1 2rs
18. 2m2 1 6mn
19. 5p2q 1 10q
20. 9a5 1 a3
21. 6w 3 2 14w2
22. m2 2 10m 5 0
23. b2 1 14b 5 0
24. 5w 2 2 5w 5 0
25. 24k 2 1 24k 5 0
26. 8r 2 2 24r 5 0
27. 9p2 1 18p 5 0
28. 6n2 2 15n 5 0
29. 28y 2 2 10y 5 0
30. 210b 2 1 25b 5 0
31. 8c 2 5 4c
32. 30r 2 5 215r
33. 224y 2 5 9y
Solve the equation.
The height of the diver is given by
h 5 216(t 2 1.5)(t 1 1)
where the height h is measured in feet, and the time t is measured in seconds. When
will the diver hit the water? Can you see a quick way to find the answer? Explain.
35. Dog To catch a frisbee, a dog leaps into the air with an initial velocity of 14 feet
per second.
a. Write a model for the height of the dog above the ground.
b. After how many seconds does the dog land on the ground?
36. Desktop Areas You have two components to the desktop
where you do your homework that fit together into an
L shape. The two components have the same area.
a. Write an equation that relates the areas of the desktop
components.
b. Find the value of w.
c. What is the combined area of the desktop components?
142
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w
3 ft
w
7 ft
Copyright © Holt McDougal. All rights reserved.
34. Diving Board A diver jumps from a diving board that is 24 feet above the water.
Name ———————————————————————
LESSON
9.4
Date ————————————
Practice C
For use with pages 5932598
5
3
1 m 2 }2 2 1 m 1 }2 2 5 0
3. (4b 1 16)(b 2 6) 5 0
4. (7a 2 14)(a 1 8) 5 0
5. (2y 1 3)(y 2 9) 5 0
6. (5z 2 8)(3z 1 2) 5 0
7. (9w 2 2)(7w 2 3) 5 0
8. (8 2 2c)(5c 1 1) 5 0
9. (9 2 8r)(10 2 4r) 5 0
1
2
2.
LESSON 9.4
Solve the equation.
2
1. (x 1 3) x 2 } 5 0
5
Factor out the greatest common monomial factor.
10. 9x 2 2 21y
11. 4m3 1 24m
12. 10p2q 2 5pq2
13. 6x 3y 1 9y 2
14. 35a2b2 2 5ab
15. 12m2n 2 8mn2
16. w 4 2 2w 3 1 w
17. 23p4 1 15p2 1 6p
18. 8r 5 2 20r 4 2 12r 2
19. 12a2 2 9a 5 0
20. 18x 2 1 12x 5 0
21. 6z 2 2 8z 5 0
22. 20p2 5 224p
23. 228m2 5 14m
24. 230r 2 5 225r
25. 100m2 5 26m
26. 15y 2 50y 2 5 0
27. 26w 1 34w 2 5 0
Solve the equation.
Find the zeros of the function.
28. f (x) 5 228x 2 1 7x
29. f (x) 5 29x 2 1 4x
30. f (x) 5 5x 2 2 3x
Copyright © Holt McDougal. All rights reserved.
31. Fish A fish jumps out of the water while swimming. The height h (in feet) of the
fish can be modeled by h 5 216t 2 1 3.5t where t is the time (in seconds) since the
fish jumped out of the water.
a. Find the zeros of the function. Explain what the zeros mean in this situation.
b. What is a reasonable domain for the function? Explain your answer.
32. Storage Structure The cross section of a wooden storage
y
Center of
structure
70
structure can be modeled by the polynomial function
3
(2x 2 40)(2x 1 40)
y 5 2}
80
where x and y are measured in feet, and the center of the
structure is where x 5 0.
a. Explain how to use the algebraic model to find the
width of the structure.
b. Use the model to find the structure’s width. Show
your work
c. Use the model to find the coordinates of the center of
the structure. Show your work.
50
30
10
215
25
5
15
x
Algebra 1
Chapter 9 Resource Book
143
Name ———————————————————————
LESSON
LESSON 9.4
9.4
Date ————————————
Review for Mastery
For use with pages 593– 598
GOAL
Solve polynomial equations.
Vocabulary
The zero-product property is used to solve an equation when one
side is zero and the other side is a product of polynomial factors.
The solutions of such an equation are also called roots.
The height of a projectile can be described by the vertical motion
model: h 5 216t 2 1 vt 1 s, where t is the time (in seconds) the
object has been in the air, v is the initial vertical velocity (in feet per
second), and s is the initial height (in feet).
EXAMPLE 1
Use the zero-product property
Solve (x 2 3)(x 1 6) 5 0.
Solution
(x 2 3)(x 1 6) 5 0
x2350
or
x53
or
x1650
x 5 26
Write original equation.
Zero-product property
Solve for x.
The roots of the equation are 3 and 26.
(3 2 3)(3 1 6) 0 0
(26 2 3)(26 1 6) 0 0
0p9 00
29 p 0 0 0
050✓
050✓
Exercises for Example 1
Solve the equation.
1. (m 2 7)(m 2 9) 5 0
EXAMPLE 2
2. (5n 1 10)(4n 1 12) 5 0
Solve an equation by factoring
Solve 6x 2 1 12x 5 0.
6x2 1 12x 5 0
Write original equation.
6x(x 1 2) 5 0
Factor left side.
6x 5 0
or
x50
or
x1250
x 5 22
Zero-product property
Solve for x.
The roots of the equation are 0 and 22.
144
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
Substitute each root into the original equation to check.
CHECK
Name ———————————————————————
LESSON
9.4
Review for Mastery
Date ————————————
continued
For use with pages 593– 598
Solve an equation by factoring
LESSON 9.4
EXAMPLE 3
Solve 9y 2 5 21y.
9y2 5 21y
Write original equation.
9y2 2 21y 5 0
Subtract 21y from each side.
3y(3y 2 7) 5 0
Factor left side.
3y 5 0
or
3y 2 7 5 0
Zero-product property
y50
or
y 5 }3
7
Solve for y.
7
The roots of the equation are 0 and }3.
Exercises for Examples 2 and 3
Solve the equation.
3. q2 1 16q 5 0
EXAMPLE 4
4.
4k 2 2 8k 5 0
5. 12h2 5 36h
Solve a multi-step problem
Jump Rope A child jumping rope leaves the ground at an initial vertical velocity
of 8 feet per second. After how many seconds does the child land on the ground?
Solution
Copyright © Holt McDougal. All rights reserved.
STEP 1
STEP 2
Write a model for the height above the ground.
h 5 216t 2 1 vt 1 s
Vertical motion model
h 5 216t 2 1 8t 1 0
Substitute 8 for v and 0 for s.
h 5 216t 2 1 8t
Simplify.
Substitute 0 for h. When the child lands, the child’s height above the
ground is 0 feet. Solve for t.
0 5 216t2 1 8t
Substitute 0 for h.
0 5 8t(22t 1 1)
Factor right side.
8t 5 0
or
22t 1 1 5 0
Zero-product property
t50
or
t 5 }2
1
Solve for t.
1
The child lands on the ground }2 second after the child jumps.
Exercise for Example 4
6. In Example 4, suppose the initial velocity is 10 feet per second. After how
many seconds will the child land on the ground?
Algebra 1
Chapter 9 Resource Book
145
Name ———————————————————————
LESSONS
9.1–9.4
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
1. Multi-Step Problem You are making a
1997–2003, the total number N (in
thousands) of mechanics employed by the
airline industry can be modeled by
a. Write a polynomial that represents
N 5 21.16t 2 1 5.51t 1 65.34
the total area of the picture with the
border.
b. Find the total area of the picture with
border when the width of the border is
2 inches.
2. Multi-Step Problem During the period
199722002, the sporting goods sales S
(in millions of dollars) and the percent P
(in decimal form) of sporting goods sales
that are for exercise equipment can be
modeled by
S 5 1990.5t 1 67,530
P 5 0.0022t 1 0.0436
where t is the number of years since 1997.
a. Write an equation that models the
sales (in millions of dollars) of
exercise equipment as a function of
the number of years since 1997.
b. Find the amount of exercise
equipment sales in 2001.
3. Open-Ended In flowers, the gene P is for
purple coloring and the gene w is for white
coloring. Any gene combination with a P
results in purple coloring.
a. Suppose one flower has the gene
combination Pw. Choose a color gene
combination for another flower. Create
a Punnett square to show the possible
gene combinations of an offspring
flower.
b. What percent of the possible gene
combinations of the offspring result in
purple coloring?
c. Show how you could use a polynomial
to model the possible color gene
combinations of the offspring.
146
4. Gridded Response During the period
scrapbook out of pictures that are 7 inches
long and 5 inches wide. You want to place a
border of equal width on each edge.
Algebra 1
Chapter 9 Resource Book
where t is the number of years since 1997.
What is the degree of the polynomial that
represents N?
5. Short Response The height h (in feet)
of a kangaroo’s jump can be modeled by
h 5 216t 2 1 18t where t is the time
(in seconds) since the kangaroo jumped
off of the ground. Find the zeros of the
function. Explain what the zeros mean in
this situation.
6. Short Response On Brian’s first vertical
jump, he has an initial vertical velocity of
40 inches per second. On his second vertical
jump, Brian has an initial vertical velocity
of 35 inches per second. For which jump
is Brian in the air for more time? Justify
your answer.
7. Extended Response During the period
1999–2003, the retail sales F (in millions of
dollars) for flower gardening and the retail
sales V (in millions of dollars) for vegetable
gardening can be modeled by
F 5 93.4t 3 2 642.5t 2 1 837.6t 1 3956.5
V 5 50.9t 3 2 198.6t 2 2 317.1t 1 2602.8
where t is the number of years since 1999.
a. Write an equation that models the
total retail sales S (in millions of
dollars) of flower gardening and
vegetable gardening as a function of
the number of years since 1999.
b. Find the total retail sales in these types
of gardening in 1999 and 2003.
c. What was the average rate of change
in total retail sales from 1999 to 2003?
Explain how you found this rate.
Copyright © Holt McDougal. All rights reserved.
LESSON 9.4
For use with pages 5722598
Name ———————————————————————
LESSON
9.4
Date ————————————
Challenge Practice
For use with pages 593–598
1. 1, 2, 3
2. 21, 0, 1
LESSON 9.4
In Exercises 1–5, find a polynomial that has these given roots. Write the
result as a polynomial with x as the variable, in both factored form and
standard form.
3. 0, 0, 1, 1
1
4. 0, } , 2
2
2
5. 21, 2} , 23
3
6. A rectangular pool whose long side is twice as long as its narrow side is being built.
There will be a paved border around all sides of the pool that is 5 feet wide around
three sides and 10 feet wide around one of the narrow ends to accommodate a diving
platform. The total area of the pool and the border is 1650 square feet. Write an
equation for the area of the pool and border where x represents the length of the
short side of the pool.
7. In Exercise 6, find the length of the sides of the pool.
8. Consider the equation x3 2 xy 2 5 0. What are the possible values of x and y that
make the equation hold true?
9. Consider the equation x4 2 x2y 2 5 0. What are the possible values of x and y that
Copyright © Holt McDougal. All rights reserved.
make the equation hold true?
10. Consider the equation (x2 1 y2)(x2 2 y2) 5 0. What are the possible values of x and
y that make the equation hold true?
Algebra 1
Chapter 9 Resource Book
147
Name ———————————————————————
LESSON
9.5
Date ————————————
Practice A
For use with pages 600–607
Match the trinomial with its correct factorization.
1. x 2 2 4x 2 12
2. x 2 2 x 2 12
3. x 2 1 4x 2 12
A. (x 1 6)(x 2 2)
B. (x 2 6)(x 1 2)
C. (x 1 3)(x 2 4)
4. x 2 1 6x 1 5
5. a2 1 10a 1 21
6. w 2 1 8w 1 15
7. p2 2 3p 2 10
8. c 2 1 10c 2 11
9. y 2 1 5y 2 14
LESSON 9.5
Factor the trinomial.
10. n2 2 4n 1 3
11. b2 2 5b 1 6
12. r 2 2 12r 1 35
13. z 2 1 7z 1 12
14. s2 2 3s 2 18
15. d 2 2 5d 2 24
16. x 2 1 5x 1 4 5 0
17. d 2 1 7d 1 10 5 0
18. p2 1 9p 1 14 5 0
19. w 2 2 12w 1 11 5 0
20. n2 2 n 2 6 5 0
21. a2 2 12a 1 35 5 0
22. y 2 2 4y 2 5 5 0
23. m2 1 2m 2 15 5 0
24. b2 1 6b 2 7 5 0
Solve the equation.
Match the equivalent equations.
25. s(s 2 6) 5 28
A. s2 2 2s 2 8 5 0
26. s(s 2 2) 5 8
B. s2 1 2s 2 8 5 0
27. s(s 1 2) 5 8
C. s2 2 6s 1 8 5 0
28. w(w 1 1) 5 12
29. x(x 2 3) 5 10
30. m(m 2 5) 5 6
31. b(b 1 4) 5 21
32. p(p 1 5) 5 36
33. r(r 2 3) 5 4
34. Boardwalk A boardwalk is being built along two sides
of a beach area. The beach area is rectangular with a
width of 80 feet and a length of 120 feet. The boardwalk
will have the same width on each side of the beach area.
If the combined area of the beach and the boardwalk is
16,500 square feet, then the area can be modeled by
(x 1 80)(x 1 120) 5 16,500. How wide should the
boardwalk be?
80 ft
x ft
x ft
120 ft
35. Note Board Design You are designing a note board
that is made of corkboard and dry erase board. The
area of the corkboard is 6 square feet.
a. Write an equation for the area of the corkboard.
b. Find the dimensions of the corkboard.
c. Find the area of the dry erase board.
148
Algebra 1
Chapter 9 Resource Book
Corkboard
(x 1 1) ft
Dry
erase x ft
board
1.5 ft
Copyright © Holt McDougal. All rights reserved.
Solve the equation.
Name ———————————————————————
LESSON
9.5
Date ————————————
Practice B
For use with pages 600–607
Factor the trinomial.
1. x 2 1 8x 1 7
2. b2 2 7b 1 10
3. w 2 2 12w 2 13
4. p2 1 10p 1 25
5. m2 2 10m 1 24
6. y 2 2 5y 2 24
7. a2 1 13a 1 36
8. n2 1 2n 2 48
9. z 2 2 14z 1 40
Solve the equation.
10. y 2 1 17y 1 72 5 0
11. a2 2 9a 2 36 5 0
12. w 2 2 13w 1 42 5 0
13. m2 2 5m 2 14 5 0
14. x 2 1 11x 1 24 5 0
15. n2 2 12n 1 27 5 0
16. d 2 1 5d 2 50 5 0
17. p2 1 16p 1 48 5 0
18. z 2 2 z 2 30 5 0
19. f(x) 5 x 2 2 5x 2 36
20. g(x) 5 x 2 1 8x 2 20
21. h(x) 5 x 2 2 11x 1 24
22. f(x) 5 x 2 1 11x 1 28
23. g(x) 5 x 2 1 11x 2 12
24. h(x) 5 x 2 1 3x 2 18
25. x(x 1 17) 5 260
26. p(p 2 4) 5 32
27. w(w 1 8) 5 215
28. n(n 1 6) 5 7
29. s2 2 3(s 1 2) 5 4
30. d 2 1 18(d 1 4) 5 29
LESSON 9.5
Find the zeros of the polynomial function.
Solve the equation.
Copyright © Holt McDougal. All rights reserved.
31. Patio Area A community center is building a patio area
along two sides of its pool. The pool is rectangular with
a width of 50 feet and a length of 100 feet. The patio area
will have the same width on each side of the pool.
a. Write a polynomial that represents the combined area
of the pool and the patio area.
b. The combined area of the pool and patio area should
be 8400 square feet. How wide should the patio area be?
x ft
50 ft
x ft
32. Area Rug You are creating your own area rug from a
square piece of remnant carpeting. You plan on cutting
4 inches from the length and 3 inches from the
width. The area of the resulting area rug is 1056 square
inches.
a. Write a polynomial that represents the area of your
area rug.
b. What is the perimeter of the original piece of remnant
carpeting?
100 ft
x in.
3 in.
Area rug
x in.
4 in.
Algebra 1
Chapter 9 Resource Book
149
Name ———————————————————————
LESSON
9.5
Date ————————————
Practice C
For use with pages 600–607
Factor the trinomial.
1. x 2 2 x 2 56
2. m2 1 14m 1 48
3. y 2 2 15y 1 54
4. p2 1 12p 1 20
5. w 2 2 14w 1 45
6. x 2 1 2x 2 24
8. z 2 1 22z 1 121 5 0
9. c 2 2 24c 1 144 5 0
Solve the equation.
7. n2 2 11n 2 60 5 0
10. x 2 1 5x 2 500 5 0
11. b2 1 b 2 132 5 0
12. m2 1 17m 1 72 5 0
13. r 2 2 4r 2 60 5 0
14. p2 2 6p 2 72 5 0
15. y 2 2 16y 1 64 5 0
16. f(x) 5 x 2 1 30x 1 225
17. h(x) 5 x 2 2 5x 2 150
18. g(x) 5 x 2 2 13x 1 30
19. g(x) 5 x 2 2 10x 2 600
20. f (x) 5 x 2 1 16x 1 28
21. f (x) 5 x 2 1 13x 1 40
22. x(x 2 4) 5 21
23. b(b 1 2) 5 24
24. n(n 2 11) 5 224
25. x 2 1 13(x 1 2) 5 210
26. x 2 2 10(x 1 2) 5 4
27. y(y 2 15) 5 256
1
28. x 2 1 2 } x 2 10 5 0
2
29. x(x 1 17) 5 242
30. c(c 2 11) 5 218
Solve the equation.
1
2
31. Zoo Exhibit A zoo is building a walkway along two sides of
x ft
400 ft
an exhibit. The exhibit is rectangular with a width of 400 feet
and a length of 200 feet. The walkway will have the same
width on each side of the exhibit.
a. Write a polynomial that represents the combined area of
the exhibit and the walkway.
b. The combined area of the exhibit and walkway should be
95,625 square feet. How wide should the walkway be?
c. If concrete costs $15 per square foot, how much will it
cost to pave the walkway?
200 ft
x ft
32. Fish Pond A rectangular fish pond is positioned in the center
of a rectangular grassy area, as shown. The area of the pond is
2000 square feet.
a. Use the dimensions given in the diagram to find the
dimensions of the pond.
b. The combined area of the pond and the surrounding grassy
area is 9900 square feet. Find the length and width of the
grassy area.
150
Algebra 1
Chapter 9 Resource Book
a ft
(a 1 15) ft
a ft
(a 1 5) ft
(a 1 25) ft
(a 1 5) ft
Copyright © Holt McDougal. All rights reserved.
LESSON 9.5
Find the zeros of the polynomial function.
Name ———————————————————————
LESSON
9.5
Date ————————————
Review for Mastery
For use with pages 600–607
GOAL
Factor trinomials of the form x 2 1 bx 1 c.
EXAMPLE 1
Factor when b and c are positive
Factor x 2 1 10x 1 24.
Solution
Find two positive factors of 24 whose sum is 10. Make an organized list.
Sum of factors
24, 1
24 1 1 5 25
✗
12, 2
12 1 2 5 14
✗
8, 3
8 1 3 5 11
✗
6, 4
6 1 4 5 10
LESSON 9.5
Factors of 24
correct sum
The factors 6 and 4 have a sum of 10, so they are the correct values of p and q.
x2 1 10x 1 24 5 (x 1 6)(x 1 4)
CHECK
(x 1 6)(x 1 4) 5 x2 1 4x 1 6x 1 24
5 x2 1 10x 1 24 ✓
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 2
Multiply binomials.
Simplify.
Factor when b is negative and c is positive
Factor w 2 2 10w 1 9.
Solution
Because b is negative and c is positive, p and q must be negative.
Factors of 9
Sum of factors
29, 21
29 1 (21) 5 210
23, 23
23 1 (23) 5 26
correct sum
✗
The factors 29 and 21 have a sum of 210, so they are the correct values of p and q.
w2 2 10w 1 9 5 (x 2 9)(x 2 1)
Exercises for Examples 1 and 2
Factor the trinomial.
1. x2 1 10x 1 16
2.
y 2 1 6y 1 5
3. z2 2 7z 1 12
Algebra 1
Chapter 9 Resource Book
151
Name ———————————————————————
LESSON
9.5
Review for Mastery
Date ————————————
continued
For use with pages 600–607
EXAMPLE 3
Factor when b is positive and c is negative
Factor k 2 1 6x 2 7.
Solution
Because c is negative, p and q must have different signs.
Factors of 7
Sum of factors
27, 1
27 1 1 5 26
7, 21
7 1 (21) 5 6
✗
correct sum
LESSON 9.5
The factors 7 and 21 have a sum of 6, so they are the correct values of p and q.
k 2 1 6k 2 7 5 (x 1 7)(x 2 1)
Exercises for Example 3
Factor the trinomial.
4. x2 2 10x 2 11
5. y 2 1 2y 2 63
6. z 2 2 5z 2 36
Solve a polynomial equation
Solve the equation h2 2 4h 5 21.
Solution
h2 2 4h 5 21
Write original equation.
h2 2 4h 2 21 5 0
Subtract 21 from each side.
(h 1 3)(h 2 7) 5 0
h1350
h 5 23
Factor left side.
or
h2750
or
h57
Zero-product property
Solve for h.
The roots of the equation are 23 and 7.
Exercise for Example 4
7. Solve the equation x2 1 30 5 11x.
152
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 4
Name ———————————————————————
LESSON
9.5
Date ————————————
Challenge Practice
For use with pages 600–607
In Exercises 1–5, use the given factor formula and the substitution method
to factor the expression.
x2 1 (a 1 b)x 1 ab 5 (x 1 a)(x 1 b)
Example: y 1 y1/2 2 6
Solution: Let x 5 y1/2. Then x2 5 y and the expression y 1 y1/2 2 6 becomes x2 1 x 2 6.
Now factor this expression using the given factor formula.
x 2 1 x 2 6 5 (x 1 3)(x 2 2)
Finally, replace x with y1/2.
(x 1 3)(x 2 2) 5 ( y1/2 1 3)( y1/2 2 2)
LESSON 9.5
1. y 2/3 1 6y1/3 1 8
2. y4 2 y 2 2 12
8
1
3. }2 2 } 2 9
y
y
}
}
4.
Ïy 2
5.
Ïy 1 12Ï4 y 1 11
5
5
1 16Ï
y 1 48
}
}
In Exercises 6–10, use substitution to factor, then solve for x.
6. x4 2 3x2 2 4 5 0
Copyright © Holt McDougal. All rights reserved.
7. x4 2 13x2 1 36 5 0
1
1
8. }2 2 } 2 12 5 0
x
x
}
9. x 2 Ï x 2 6 5 0
10. x4 2 16x2 1 48 5 0
Algebra 1
Chapter 9 Resource Book
153
Name ———————————————————————
LESSON
9.6
Date ————————————
Practice A
For use with pages 610–617
Match the trinomial with its correct factorization.
1. 4x 2 2 2x 2 2
A. (4x 1 1)(x 2 2)
2. 4x 2 2 7x 2 2
B. (2x 1 1)(2x 2 2)
3. 4x 2 1 7x 2 2
C. (4x 2 1)(x 1 2)
Factor the trinomial.
4. 2x 2 2 2x 1 15
5. 2m2 1 3m 2 2
6. 2p 2 1 5p 1 14
7. 2w 2 1 7w 1 3
8. 3y 2 1 5y 1 2
9. 2b2 1 b 2 1
10. 3n2 2 3
11. 5a 2 1 13a 2 6
12. 2z 2 1 9z 2 5
13. 7d 2 2 15d 1 2
14. 2r 2 2 12r 1 10
15. 6s2 2 13s 1 2
16. 2x 2 1 7x 2 15 5 0
17. 3n2 1 13n 1 4 5 0
18. 4b2 1 2b 2 2 5 0
19. 2m2 1 5m 2 3 5 0
20. 3p2 1 11p 2 4 5 0
21. 3y 2 1 11y 1 10 5 0
22. 4r 2 1 8r 1 3 5 0
23. 9w 2 1 3w 2 2 5 0
24. 5a 2 2 8a 2 4 5 0
25. 3c 2 1 19c 2 14 5 0
26. 8z 2 1 6z 1 1 5 0
27. 12d 2 1 14d 2 6 5 0
Solve the equation.
28. f(x) 5 2x 2 2 4x 1 5
29. g(x) 5 3x 2 2 13x 2 10
30. h(x) 5 22x 2 1 9x 1 5
31. g(x) 5 2x 2 1 5x 2 6
32. f (x) 5 4x 2 2 9x 1 2
33. g(x) 5 22x 2 2 9x 1 18
34. h(x) 5 2x 2 1 7x 2 4
35. h(x) 5 6x 2 1 3x 2 9
36. f (x) 5 24x 2 2 9x 2 2
37. Ball Toss A ball is tossed into the air from a height of 8 feet with an initial
velocity of 8 feet per second. Find the time t (in seconds) it takes for the
object to reach the ground by solving the equation 216t 2 1 8t 1 8 5 0.
38. Wallpaper You trimmed a large strip of wallpaper from a scrap
to fit into the corner of a wall you are wallpapering. You trimmed
15 inches from the length and 6 inches from the width. The area
of the resulting strip of wallpaper is 684 square inches.
a. If the length of the original strip of wallpaper is four times the
original width, write a polynomial that represents the area of
the trimmed strip of wallpaper.
b. What are the dimensions of the original scrap of wallpaper?
(4x 2 15) in.
4x in.
(x 2 6) in.
x in.
154
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 9.6
Find the zeros of the polynomial function.
Name ———————————————————————
LESSON
9.6
Date ————————————
Practice B
For use with pages 610–617
Factor the trinomial.
1. 2x 2 2 3x 1 28
2. 2p2 1 8p 2 12
3. 2m2 2 13m 2 40
4. 2y 2 1 15y 1 7
5. 3a 2 2 13a 1 4
6. 5d 2 2 18d 2 8
7. 6c 2 1 7c 1 2
8. 10n2 2 26n 1 12
9. 12w 2 1 8w 2 15
10. 22b2 2 5b 1 12
11. 23r 2 2 17r 2 10
12. 24s2 1 6s 1 4
13. 2x 2 1 x 1 20 5 0
14. 2m2 2 10m 2 16 5 0
15. 2p2 1 13p 2 42 5 0
16. 2c 2 2 11c 1 5 5 0
17. 2y 2 1 y 2 10 5 0
18. 16r 2 1 18r 1 5 5 0
19. 3w 2 1 19w 1 6 5 0
20. 12n2 2 11n 1 2 5 0
21. 15a2 2 2a 2 8 5 0
22. 22x 2 2 9x 2 4 5 0
23. 23s2 2 s 1 10 5 0
24. 8d 2 2 6d 2 5 5 0
Solve the equation.
Find the zeros of the polynomial function.
25. f(x) 5 2x 2 1 6x 1 27
26. f (x) 5 6x2 1 45x 2 24
27. f (x) 5 23x 2 2 14x 1 24
28. f(x) 5 22x 2 1 2x 1 4
29. f (x) 5 3x 2 2 17x 1 20
30. f (x) 5 8x 2 1 53x 2 21
31. f(x) 5 4x 2 1 29x 1 30
32. f (x) 5 22x 2 2 17x 1 30
33. f (x) 5 10x 2 1 5x 2 5
business can be modeled by
R 5 22t 2 1 87t 1 90
where t represents the number of weeks since the first week you started selling
T-shirts. How much did you make your first week?
LESSON 9.6
Copyright © Holt McDougal. All rights reserved.
34. Summer Business Your weekly revenue R (in dollars) from your tie-dye T-shirt
35. Cliff Diving A cliff diver jumps from a ledge 96 feet above the ocean with an initial
upward velocity of 16 feet per second. How long will it take until the diver enters
the water?
2 in.
36. Wall Mirror You plan on making a wall hanging that contains two
small mirrors as shown.
a. Write a polynomial that represents the area of the wall hanging.
b. The area of the wall hanging will be 480 square inches. Find the
length and width of the mirrors you will use.
2 in.
2x in.
x in.
4 in.
2x in.
x in.
Algebra 1
Chapter 9 Resource Book
155
Name ———————————————————————
LESSON
9.6
Date ————————————
Practice C
For use with pages 610–617
Factor the trinomial.
1. 2x 2 2 11x 1 180
2. 22m2 1 19m 2 24
3. 23p2 1 26p 1 40
4. 8r 2 1 26r 1 15
5. 14b2 1 38b 2 12
6. 10y 2 2 36y 1 18
8. 28n2 2 16n 2 6 5 0
9. 215s2 1 4s 1 4 5 0
Solve the equation.
7. 232x 2 2 28x 1 15 5 0
10. 26p2 2 17p 2 5 5 0
11. 63m2 2 31m 2 10 5 0
12. 40r 2 2 42r 1 9 5 0
13. 16a2 2 2a 2 3 5 0
14. 215d 2 2 2d 1 8 5 0
15. 26y 2 1 32y 2 10 5 0
Find the zeros of the polynomial function.
16. f(x) 5 216x 2 1 50x 2 25
17. h(x) 5 220x 2 1 44x 2 21 18. h(x) 5 20x 2 1 18x 2 44
19. g(x) 5 236x 2 2 30x 2 6
20. f (x) 5 12x 2 1 8x 2 15
21. g(x) 5 21x 2 1 14x 2 7
Multiply each side of the equation by an appropriate power of 10 to obtain
integer coefficients. Then solve the equation.
22. 0.2x 2 2 0.3x 2 3.5 5 0
23. r 2 1 0.6r 2 0.4 5 0
24. 0.8m2 1 m 2 0.3 5 0
25. 20.5x2 1 1.2x 5 0.4
26. 1.2(p2 1 1) 5 2.5p
27. 20.36n2 1 0.6n 2 0.25 5 0
28. Baseball A baseball player releases a baseball at a height of 7 feet with an initial
29. Rocket Launch A miniature rocket is launched off a roof 20 feet above the ground
with an initial velocity of 22 feet per second. How much time will elapse before the
rocket reaches the ground?
30. Frog Jump A frog jumps from the ground into the air with an initial vertical
velocity of 8 feet per second.
a. Write an equation that gives the frog’s height (in feet) as a function of the time
(in seconds) since it left the ground.
b. After how many seconds is the frog 12 inches above the ground?
c. Does the frog go any higher than 12 inches? Explain your reasoning using your
answer from part (b).
d. Suppose the frog now jumps from 4 feet above the ground with the same initial
vertical velocity. Write an equation that gives the frog’s height (in feet) as a
function of the time (in seconds) since it left the ground.
e. Should the frog reach the ground in the same time in both jumps? Explain why
or why not.
156
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 9.6
velocity of 54 feet per second. How long will it take the ball to reach the ground?
Name ———————————————————————
LESSON
9.6
Date ————————————
Review for Mastery
For use with pages 610–617
GOAL
EXAMPLE 1
Factor trinomials of the form ax 2 1 bx 1 c.
Factor when b is negative and c is positive
Factor 5n 2 2 12n 1 7.
Solution
Because b is negative and c is positive, both factors of c must be negative. Make a
table to organize your work.
You must consider the order of the factors of 7, because the x-terms of the possible
factorization are different.
Factors
of 5
Factors
of 7
Possible
factorization
Middle term
when multiplied
1, 5
21, 27
(n 2 1)(5n 2 7)
25n 2 7n 5 212n
1, 5
27, 21
(n 2 7)(5n 2 1)
2n 2 35n 5 236n
correct
✗
5n2 2 12n 1 7 5 (n 2 1)(5n 2 7)
EXAMPLE 2
Factor when b is negative and c is negative
Factor 3m2 2 5m 2 22.
LESSON 9.6
Copyright © Holt McDougal. All rights reserved.
Solution
Because b is negative and c is negative, p and q must have different signs.
Factors
of 3
Factors
of 22
Possible
factorization
Middle term
when multiplied
1, 3
1, 222
(m 1 1)(3m 2 22)
3m 2 22m 5 219m
✗
1, 3
21, 22
(m 2 1)(3m 1 22)
22m 2 3m 5 19m
✗
1, 3
2, 211
(m 1 2)(3m 2 11)
211m 1 6m 5 25m
1, 3
211, 2
(m 2 11)(3m 1 2)
2m 2 33m 5 231m
correct
✗
3m2 2 5m 2 22 5 (m 1 2)(3m 2 11)
Exercises for Examples 1 and 2
Factor the trinomial.
1. 7a2 2 50a 1 7
2.
4b2 2 8b 2 5
3. 6c2 1 5c 2 14
Algebra 1
Chapter 9 Resource Book
157
Name ———————————————————————
LESSON
9.6
Review for Mastery
Date ————————————
continued
For use with pages 610–617
EXAMPLE 3
Factor when a is negative
Factor 22x 2 1 9x 2 9.
Solution
STEP 1
Factor 21 from each term of the trinomial.
22x2 1 9x 2 9 5 2(2x2 2 9x 1 9)
STEP 2
Factor the trinomial 2x2 2 9x 1 9. Because b is negative and c is positive,
both factors of c must be negative. Use a table to organize information about
the factors of a and c.
Factors
of 2
Factors
of 9
Possible
factorization
Middle term
when multiplied
1, 2
21, 29
(x 2 1)(2x 2 9)
29x 2 2x 5 211x
✗
1, 2
29, 21
(x 2 9)(2x 2 1)
2x 2 18x 5 219x
✗
1, 2
23, 23
(x 2 3)(2x 2 3)
23x 2 6x 5 29x
correct
22x2 1 9x 2 9 5 2(x 2 3)(2x 2 3)
158
Factor the trinomial.
4. 23r 2 2 7r 2 4
5. 23s2 1 8s 1 16
6. 28t2 1 6t 2 1
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 9.6
Exercises for Example 3
Name ———————————————————————
LESSON
9.6
Date ————————————
Problem Solving Workshop:
Using Alternative Methods
For use with pages 6102617
Another Way to Solve Example 4 on page 613
Multiple Representations In Example 4 on page 613, you saw how to solve a
problem about a discus by factoring a quadratic equation. You can also solve the problem by
using a graph.
PROBLEM
Discus An athlete throws a discus from an initial height of 6 feet and with an initial
vertical velocity of 46 feet per second. Write an equation that gives the height
(in feet) of the discus as a function of the time (in seconds) since it left the athlete’s
hand. After how many seconds does the discus hit the ground?
METHOD
Using a Graph You can solve the problem by using a graph.
STEP 1
STEP 2
Copyright © Holt McDougal. All rights reserved.
PRACTICE
h 5 216t 2 1 vt 1 s
Vertical motion model
h 5 216t 2 1 46t 1 6
Substitute 46 for v and 6 for s.
Graph the equation for the height of the discus
using a graphing calculator. Graph
y1 5 216x 2 1 46x 1 6. Because you are
looking for when the discus hits the ground,
you need to find the time when the height is 0.
Zero
X=3
Y=0
Find the zeros of the graph by using the zero feature on your calculator. You only
need to consider positive values of x because a negative solution does not make
sense in this situation. There is a zero at (3, 0). The discus hits the ground after
3 seconds.
1. Cliff Diving A cliff diver jumps from
a ledge 88 feet above the ocean with an
initial upward velocity of 12 feet per
second. How long will it take until the
diver enters the water?
2. Error Analysis Describe and correct
the error made in Exercise 1.
216t2 2 12t 1 88 5 0
24(4t2 1 3t 2 22) 5 0
24(4t 1 11)(t 2 2) 5 0
The cliff diver enters the water after
2 seconds.
LESSON 9.6
STEP 3
Use the vertical motion model to write an equation for the height h (in feet) of
the discus. In this case, v 5 46 and s 5 6.
3. Tennis A tennis ball is hit when it is
6 feet off the ground with an initial
upward velocity of 20 feet per second.
How long does it take for the tennis ball
to hit the ground?
4. Football You throw a football from a
height of 6 feet into the air with an
initial vertical velocity of 12 feet per
second. The football is caught at a
height of 2 feet. After how many
seconds is the football caught?
5. What If? Suppose in Exercise 4 the
football is thrown with an initial vertical
velocity of 30 feet per second. After how
many seconds is the football caught?
Algebra 1
Chapter 9 Resource Book
159
Name ———————————————————————
Date ————————————
Challenge Practice
LESSON
9.6
For use with pages 610–617
In Exercises 1–5, use the substitution method to factor the expression.
Example: 3y 1 11y1/2 2 4
Solution: Let x 5 y1/2. Then x2 5 y and the expression 3y 1 11y1/2 2 4 becomes
3x2 1 11x 2 4. Now factor this expression.
3x2 1 11x 2 4 5 (3x 2 1)(x 1 4)
Finally, replace x with y1/2.
(3x 2 1)(x 1 4) 5 (3y1/2 2 1)( y1/2 1 4)
1. 4y2/3 1 12y1/3 1 5
2. 8y4 2 10y2 2 3
9 12
3. }2 2 } 2 5
y
y
3
}
3}
4. 7Ï y2 1 36Ï y 1 5
}
}
4
5. 28Ï
y 1 8Ï y 1 6
In Exercises 6 –10, use substitution to factor, then solve for x.
6. 6x6 1 x3 2 2 5 0
7. 9x4 2 12x2 2 5 5 0
160
}
9. 3x 2 Ï x 2 14 5 0
10. 5x4 1 21x2 2 20 5 0
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 9.6
5
28
8. }2 1 } 1 15 5 0
x
x
Name ———————————————————————
LESSON
9.7
Date ————————————
Practice A
For use with pages 6182623
Match the trinomial with its correct factorization.
1. x 2 2 25
2. x 2 1 10x 1 25
3. x 2 2 10x 1 25
A. (x 1 5)2
B. (x 2 5)(x 1 5)
C. (x 2 5)2
Factor the difference of two squares.
4. x 2 2 1
5. b2 2 81
6. m2 2 100
7. p2 2 225
8. 4y 2 2 1
9. 16n2 2 25
10. 9w 2 2 100
11. 64z 2 2 36
12. 49d 2 2 25
13. 4r 2 2 121
14. 9s2 2 144
15. c 2 2 625
Factor the perfect square trinomial.
16. x 2 1 6x 1 9
17. b2 1 10b 1 25
18. w 2 2 12w 1 36
19. m2 2 8m 1 16
20. r 2 2 20r 1 100
21. z 2 1 16z 1 64
22. s2 1 22s 1 121
23. x 2 2 16x 1 64
24. 4c 2 1 4c 1 1
25. 16d 2 1 8d 1 1
26. 9y 2 2 6y 1 1
27. 9p2 2 12p 1 4
28. x 2 2 9 5 0
29. p2 1 14p 1 49 5 0
30. d 2 2 10d 1 25 5 0
31. 25m2 2 1 5 0
32. r 2 2 2r 1 1 5 0
33. n2 1 20n 1 100 5 0
34. 4y 2 2 9 5 0
35. 36x 2 2 64 5 0
36. w 2 1 4w 1 4 5 0
37. Washers Washers are available in many different sizes.
a. Write and factor an expression for the area of one side
of the washer. Leave your answer in terms of π.
b. Find the area of one side of the washer when
x 5 8 centimeters and y 5 3 centimeters.
y
x
38. Cherry Tree A cherry falls from a tree branch that is 9 feet above the ground.
LESSON 9.7
Copyright © Holt McDougal. All rights reserved.
Solve the equation.
a. How far above the ground is the cherry after 0.2 second?
b. After how many seconds does the cherry reach the ground?
39. Wind Chime A wind chime falls from a roof that is 10 feet above the ground.
a. How far above the ground is the wind chime after 0.5 second?
b. After how many seconds does the wind chime reach the ground?
Algebra 1
Chapter 9 Resource Book
161
Name ———————————————————————
LESSON
9.7
Date ————————————
Practice B
For use with pages 6182623
Factor the polynomial.
1. x 2 2 36
2. 25p2 2 144
3. 4b2 2 100
4. 36m2 2 81
5. 22x 2 1 32
6. 24r 2 1 100s2
7. y 2 1 24y 1 144
8. 9c 2 1 24c 1 16
9. 25w 2 2 20w 1 4
10. 16n2 2 56n 1 49
11. 218a2 2 12a 2 2
12. 20z 2 2 140z 1 245
13. x 2 1 14x 1 49 5 0
14. 8w 2 5 50
15. 64p2 2 16p 1 1 5 0
16. 8a 2 2 72 5 0
17. 3m2 1 30m 1 75 5 0
18. 24y 2 1 32y 2 64 5 0
19. 25x 2 1 125 5 0
20. 27r 2 1 140r 2 700 5 0
21. 24w 2 2 24w 1 6 5 0
22. 18n2 1 60n 1 50 5 0
9
25
23. } x 2 1 15x 1 } 5 0
2
2
9
24. 4x 2 5 }
16
Solve the equation.
Find the value of x in the geometric shape.
25. Area 5 144π cm2
26. Area 5 225 in.2
(x 1 4) cm
27. Measuring Tape A measuring tape drops from a roof that is 16 feet above the
ground. After how many seconds does the measuring tape land on the ground?
28. Playground A curved ladder that children can
climb on can be modeled by the equation
1
y 5 2} x 2 1 x
20
where x and y are measured in feet.
and 20 feet from the left end.
b. For what additional values of x does the equation make sense? Explain.
c. Plot the ordered pairs in the table from part (a) as
y
points in the coordinate plane. Connect the points
5
4
with a smooth curve.
3
d. At approximately what distance from the left end
2
does the ladder reach a height of 5 feet? Check
1
your answer algebraically.
Height (feet)
LESSON 9.7
a. Make a table of values that shows the height of the ladder for x 5 0, 5, 10, 15,
0
162
Algebra 1
Chapter 9 Resource Book
0
5
10
15
20 x
Distance from left end (feet)
Copyright © Holt McDougal. All rights reserved.
(4x 1 3) in.
Name ———————————————————————
LESSON
9.7
Date ————————————
Practice C
For use with pages 6182623
Factor the polynomial.
1. 25x 2 2 81
2. 225p2 2 100
3. 121w 2 2 625
4. 36m2 2 64
1
9
5. } r 2 2 }
16
16
6. 81x 2 2 49y 2
7. 23y 2 2 48y 2 192
8. 4n2 2 40n 1 100
9. 12z 2 1 12z 1 3
11. 218s2 2 48st 2 32t 2
1
12. 5z 2 1 2z 1 }
5
13. 25m2 2 64 5 0
14. 2p2 1 36p 1 162 5 0
15. 216r 2 1 196 5 0
16. 3w 2 2 60w 1 300 5 0
17. 36x 2 2 132x 1 121 5 0
18. 225a 2 2 120a 1 16 5 0
19. 275y 2 2 90y 2 27 5 0
20. 196n2 2 224n 1 64 5 0
21. 160z 2 5 640
22. 0.9r 2 2 4.8r 1 6.4 5 0
1
25
23. } b2 1 5b 1 } 5 0
2
2
24. 296d 2 1 144d 2 54 5 0
10. 24a 2 2 120ab 1 150b2
Solve the equation.
Determine the value(s) of k that make the expression a perfect
square trinomial.
25. 81x 2 1 kx 1 25
26. 100x 2 1 kx 1 49
27. 25x 2 2 60x 1 k
28. kx 2 1 72x 1 81
29. 4x 2 2 12x 1 k
30. 49x 2 1 kxy 1 4y 2
31. Squirrel A squirrel jumps straight up with an initial vertical velocity of 16 feet
32. Foot Bridge A foot bridge that spans a small creek can be
modeled by the equation
3
3
y 5 2} x 2 1 } x
800
10
Height (feet)
where x and y are measured in feet.
a. Make a table of values that shows the height of the bridge for x 5 0, 20, 40, 60,
and 80 feet from the left end.
b. For what additional values of x does the equation make sense? Explain.
c. Plot the ordered pairs in the table from part (a)
y
as points in the coordinate plane. Connect the
6
5
points with a smooth curve.
4
d. At approximately what distance from the left end
3
does the bridge reach a height of 6 feet? Check
2
your answer algebraically.
1
0
LESSON 9.7
Copyright © Holt McDougal. All rights reserved.
per second. How many times does the squirrel reach a height of 4 feet? Explain
your answer.
0 10 20 30 40 50 60 70 80 x
Distance from left end (feet)
Algebra 1
Chapter 9 Resource Book
163
Name ———————————————————————
LESSON
9.7
Date ————————————
Review for Mastery
For use with pages 6182623
GOAL
Factor special products.
Vocabulary
The pattern for finding the square of a binomial gives you the pattern
for factoring trinomials of the form a2 1 2ab 1 b2 and a 2 2 2ab 1 b2.
These are called perfect square trinomials.
EXAMPLE 1
Factor the difference of squares
Factor the polynomial.
5 (r 2 9)(r 1 9)
b. 9s2 2 4t 2 5 (3s)2 2 (2t)2
5 (3s 2 2t)(3s 1 2t)
c. 80 2 125q2 5 5(16 2 25q2)
Difference of two squares
pattern
Factor out common factor.
5 5(2 2 5q)(2 1 5q)
Difference of two squares
pattern
1. m2 2 121
2. 9n2 2 64
3. 3y2 2 147z 2
LESSON 9.7
Write as a2 2 b2.
Write 16 2 25q2 as a2 2 b2.
Factor the polynomial.
Algebra 1
Chapter 9 Resource Book
Difference of two squares
pattern
5 5[42 2 (5q)2]
Exercises for Example 1
164
Write as a2 2 b2.
Copyright © Holt McDougal. All rights reserved.
a. r 2 2 81 5 r 2 2 92
Name ———————————————————————
LESSON
9.7
Review for Mastery
Date ————————————
continued
For use with pages 618–623
EXAMPLE 2
Factor perfect square trinomials
Factor the polynomial.
a. x2 1 14x 1 49 5 x2 1 2(x)(7) 1 72
Write as a2 1 2ab 1 b2.
5 (x 1 7)2
Perfect square trinomial
pattern
b. 144y2 2 120y 1 25 5 (12y)2 2 2(12y)(5) 1 52
5 (12y 2 5)2
Write as a2 2 2ab 1 b2.
Perfect square trinomial
pattern
c. 150z2 2 60z 1 6 5 6(25z2 2 10z 1 1)
Factor out common factor.
5 6[(5z)2 2 2(5z)(1) 1 12]
Write 25z 2 2 10z 1 1
as a2 2 2ab 1 b2.
5 6(5z 2 1)2
Perfect square trinomial
pattern
Exercises for Example 2
Factor the polynomial.
1
1
4. m2 2 } m 1 }
2
16
16r 2 1 40rs 1 25s2
6. 36x2 2 36x 1 9
Solve a polynomial equation
Solve the equation q 2 2 100 5 0.
Solution
q2 2 100 5 0
Write original equation.
q2 2 102 5 0
Write left side as a2 2 b2.
(q 1 10)(q 2 10) 5 0
q 1 10 5 0
q 5 210
or
Difference of two squares pattern
q 2 10 5 0
or
q 5 10
Zero-product property
Solve for q.
The roots of the equation are 210 and 10.
LESSON 9.7
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 3
5.
Exercises for Example 3
Solve the equation.
7. r 2 2 10r 1 25 5 0
8. 16m2 2 81 5 0
Algebra 1
Chapter 9 Resource Book
165
Name ———————————————————————
LESSON
9.7
Date ————————————
Challenge Practice
For use with pages 618–623
In Exercises 1–3, factor the expression.
1. x2 2 6xy 1 9y2
2. 4x2 2 20xy 1 25y2
3. 25x2y2 1 40xy 116
In Exercises 4 and 5, use the substitution method to factor the expression.
Example: 16(y 1 3)2 2 40(y 1 3) 1 25
Solution: Let x 5 y 1 3. Then the expression 16(y 1 3)2 2 40(y 1 3) 1 25 becomes
16x2 2 40x 1 25. Now factor this expression.
16x 2 2 40x 1 25 5 (4x 2 5)2
Finally, replace x with (y 1 3).
(4x 2 5)2 5 [4(y 1 3) 2 5]2 5 (4y 1 7)2
4. 4(x 2 7)2 2 24(x 2 7) 1 36
5. 25(x 1 3)2 2 20(x 1 3) 1 4
In Exercises 6–10, use substitution to factor, then solve for x.
6. (x 2 5)4 2 10(x 2 5)2 1 25 5 0
7. 4(2x 2 7)6 2 28(2x 2 7)3 1 49 5 0
16
56
9. }2 1 } 1 49 5 0
x
x
LESSON 9.7
9
12
10. }2 1 } 1 4 5 0
x11
(x 1 1)
166
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
8. 25(x 1 2)2 1 30(x 1 2) 1 9 5 0
Name ———————————————————————
LESSON
9.8
Date ————————————
Practice A
For use with pages 6242631
1. 2x(x 1 5) 2 (x 1 5)
2. 2x(x 1 5) 1 (x 1 5)
3. 2x(x 2 5) 2 (x 2 5)
A. (2x 1 1)(x 1 5)
B. (2x 2 1)(x 2 5)
C. (2x 2 1)(x 1 5)
4. x(x 1 4) 1 (x 1 4)
5. b(b 1 3) 2 (b 1 3)
6. 2m(m 1 1) 1 (m 1 1)
7. 5r(r 1 2) 2 (r 1 2)
8. w(w 1 6) 1 3(w 1 6)
9. y(y 1 4) 2 6(y 1 4)
Factor the expression.
10. n(n 2 3) 2 7(n 2 3)
11. 3z(z 2 4) 1 8(z 2 4)
LESSON 9.8
Match the trinomial with its correct factorization.
12. 2p(p 1 5) 2 3(p 1 5)
Factor the polynomial by grouping.
13. x 2 1 x 1 3x 1 3
14. x 2 2 x 1 2x 2 2
15. x 2 1 8x 2 x 2 8
16. x 3 2 5x 2 1 2x 2 10
17. x 3 2 4x 2 2 6x 1 24
18. x 3 1 3x 2 1 5x 1 15
19. x 3 2 x 2 1 7x 2 7
20. x 3 1 3x 2 2 3x 2 9
21. x 3 1 3x 2 2 x 2 3
Determine whether the polynomial has been completely factored.
22. x 4 1 x 3
23. x 2 1 1
24. 2x 2 1 4
Factor the polynomial completely.
25. x 5 2 x 3
26. 4a 4 2 25a 2
27. 5y 6 2 125y 4
28. x 3 1 x 2 2 25x 2 25 5 0
29. x 3 1 x 2 2 16x 2 16 5 0
30. x 3 2 x 2 2 4x 1 4 5 0
31. x 3 2 x 2 2 9x 1 9 5 0
32. z 3 2 4z 5 0
33. c 4 2 64c 2 5 0
Copyright © Holt McDougal. All rights reserved.
Solve the equation.
34. Metal Plate You have a metal plate that you have drilled a
hole into. The entire area enclosed by the metal plate is given
by 5x 2 1 12x 1 10 and the area of the hole is given by x 2 1 2.
Write an expression for the area in factored form of the plate
that is left after the hole is drilled.
35. Storage Container A plastic storage container in the shape of a cylinder has a
height of 8 inches and a volume of 72π cubic inches.
a. Write an equation for the volume of the storage container.
b. What is the radius of the storage container?
36. Tennis Ball For a science experiment, you toss a tennis ball from a height of 32 feet
with an initial upward velocity of 16 feet per second. How long will it take the tennis
ball to reach the ground?
Algebra 1
Chapter 9 Resource Book
167
Name ———————————————————————
LESSON
LESSON 9.8
9.8
Date ————————————
Practice B
For use with pages 6242631
Factor the expression.
1. 4x(x 1 5) 2 3(x 1 5)
2. 12(a 2 3) 2 2a(a 2 3)
3. w 2(w 1 8) 2 5(w 1 8)
4. 2b2(b 1 6) 1 3(b 1 6)
5. y(15 1 x) 2 (x 1 15)
6. 3x(4 1 y) 2 6(4 1 y)
Factor the polynomial by grouping.
7. x 3 1 x 2 1 5x 1 5
8. y 3 2 14y 2 1 y 2 14
9. m3 2 6m2 1 2m 2 12
10. p3 1 9p2 1 4p 1 36
11. t 3 1 12t 2 2 2t 2 24
12. 3n3 2 3n2 1 n 2 1
Factor the polynomial completely.
13. 7x 3 1 28x 2
14. 4m3 2 16m
15. 216p3 2 2p
16. 48r 3 2 30r 2
17. 15y 2 60y 2
18. 18xy 2 24x 2
19. 5m2 1 20m 1 40
20. 6x 2 1 6x 2 120
21. 4z 3 2 4z 2 2 8z
22. 9x 3 1 36x 2 1 36
23. x 3 1 x 2 1 5x 1 5
24. d 3 1 4d 2 1 5d 1 20
25. 3x 2 1 18x 1 24 5 0
26. 10x 2 5 250
27. 4m2 2 28m 1 49 5 0
28. 12x 2 1 18x 1 6 5 0
29. 18x 2 2 48x 1 32 5 0
30. 218x 2 2 60x 2 50 5 0
31. Countertop A countertop will have a hole drilled in it to hold
a cylindrical container that will function as a utensil holder.
The area of the entire countertop is given by 5x 2 1 12x 1 7.
The area of the hole is given by x 2 1 2x 1 1. Write an
expression for the area in factored form of the countertop
that is left after the hole is drilled.
32. Film Canister A film canister in the shape of a cylinder has a height of
8 centimeters and a volume of 32π cubic centimeters.
a. Write an equation for the volume of the film canister.
b. What is the radius of the film canister?
33. Badminton You hit a badminton birdie upward with a racket from a height
of 4 feet with an initial velocity of 12 feet per second.
a. Write an equation that models this situation.
b. How high is the birdie at 0.1 second?
c. How high is the birdie at 0.25 second?
d. How long will it take the birdie to reach the ground?
168
Algebra 1
Chapter 9 Resource Book
Copyright © Holt McDougal. All rights reserved.
Solve the equation.
Name ———————————————————————
LESSON
9.8
Date ————————————
Practice C
For use with pages 6242631
1. 13a 2 26a 2
2. 30xy 2 45x 2
3. 22m2 2 16m 2 14
4. 14p2 2 35p 1 21
5. r 3 1 10r 2 1 25r
6. 5b4 1 40b3 1 80b2
7. 4n5 1 4n4 2 120n3
8. 7c 3 2 28c 2 1 28c
9. 210t 2 2 5t 1 75
10. x 2 1 9x 2 xy 2 9y
11. x 3 1 5x 2 2 8x 2 40
12. 9x 2 2 64y 2
13. 3x5y 2 243x 3y
14. 8r 3s4 2 72rs4
15. 25x 3y 2 100x 2y
16. 5x 3 1 20x 2 1 15x 5 0
17. 219x 2 1 76 5 0
18. 218p3 2 21p2 1 15p 5 0
19. 48p2 2 675 5 0
20. 14x 3 2 68x 2 2 10x 5 0
21. 23n4 2 36n3 2 108n2 5 0
22. 20t 4 1 28t 3 5 24t 2
23. 64t 5 12t 2 1 45
24. 900x 2 5 625
25. 16m4 2 81m2 5 0
26. 16x 1 280 5 8x 2
27. 2r 2 1 392 5 56r
28. 75a3 1 90a2 1 27a 5 0
29. 2p2 5 12p 1 54
30. 81x 3 5 100x
LESSON 9.8
Factor the polynomial completely.
Solve the equation.
31. Use factoring by grouping to show that a trinomial of the form a2 2 2ab 1 b2 can be
factored as (a 2 b)2. Justify your steps.
Copyright © Holt McDougal. All rights reserved.
32. Work Bench You are drilling holes into your work bench
that will hold caddies for some of your gardening equipment.
The area of the entire work bench before the holes are drilled
is given by 24x 2 1 5x. The area of one hole is given by
3x 2 1 x 1 3. Write an expression for the area in factored form
of the work bench that is left after the holes are drilled.
33. Poster Tube A poster tube in the shape of a cylinder has a height of 2 feet and
1
a volume of }2 π cubic feet.
a. Write an equation for the volume of the poster tube.
b. What is the radius of the poster tube?
16
34. Moon On the moon, the vertical motion model is given by h 5 2} t 2 1 vt 1 s
6
where h is the height (in feet), v is the initial velocity (in feet per second), t is
the time (in seconds), and s is the initial height (in feet). On the moon, an astronaut
2
tosses a baseball from a height of 64 feet with an initial upward velocity of 23 }3 feet
per second. How long does it take the ball to reach the ground?
Algebra 1
Chapter 9 Resource Book
169
Name ———————————————————————
LESSON
LESSON 9.8
9.8
Date ————————————
Review for Mastery
For use with pages 624– 631
GOAL
Factor polynomials completely.
Vocabulary
Factoring a common monomial from pairs of terms, then looking for a
common binomial factor is called factor by grouping.
A polynomial of two or more terms is prime if it cannot be written
as the product of polynomials of lesser degree using only integer
coefficients and constants, and if the only common factors of its
terms are 1 and –1.
A polynomial is factored completely if it is written as a monomial or
as the product of a monomial (possibly 1 or 21) and one or more
prime polynomials.
EXAMPLE 1
Factor out a common binomial
Factor the expression.
a. 5x2(x 2 2) 2 3(x 2 2)
b. 7y(5 2 y) 1 3( y 2 5)
Solution
a. 5x2(x 2 2) 2 3(x 2 2) 5 (x 2 2)(5x2 2 3)
5 2 y to obtain a common binomial factor.
7y(5 2 y) 1 3( y 2 5) 5 27y( y 2 5) 1 3( y 2 5)
5 ( y 2 5)(27y 1 3)
EXAMPLE 2
Factor 21 from
(5 2 y).
Distributive property
Factor by grouping
Factor the polynomial.
a. m3 1 7m2 2 2m 2 14
b. n3 1 30 1 6n2 1 5n
Solution
a. m3 1 7m2 2 2m 2 14 5 (m3 1 7m2) 1 (22m 2 14)
5 (m 1 7) 2 2(m 1 7)
5 (m 1 7)(m2 2 2)
m2
b. n3 1 30 1 6n2 1 5n 5 n3 1 6n2 1 5n 1 30
5 (n 1 6n ) 1 (5n 1 30)
5 n2(n 1 6) 1 5(n 1 6)
5 (n 1 6)(n2 1 5)
3
170
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Chapter 9 Resource Book
2
Group terms.
Factor each group.
Distributive property
Rearrange terms.
Group terms.
Factor each group.
Distributive property
Copyright © Holt McDougal. All rights reserved.
b. The binomials 5 2 y and y 2 5 are opposites. Factor 21 from
Name ———————————————————————
LESSON
9.8
Review for Mastery
For use with pages 624–631
Date ————————————
continued
LESSON 9.8
Exercises for Examples 1 and 2
Factor the expression.
1. 11x(x 2 8) 1 3(x 2 8)
2. 9x3 1 9x2 2 7x 2 7
3. 10x3 1 21y 2 35x2 2 6xy
EXAMPLE 3
Solve a polynomial equation
Solve the equation 7x 3 1 14x 2 5 105x.
Solution
7x3 1 14x2 5 105x
Write original equation.
7x3 1 14x2 2 105x 5 0
Subtract 105x from each side.
2
7x(x 1 2x 2 15) 5 0
Factor out 7x.
7x(x 1 5)(x 2 3) 5 0
Factor the trinomial.
7x 5 0
or
x50
or
x1550
or
x2350
x 5 25 or
x53
Zero-product property
Solve for x.
The roots of the equation are 0, 25, and 3.
Copyright © Holt McDougal. All rights reserved.
Exercises for Example 3
Solve the equation.
4. 2c 3 1 8c 2 2 42c 5 0
5. 4x3 1 48x2 1 144x 5 0
6. 5r 3 1 15r 5 20r 2
Algebra 1
Chapter 9 Resource Book
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Name ———————————————————————
LESSONS
9.5–9.8
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
LESSON 9.8
For use with pages 6002631
1. Multi-Step Problem The length of a box
is 12 inches more than its height. The width
of the box is 3 inches less than its height.
a. Draw a diagram of the box. Label its
dimensions in terms of the height h.
b. Write a polynomial that represents the
volume of the box.
c. The box has a volume of 324 cubic
inches. What are the length, width,
and height of the box?
2. Open-Ended Describe a situation that can
be modeled using the vertical motion model
h 5 216t2 1 24t. Then find the value of t
when h 5 0. Explain what this value of t
means in this situation.
3. Multi-Step Problem A block of wood
has the dimensions shown.
x in.
5. Extended Response You hit a baseball
straight up into the air. The baseball is hit
with an initial vertical velocity of 60 feet per
second when it is 4 feet off the ground.
a. Write an equation that gives the height
(in feet) of the baseball as a function
of the time (in seconds) since it
was hit.
b. After how many seconds does the ball
reach a height of 54 feet?
c. Does the ball reach a height of 54 feet
more than once? Justify your answer.
6. Gridded Response While standing on a
ladder, you drop a paintbrush from a height
of 9 feet. After how many seconds does the
paintbrush land on the ground?
7. Extended Response You want to make
a box with no lid out of a 9 inch by 13 inch
piece of cardboard. You cut out squares of
the same size from each corner. Then you
fold up the sides and tape them together.
(x 2 5) in.
(x 1 3) in.
x in.
9 in.
a. Write a polynomial that represents the
surface area of the wood.
b. The wood has a surface area of
384 square inches. What are the
length, width, and height of the block?
4. Short Response The shape of an
underpass for cars can be modeled by the
graph of the equation y 5 20.4x(x 2 14)
where x and y are measured in feet. On a
coordinate plane, the ground is represented
by the x-axis. How wide is the underpass at
its base? Explain how you found
your answer.
172
Algebra 1
Chapter 9 Resource Book
13 in.
a. Write a polynomial that represents the
volume of the box.
b. Find the volume of the box for cut out
square side lengths of 1 inch, 2 inches,
3 inches, and 4 inches. Which cut out
side square length gives the largest
volume?
c. Could a box be formed using cut out
squares with side lengths of 5 inches?
Explain why or why not.
Copyright © Holt McDougal. All rights reserved.
x in.
Name ———————————————————————
LESSON
9.8
Date ————————————
Challenge Practice
For use with pages 624–631
LESSON 9.8
In Exercises 1–5, factor the expression completely.
1. 8(y 1 3)3 1 22(y 1 3)2 1 15(y 1 3)
2. (y 2 1)4 2 16
3. (9x2 2 12x 1 4) 2 9
4. 21x2 1 15x 114x 1 10
5. 2y5 2 32y
In Exercises 6– 10, factor completely to solve for x.
6. (x 1 3)2 1 3(x 1 3) 5 10
7. x5 5 81x
8. 8x2 1 14x 1 21 5 212x
9. 2x2 2 5x 1 30 5 12x
9
6
1
10. }3 2 }2 52}
x
x
x
In Exercises 11 and 12, use the following information.
A roller coaster has a velocity v (in miles per hours) described by the polynomial
v(t) 5 210t 4 1 100t 2 2 90 for times from t 5 1 to t 5 3 minutes.
11. Find the velocity of the roller coaster when t 5 2 minutes.
Copyright © Holt McDougal. All rights reserved.
12. For what times on the interval from t 5 1 to t 5 3 minutes does the roller coaster
have a velocity of 0?
Algebra 1
Chapter 9 Resource Book
173
Name ———————————————————————
CHAPTER
9
Date ————————————
Chapter Review Game
For use after Chapter 9
Crossword Puzzle
Use the clues at the bottom of the page to fill in the correct vocabulary
word from Chapter 9 in the crossword puzzle.
1
2
3
4
5
6
7
8
9
10
11
12
13
Across
2. x 2 4x 1 4 is an example of a _______
2
CHAPTER REVIEW GAME
square trinomial.
174
Down
1. Using the distributive property to factor
polynomials with four terms is called
factoring by _______.
4. Solutions of an equation
3. 4x 2 2 2x 1 1
7. A monomial or sum of monomials
5. Use this to multiply binomials
9. The number 6 in the polynomial 6x2 2 24
6. Writing a polynomial as a product of
is called the _______ coefficient.
11. 2x
13. Sum of the exponents of the variables
in a monomial
14. An object propelled into the air but has
no power to keep itself in the air
Algebra 1
Chapter 9 Resource Book
other polynomials
8. A polynomial with two terms
10. A polynomial that cannot be factored
using integer coefficients
12. The height of a projectile can be described
by the _______ motion model.
Copyright © Holt McDougal. All rights reserved.
14
Name ———————————————————————
CHAPTER
10
Date ————————————
Family Letter
For use with Chapter 10
Lesson Title
Lesson Goals
Key Applications
10.1: Graph y 5 ax 2 1 c
Graph simple quadratic
functions.
• Solar Energy • Astronomy
• Sailing
10.2: Graph y 5 ax 2 1 bx 1 c
Graph general quadratic
functions.
• Suspension Bridges
• Spiders
• Architecture
Focus on Functions
10.3: Solve Quadratic Equations
by Graphing
Solve quadratic equations
by graphing.
• Sports
• Diving
10.4: Use Square Roots to Solve
Quadratic Equations
Solve a quadratic equation
by finding square roots.
• Sports Event • Gemology
• Internet Usage
10.5: Solve Quadratic Equations
by Completing the Square
Solve quadratic equations
by completing the square.
• Crafts
• Landscaping
• Snowboarding
Focus on Functions
Copyright © Holt McDougal. All rights reserved.
Graph quadratic functions
in intercept form.
Graph quadratic functions
in vertex form.
CHAPTER SUPPORT
Chapter Overview One way you can help your student succeed in Chapter 10
is by discussing the lesson goals in the chart below. When a lesson is completed, ask your
student the following questions. “What were the goals of the lesson? What new words and
formulas did you learn? How can you apply the ideas of the lesson to your life?”
• Soccer
10.6: Solve Quadratic Equations
by the Quadratic Formula
Solve quadratic equations
using the quadratic formula.
• Film Production
• Advertising
• Cell Phones
10.7: Interpret the Discriminant
Use the value of the
discriminant.
• Fountains
• Food
10.8: Compare Linear,
Exponential, and
Quadratic Models
Compare linear, exponential, and quadratic models.
• Cycling
• Lizards
• Nautilus
• Biology
Key Ideas for Chapter 10
In Chapter 10, you will apply the key ideas listed in the Chapter Opener
(see page 647) and reviewed in the Chapter Summary (see page 718).
1. Graphing quadratic functions
2. Solving quadratic functions
3. Comparing linear, exponential, and quadratic models
Algebra 1
Chapter 10 Resource Book
175
Name ———————————————————————
Family Letter
CHAPTER
continued
For use with Chapter 10
Key Ideas Your student can demonstrate understanding of key concepts by working
through the following exercises with you.
Lesson
Exercise
1
10.1
What is the vertex of the graph of the function y 5 2}2 x 2 1 5?
10.2
Find the axis of symmetry and the vertex of the graph of the function
y 5 22x 2 1 6x 2 4.
Focus on
Functions
Graph y 5 2(x 1 1)(x 2 3). Label the vertex, axis of symmetry, and x-intercepts.
10.3
Find the zeros of f(x) 5 2x 2 2 2x 1 8.
10.4
Solve the equation 2x 2 2 25 5 103 by using square roots. Round your solution to
the nearest hundredth, if necessary.
10.5
Solve the equation x 2 1 16x 1 20 5 0 by completing the square. Round your
solution to the nearest hundredth, if necessary.
Write y 5 x 2 2 14x 1 48 in vertex form. Then graph the function. Label the vertex
and axis of symmetry.
Focus on
Functions
10.6
Use the quadratic formula to solve the equation 5x 2 1 12x 1 4 5 0.
10.7
Tell whether the equation 8x 2 2 8x 5 22 has two, one, or no solution.
10.8
Tell whether the table of values represents
a linear function, an exponential function,
or a quadratic function.
x 23 22 21 0 1 2
y
4
1
0
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
10
Date ————————————
1 4 9
Home Involvement Activity
Directions Write a function that gives the surface area (in square feet) of a room
in your home. Investigate how much it would cost to paint five of the surfaces and
carpet (or tile) the sixth. (Remember to subtract the area of doors and windows.)
X
Functions:
X Y
10.6: 22 and 2}5 10.7: one solution 10.8: quadratic; y 5 (x 1 1)2
2
10.1: (0, 5) 10.2: axis of symmetry: x 5 }2; vertex: 1 }2, }2 2 Focus on Functions:
3
1
O
10.3: 2, 24 10.4: 8, 8 10.5: 21.37 and 214.63 Focus on
3 1
( 1, 0)
x
(3, 0)
1
x
y
1
(1, 4)
Answers
176
Algebra 1
Chapter 10 Resource Book
Nombre ——————————————————————
CAPÍTULO
10
Fecha ———————————
Carta para la familia
Usar con el Capítulo 10
Título de la lección
Aplicaciones clave
10.1: Graficar y 5 ax 2 1 c
Graficar funciones
cuadráticas simples
• Energía solar • Astronomía
• Navegación
10.2: Graficar y 5 ax 2 1 bx 1 c
Graficar funciones
cuadráticas generales
• Puentes colgantes
• Arañas
• Arquitectura
Enfoque en las funciones
Graficar funciones
cuadráticas en forma de
intersección
10.3: Resolver ecuaciones
cuadráticas al graficar
Resolver ecuaciones
cuadráticas al graficar
• Deportes
• Fútbol
• Zambullidas
10.4: Usar raíces cuadradas
para resolver ecuaciones
cuadráticas
Resolver ecuaciones
cuadráticas al elevar al
cuadrado
• Evento deportivo • Joyería
• Uso de Internet
10.5: Resolver ecuaciones
cuadráticas al elevar al
cuadrado
Graficar funciones
cuadráticas en forma
de vértice
• Manualidades
• Jardinería
• Hacer snowboard
Enfoque en las funciones
Copyright © Holt McDougal. All rights reserved.
Objetivos de la lección
CHAPTER SUPPORT
Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener éxito en el Capítulo
10 es hablar sobre los objetivos de la lección en la tabla a continuación. Cuando se termina una
lección, pregúntele a su hijo lo siguiente: “¿Cuáles fueron los objetivos de la lección? ¿Qué
palabras y fórmulas nuevas aprendiste? ¿Cómo puedes aplicar a tu vida las ideas de la lección?”
Graficar funciones
cuadráticas en forma de
vértice
10.6: Resolver ecuaciones
cuadráticas por la fórmula
cuadrática
Resolver ecuaciones
cuadráticas usando la
fórmula cuadrática
• Rodaje de película
• Publicidad
• Teléfonos celulares
10.7: Interpretar el
discriminante
Usar el valor del
discriminante
• Fuentes
• Comida
10.8: Comparar modelos
lineales, exponenciales
y cuadráticos
Comparar modelos
lineales, exponenciales
y cuadráticos
• Ciclismo
• Lagartijas
• Sepia
• Biología
Ideas clave para el Capítulo 10
En el Capítulo 10, aplicarás las ideas clave enumeradas en la Presentación del capítulo
(ver la página 647) y revisadas en el Resumen del capítulo (ver la página 718).
1. Graficar funciones cuadráticas
2. Resolver funciones cuadráticas
3. Comparar modelos lineales, exponenciales y cuadráticos
Algebra 1
Chapter 10 Resource Book
177
Nombre ——————————————————————
Carta para la familia
CAPÍTULO
continúa
Usar con el Capítulo 10
Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer los
siguientes ejercicios con usted.
Lección
Ejercicio
1
10.1
¿Qué es el vértice de la gráfica de la función y 5 2}2 x 2 1 5?
10.2
Halla el eje de simetría y el vértice de la gráfica de la función
y 5 22x 2 1 6x 2 4.
Enfoque en
las funciones
Grafica y 5 2(x 1 1)(x 2 3). Rotula el vértice, el eje de simetría y los interceptos
en x.
10.3
Halla los ceros de f(x) 5 2x 2 2 2x 1 8.
10.4
Resuelve la ecuación 2x 2 2 25 5 103 al usar raíces cuadradas. Redondea tu
solución al centésimo más próximo, si es necesario.
10.5
Resuelve la ecuación x 2 1 16x 1 20 5 0 en forma de vértice. Luego grafica la
función. Rotula el vértice y el eje de simetría.
Enfoque en
las funciones
Escribe y 5 x 2 2 14x 1 48 en forma de vértice. Luego grafica la función. Rotula el
vértice y el eje de simetría.
10.6
Usa la fórmula cuadrática para resolver la ecuación 5x 2 1 12x 1 4 5 0.
10.7
Indica si la ecuación 8x 2 2 8x 5 22 tiene una, dos o ninguna solución.
10.8
Indica si la tabla de valores representa
una función lineal, una función exponencial
o una función cuadrática.
x 23 22 21 0 1 2
y
4
1
0
1 4 9
Actividad para la familia
Instrucciones Escribe una función que da el área de la superficie (en pies
cuadrados) de una habitación en tu casa. Investiga cuánto costaría pintar cinco de las
superficies y ponerle alfombra (o baldosas) a la sexta superficie. (Recuérdate de restar
el área de puertas y ventanas.)
X
Enfoque en las funciones:
X 10.6: 22 y 2}5 10.7: una solución
10.8: cuadrática; y 5 (x 1 1)2
Y
2
10.1: (0, 5) 10.2: eje de simetría: x 5 }2 ; vértice: 1 }2 , }2 2 Enfoque en las funciones:
3
3 1
1
O
( 1, 0)
10.3: 2, 24 10.4: 8, 8 10.5: 21.37 y 214.63
x
(3, 0)
1
x
y
1
(1, 4)
Respuestas
178
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
10
Fecha ———————————
Name ———————————————————————
LESSON
10.1
Date ————————————
Practice A
For use with pages 6482654
Use the quadratic function to complete the table of values.
1. y 5 5x 2
2. y 5 24x 2
x
22
21
0
1
2
x
22
21
0
1
2
y
?
?
?
?
?
y
?
?
?
?
?
3. y 5 x 2 1 6
4. y 5 x 2 2 8
x
22
21
0
1
2
x
22
21
0
1
2
y
?
?
?
?
?
y
?
?
?
?
?
A.
Copyright © Holt McDougal. All rights reserved.
23
1
7. y 5 } x 2
4
B.
y
3
1
1
1
3 x
C.
y
3
21
21
23
LESSON 10.1
Match the function with its graph.
1
5. y 5 2} x 2
6. y 5 2x 2
2
y
1
23
21
21
1
3 x
3 x
21
23
Graph the function and identify its domain and range. Compare the graph
with the graph of y 5 x 2.
1
8. y 5 5x 2
9. y 5 2} x 2
10. y 5 26x 2
3
y
y
y
5
23
1
3
23
21
21
1
3 x
21
21
1
3 x
1
3 x
23
1
23
21
21
1
23
3 x
25
Identify the vertex and axis of symmetry of the graph.
11.
12.
y
23
10
21
21
23
6
13.
y
1
y
3 x
0.5
23
2
23
21
1
3 x
Algebra 1
Chapter 10 Resource Book
179
Name ———————————————————————
LESSON
10.1
Practice A
For use with pages 6482654
Date ————————————
continued
Match the function with its graph.
14. y 5 x 2 2 3
A.
15. y 5 3x 2 2 1
B.
y
16. y 5 2x 2 1 3
C.
y
y
1
23
3
1
21
3 x
1
LESSON 10.1
23
1
21
21
3 x
1
23
21
3 x
1
Graph the function and identify its domain and range. Compare the graph
with the graph of y 5 x 2.
17. y 5 x 2 2 5
18. y 5 x 2 1 7
y
21
21
y
y
1
23
19. y 5 2x 2 2 3
1
3 x
23
10
3
6
1
23
2
23
25
21
22
21
21
3 x
1
3 x
1
23
Complete the statement.
graph of y 5 x 2
? .
21. The graph of y 5 10x 2 can be obtained from the graph of y 5 x 2 by
2
of y 5 x by a factor of
?
the graph
? .
22. Pot Rack A cross section of the pot rack shown can be modeled
y
by the graph of the function y 5 20.08x 2 1 8 where x and y are
measured in inches.
a. Find the domain of the function in this situation.
b. Find the range of the function in this situation.
6
2
26
22
23. Drawer Handle A cross section of the drawer handle shown
y
3
1
can be modeled by the graph of the function y 5 2}
x2 1 2
18
where x and y are measured in centimeters.
a. Find the domain of the function in this situation.
b. Find the range of the function in this situation.
180
Algebra 1
Chapter 10 Resource Book
x
6
2
1
25
23
21
x
1
3
5
Copyright © Holt McDougal. All rights reserved.
20. The graph of y 5 x 2 1 5 can be obtained from the graph of y 5 x 2 by shifting the
Name ———————————————————————
LESSON
10.1
Date ————————————
Practice B
For use with pages 6482654
Use the quadratic function to complete the table of values.
1. y 5 9x 2
2. y 5 25x 2
x
22
21
0
1
2
x
22
21
0
1
2
y
?
?
?
?
?
y
?
?
?
?
?
1
4. y 5 2} x 2 2 2
8
5
3. y 5 } x 2 1 1
2
x
24
22
0
2
4
x
216
28
0
8
16
y
?
?
?
?
?
y
?
?
?
?
?
LESSON 10.1
5. y 5 24x 2 1 3
6. y 5 6x 2 2 5
x
22
21
0
1
2
x
22
21
0
1
2
y
?
?
?
?
?
y
?
?
?
?
?
Match the function with its graph.
7. y 5 24x 2 1 3
1
10. y 5 } x 2 2 3
4
A.
1
9. y 5 } x 2 2 4
3
8. y 5 3x 2 1 4
11. y 5 23x 2 1 4
B.
y
12. y 5 4x 2 1 3
C.
y
y
1
Copyright © Holt McDougal. All rights reserved.
21
21
1
x
1
3
23
1
23
21
1
D.
25
3 x
E.
y
3 x
21
F.
y
y
1
21
21
1
x
1
3
3 x
23
23
1
25
23
21
1
3 x
23
Describe how you can use the graph of y 5 x 2 to graph the given function.
13. y 5 x 2 2 8
14. y 5 2x 2 1 4
15. y 5 2x 2 1 3
16. y 5 25x 2 1 1
1
17. y 5 } x 2 2 2
2
3
18. y 5 2} x 2 1 5
4
Algebra 1
Chapter 10 Resource Book
181
Name ———————————————————————
LESSON
10.1
Date ————————————
Practice B
continued
For use with pages 6482654
Graph the function and identify its domain and range. Compare the graph
with the graph of y 5 x 2.
3
1
19. y 5 x 2 1 9
20. y 5 2} x 2
21. y 5 2} x 2
5
2
y
y
23
3
LESSON 10.1
29
23
23
3
9 x
22. y 5 x 2 2 3.5
3
1
1
23
21
24. y 5 25x 2 1 2
y
y
9
1
3 x
3
3
29
25
3 x
1
23
23. y 5 2x 2 2 9
y
21
21
23
3 x
21
1
23
y
3
23
3
9 x
23
21
3 x
1
29
25. Serving Plate The top view of a freeform serving plate you made in
y
a ceramics class is shown in the graph. One edge of the plate can be
5
modeled by the graph of the function y 5 2}
x 2 1 20 where x and y
81
4
212
100 feet above the ground. The height y (in feet) of the dropped
roof shingle is given by the function y 5 216t 2 1 100 where t
is the time (in seconds) since the shingle is dropped.
a. Graph the function.
b. Identify the domain and range of the function in this situation.
c. Use the graph to estimate the shingle’s height at 1 second.
d. Use the graph to estimate when the shingle is at a height of 50 feet.
e. Use the graph to estimate when the shingle is at a height of 0 feet.
182
Algebra 1
Chapter 10 Resource Book
Height (feet)
26. Roof Shingle A roof shingle is dropped from a rooftop that is
y
100
80
60
40
20
0
24
24
0
t
12
t
1
2
Time (seconds)
Copyright © Holt McDougal. All rights reserved.
are measured in inches.
a. Find the domain of the function in this situation.
b. Find the range of the function in this situation.
12
Name ———————————————————————
LESSON
10.1
Date ————————————
Practice C
For use with pages 6482654
Use the quadratic function to complete the table of values.
1. y 5 10x 2 2 4
2. y 5 21.5x 2 1 3
x
22
21
0
1
2
x
22
21
0
1
2
y
?
?
?
?
?
y
?
?
?
?
?
Graph the function and identify its domain and range. Compare the graph
with the graph of y 5 x 2.
1
3. y 5 } x 2 1 2
6
2
23
1
21
21
1
21
22
1
6
3 x
2
3 x
26
23
21
22
1
3 x
1
3 x
LESSON 10.1
y
y
y
3
23
7
5. y 5 9x 2 2 }
2
4. y 5 24x 2 2 3
210
23
3
1
6. y 5 } x 2 1 }
5
5
3
8. y 5 6x 2 1 }
4
1
7. y 5 2} x 2 1 4
2
y
y
y
5
30
3
Copyright © Holt McDougal. All rights reserved.
3
18
1
1
23
23
21
21
1
3 x
2
9. y 5 4x 2 2 }
3
21
21
23
21
26
11. y 5 25x 2 1 15
y
y
y
2
23
9
21
22
26
3
21
23
6
3 x
1
10. y 5 22x 2 2 }
2
15
23
1
1
3 x
210
15
1
3 x
5
23
21
25
1
3 x
215
Algebra 1
Chapter 10 Resource Book
183
Name ———————————————————————
Practice C
LESSON
10.1
For use with pages 6482654
Date ————————————
continued
Tell how you can obtain the graph of g from the graph of f by
using transformations.
12. f(x) 5 x 2 1 6
1
14. f (x) 5 2} x 2 2 3
2
13. f (x) 5 2x 2 1 14
g(x) 5 x 2 2 2
g(x) 5 2x 2 1 9
15. f(x) 5 3x 2 2 5
g(x) 5 3x 2 1 11
1
g(x) 5 2}2 x 2 2 7
16. f (x) 5 3x 2
17. f (x) 5 8x 2
g(x) 5 9x 2
g(x) 5 4x 2
18. (0, 6), (2, 10)
19. (0, 1), (21, 0)
y
20. (0, 24), (23, 5)
y
y
6
1
10
23
6
21
1
3
2
x
23
21
22
1
3
x
23
2
23
21
21
1
3
26
x
rope can be modeled by the function w 5 22,210d 2 where
d is the diameter (in inches) of the rope.
a. Graph the function.
b. Use the graph to estimate the diameter of a nylon rope
that has a breaking weight of 50,000 pounds.
22. Foam Ball A foam ball is dropped from a
Weight (pounds)
21. Nylon Rope The breaking weight w (in pounds) of a nylon
w
100,000
80,000
60,000
40,000
20,000
0
0
0.5
1.0
1.5
2.0 d
Diameter (inches)
y
y
184
Algebra 1
Chapter 10 Resource Book
Height (feet)
Distance (feet)
deck that is 20 feet above the ground.
20
20
16
16
a. The distance y (in feet) that the ball falls
12
12
is given by the function y 5 16t 2 where t
8
8
is the time (in seconds) since the ball was
4
4
dropped. Graph the function.
0
0
t
t
0
0.4
0.8
0
0.4
0.8
b. The height y (in feet) of the dropped ball
Time (seconds)
Time (seconds)
2
is given by the function y 5 216t 1 20
where t is the time (in seconds) since the
ball was dropped. Graph the function.
c. How are the graphs from part (a) and part (b) related? Explain how you can use each graph to
find the number of seconds after which the ball has dropped 8 feet.
Copyright © Holt McDougal. All rights reserved.
LESSON 10.1
Write a function of the form y 5 ax 2 1 c whose graph passes through the two
given points. Then graph the function.
Name ———————————————————————
LESSON
10.1
Date ————————————
Review for Mastery
For use with pages 6482654
GOAL
Graph simple quadratic functions.
Vocabulary
A quadratic function is a nonlinear function that can be written in the
standard form y 5 ax 2 1 bx 1 c where a Þ 0.
Every quadratic function has a U-shaped graph called a parabola.
The most basic quadratic function in the family of quadratic functions,
called the parent quadratic function, is y 5 x 2.
The lowest or highest point on a parabola is the vertex.
EXAMPLE 1
LESSON 10.1
The line that passes through the vertex and divides the parabola into
two symmetric parts is called the axis of symmetry.
Graph y 5 ax 2 when a > 1
Graph y 5 26x 2. Compare the graph with the graph of y 5 x 2.
Solution
STEP 1
y
Make a table of values for y 5 26x 2.
x
22
21
0
1
2
y
224
26
0
26
224
9
3
Copyright © Holt McDougal. All rights reserved.
23
STEP 2
Plot the points from the table.
STEP 3
Draw a smooth curve through the points.
STEP 4
Compare the graphs of y 5 26x 2 and y 5 x 2.
Both graphs have the same vertex, (0, 0), and
the same axis of symmetry, x 5 0. However,
the graph of y 5 26x 2 is narrower than the
graph of y 5 x 2 and it opens down. This is
because the graph of y 5 26x 2 is a vertical
stretch (by a factor of 6) of the graph of y 5 x 2
and a reflection in the x-axis of the graph of y 5 x 2.
y 5 x2
21
29
1
3
x
y 5 26x 2
215
221
Algebra 1
Chapter 10 Resource Book
185
Name ———————————————————————
LESSON
10.1
Review for Mastery
Date ————————————
continued
For use with pages 6482654
EXAMPLE 2
Graph y 5 ax 2 when⏐a⏐< 1
2 2
Graph y 5 }
x . Compare the graph with the graph of y 5 x 2.
5
STEP 1
2
y
Make a table of values for y 5 }5 x 2.
35
x
210
25
0
5
10
y
40
10
0
10
40
25
2
y 5 5 x2
y 5 x2
Plot the points from the table.
STEP 3 Draw a smooth curve through the points.
STEP 2
LESSON 10.1
STEP 4
2
Compare the graphs of y 5 }5 x 2 and y 5 x 2.
215
25
x
15
5
Both graphs have the same vertex, (0, 0), and the same axis of symmetry,
2
x 5 0. Both graphs open upward. However, the graph of y 5 }5 x 2 is wider
than the graph of y 5 x 2.
2
2
This is because the graph of y 5 }5 x 2 is a vertical shrink 1 by a factor of }5 2
of the graph of y 5 x 2.
EXAMPLE 3
Graph y 5 ax 2 1 c
Graph y 5 3x 2 2 1. Compare the graph with the graph of y 5 x 2.
Make a table of values for y 5 3x 2 2 1.
y
x
22
21
0
1
2
10
y
11
2
21
2
11
6
y 5 x2
2
STEP 2
Plot the points from the table.
STEP 3
Draw a smooth curve through the points.
STEP 4
Compare the graphs of y 5 3x 2 2 1 and
y 5 x 2. Both graphs open up and have the same axis of symmetry, x 5 0.
However, the graph of y 5 3x 2 2 1 is narrower and has a lower vertex than
the graph of y 5 x 2. This is because the graph of y 5 3x 2 2 1 is a vertical
stretch (by a factor of 3) and a vertical translation (1 unit down) of the graph
of y 5 x 2.
23
21
22
1
Exercises for Examples 1, 2, and 3
Graph the function. Compare the graph with the graph of y 5 x 2.
186
1
1
3. y 5 2} x 2
3
1
1
6. y 5 2} x 2 2 1
2
1. y 5 28x 2
2.
y 5 }7 x 2
4. y 5 x 2 2 3
5.
y 5 }4 x 2 1 2
Algebra 1
Chapter 10 Resource Book
3
x
y 5 3x 2 2 1
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
Date ————————————
Challenge Practice
LESSON
10.1
For use with pages 6482654
In Exercises 1–5, write the function of the form y 5 ax 2 1 c whose graph
passes through the given points.
1. (0, 4), (21, 7), (1, 7)
2. (1, 21), (21, 21), (3, 217)
3. (1, 26), (2, 6), (3, 26)
4. (21, 4), (2, 1), (3, 24)
5.
1 1, }2 2, (0, 2), (22, 0)
3
In Exercises 6 –10, use the following information.
6. What is the mass (in kilograms) of an object containing 9.61 3 1016 joules
of energy?
LESSON 10.1
Einstein’s famous formula E 5 mc 2 relates mass m (in kilograms) to the energy E
(in joules) contained within the mass. The constant c is equal to the speed of light in a
vacuum (in meters per second), c ø 3.1 3 108 meters per second.
7. The average automobile uses 5 3 1010 joules of energy per year. What is the mass
represented by this energy?
8. Suppose Einstein’s formula holds true in an alternate universe where the speed of
Copyright © Holt McDougal. All rights reserved.
light is not the same as in our universe. If an experiment is conducted in which
1 kilogram of mass is equivalent to 1 3 1020 joules of energy, then what is the speed
of light in the alternate universe?
9. The average home uses 1 3 108 joules of energy per year. What is the mass
represented by this energy?
10. Suppose Einstein’s formula holds true in an alternate universe where the speed of
light is not the same as in our universe. If the speed of light in the alternate universe
is 4 3 105 meters per second, then how much mass would be needed to produce
5 3 1011 joules of energy?
Algebra 1
Chapter 10 Resource Book
187
Name ———————————————————————
LESSON
10.2
Activity Support Master
For use with pages 655–656
Equation graphed
y-intercept
x-intercept(s)
Axis of symmetry
y 5 2x2
0
0
x50
y 5 2x2 2 4x
0
0, 2
x51
Equation graphed
y-intercept
x-intercept(s)
Axis of symmetry
y 5 2x2 2 4x
0
0, 2
x51
y 5 2x2 2 4x 2 6
26
21, 3
x51
Copyright © Holt McDougal. All rights reserved.
LESSON 10.2
188
Date ————————————
Algebra 1
Chapter 10 Resource Book
Name ———————————————————————
LESSON
10.2
Date ————————————
Practice A
For use with pages 657–662
Identify the values of a, b, and c in the quadratic function.
1. y 5 7x 2 1 2x 1 11
2. y 5 3x 2 2 5x 1 1
3. y 5 4x 2 1 2x 2 2
4. y 5 23x 2 1 9x 1 4
1
5. y 5 } x 2 2 x 2 5
2
6. y 5 2x 2 1 7x 2 6
Tell whether the graph opens upward or downward. Then find the axis of
symmetry of the graph of the function.
7. y 5 x 2 1 6
8. y 5 2x 2 2 1
9. y 5 x 2 1 6x 1 1
10. y 5 x 2 2 4x 1 5
11. y 5 2x 2 1 4x 2 5
12. y 5 2x 2 1 8x 1 3
13. y 5 x 2 1 3x 2 6
14. y 5 2x 2 1 7x 2 2
15. y 5 3x 2 1 6x 1 10
Find the vertex of the graph of the function.
16. y 5 x 2 1 5
17. y 5 2x 2 1 3
18. y 5 x 2 1 10x 1 3
19. y 5 2x 2 1 4x 2 2
20. y 5 3x 2 1 6x 1 1
21. y 5 22x 2 1 8x 2 3
22. y 5 10x 2 2 10x 1 7
23. y 5 x 2 1 x 1 3
24. y 5 x 2 2 x 1 1
Use the quadratic function to complete the table of values.
26. y 5 2x 2 1 12x 2 5
x
1
2
3
4
5
x
4
5
6
7
8
y
?
?
?
?
?
y
?
?
?
?
?
27. y 5 7x 2 1 14x 1 2
LESSON 10.2
Copyright © Holt McDougal. All rights reserved.
25. y 5 x 2 2 6x 1 8
28. y 5 22x 2 2 4x 1 1
x
23
22
21
0
1
x
23
22
21
0
1
y
?
?
?
?
?
y
?
?
?
?
?
Match the function with its graph.
29. y 5 8x 2 1 2x 1 3
A.
B.
y
4
220
24
1
31. y 5 } x 2 1 8x 1 5
2
30. y 5 2x 2 1 8x 1 1
25
4
C.
y
23
21
1
y
x
x
1
27
23
21
1
3 x
Algebra 1
Chapter 10 Resource Book
189
Name ———————————————————————
Practice A
LESSON
10.2
For use with pages 657–662
Date ————————————
continued
Graph the function. Label the vertex and axis of symmetry.
32. y 5 2x 2 2 6
33. y 5 x 2 1 7
34. y 5 x 2 1 2x 1 5
y
y
y
7
2
23
21
22
1
3
10
x
5
6
26
3
2
210
23
35. y 5 x 2 2 8x 1 1
21
22
3
1
x
25
36. y 5 22x 2 1 x 2 3
y
22
22
1
23
21
37. y 5 2x 2 2 4x 1 3
y
y
2
6
10
x
23
21
21
3 x
1
3
1
7
x
5
26
23
210
25
214
27
3
1
25
23
21
1
x
38. f(x) 5 x 2 2 7
39. f (x) 5 2x 2 1 9
40. f (x) 5 2x 2 1 4x
41. Greenhouse The dome of the greenhouse shown can be modeled by the graph of
the function y 5 20.15625x 2 1 2.5x where x and y are measured in feet. What is
the height h at the highest point of the dome as shown in the diagram?
y
10
6
h
2
2
6
42. Fencing A parabola forms the top of a fencing panel
as shown. This parabola can be modeled by the graph
of the function y 5 0.03125x 2 2 0.25x 1 4 where x
and y are measured in feet and y represents the number
of feet the parabola is above the ground. How far above
the ground is the lowest point of the parabola formed
by the fence?
190
Algebra 1
Chapter 10 Resource Book
10
14
x
Copyright © Holt McDougal. All rights reserved.
LESSON 10.2
Tell whether the function has a minimum value or a maximum value.
Then find the minimum or maximum value.
Name ———————————————————————
LESSON
10.2
Date ————————————
Practice B
For use with pages 657–662
Identify the values of a, b, and c in the quadratic function.
1. y 5 6x 2 1 3x 1 5
3
2. y 5 } x 2 2 x 1 8
2
3. y 5 7x 2 2 3x 2 1
4. y 5 22x 2 1 9x
3
5. y 5 } x 2 2 10
4
6. y 5 28x 2 1 3x 2 7
Tell whether the graph opens upward or downward. Then find the axis of
symmetry and vertex of the graph of the function.
7. y 5 x 2 2 5
8. y 5 2x 2 1 9
9. y 5 22x 2 1 6x 1 7
10. y 5 3x 2 2 12x 1 1
11. y 5 3x 2 1 6x 2 2
12. y 5 22x 2 1 7x 2 21
1
13. y 5 } x 2 1 5x 2 4
2
1
14. y 5 2} x 2 2 24
4
15. y 5 23x 2 1 9x 2 8
16. y 5 3x 2 2 2x 1 3
17. y 5 22x 2 1 7x 1 1
18. y 5 3x 2 1 2x 2 5
Find the vertex of the graph of the function. Make a table of values using
x-values to the left and right of the vertex.
19. y 5 x 2 2 10x 1 3
20. y 5 2x 2 1 6x 2 2
x
?
?
?
?
?
x
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
1
22. y 5 } x 2 2 2x 1 3
3
x
?
?
?
?
?
x
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
Graph the function. Label the vertex and axis of symmetry.
23. y 5 2x 2 2 10
24. y 5 2x 2 1 3
y
215
25
25
25. y 5 22x 2 1 2x 1 1
y
5
y
3
5
5
LESSON 10.2
Copyright © Holt McDougal. All rights reserved.
1
21. y 5 } x 2 2 x 1 7
2
15 x
1
23
1
23
21
21
1
21
1
3 x
3 x
Algebra 1
Chapter 10 Resource Book
191
Name ———————————————————————
Practice B
LESSON
10.2
Date ————————————
continued
For use with pages 657–662
26. y 5 5x 2 1 2x
27. y 5 22x 2 1 x 2 4
28. y 5 x 2 2 8x 1 5
y
2
y
y
3
23
21
21
3 x
1
22
22
1
23
21
10 x
6
23
3 x
1
2
26
210
1
29. y 5 2} x 2 2 8x 1 3
2
3
31. y 5 2} x 2 2 2x 1 2
4
1
30. y 5 } x 2 1 3x 2 1
4
y
2
y
y
22
210
x
1
10
230
10
25
21
x
210
x
23
Tell whether the function has a minimum value or a maximum value.
Then find the minimum or maximum value.
33. f (x) 5 25x 2 1 10x 2 2
34. f (x) 5 8x 2 2 4x 1 4
35. Storage Building The storage building shown can be modeled by the graph of the
function y 5 20.12x 2 1 2.4x where x and y are measured in feet. What is the height
h at the highest point of the building as shown in the diagram?
y
10
h
6
2
2
6
10
36. Velvet Rope A parabola is formed by a piece of velvet rope
found around a museum display as shown. This parabola can be
4
16
modeled by the graph of the function y 5 }
x2 2 }
x 1 40
225
15
where x and y are measured in inches and y represents the
number of inches the parabola is above the ground. How far
above the ground is the lowest point on the rope?
192
Algebra 1
Chapter 10 Resource Book
14
18
x
Copyright © Holt McDougal. All rights reserved.
LESSON 10.2
32. f(x) 5 8x 2 2 40
Name ———————————————————————
Date ————————————
Practice C
LESSON
10.2
For use with pages 657–662
Tell whether the graph opens upward or downward. Then find the axis of
symmetry and vertex of the graph of the function.
1. y 5 23x 2 1 3x 1 5
5
2. y 5 } x 2 2 2x 1 1
2
3. y 5 8x 2 2 2x 1 3
4. y 5 29x 2 1 9x
2
5. y 5 } x 2 2 9
3
6. y 5 25x 2 1 2x 2 3
1
7. y 5 } x 2 2 2x
8
1
8. y 5 2} x 2 1 7
5
9. y 5 26x 2 1 12x 1 5
10. y 5 4x 2 2 12x 1 8
11. y 5 5x 2 1 10x 2 3
12. y 5 26x 2 1 8x 2 10
Find the vertex of the graph of the function. Make a table of values using
x-values to the left and right of the vertex.
5
14. y 5 2} x 2 1 10x 2 1
2
1
13. y 5 } x 2 2 2x 1 5
4
x
?
?
?
?
?
x
?
?
?
?
?
y
?
?
?
?
?
y
?
?
?
?
?
Graph the function. Label the vertex and axis of symmetry.
15. y 5 2x 2 2 15
16. y 5 6x 2 1 8
Copyright © Holt McDougal. All rights reserved.
y
14
25
25
5
15
3
x
10
1
215
6
225
2
26
18. y 5 2x 2 1 20
22
2
6
21
21
y
20
5
3
12
1
4
215
15
x
23
21
21
12
x
y
5
5
x
20. y 5 23x 2 1 18x 2 4
15
25
25
3
1
23
x
19. y 5 7x 2 2 14x 1 6
y
215
23
LESSON 10.2
y
y
5
215
17. y 5 24x 2 1 4x 1 3
1
3
x
212
24
24
4
Algebra 1
Chapter 10 Resource Book
193
Name ———————————————————————
Practice C
LESSON
10.2
Date ————————————
continued
For use with pages 657–662
7
21. y 5 2} x 2 1 21x 2 5
2
1
22. y 5 } x 2 2 2x 1 10
4
23. y 5 6x 2 2 12x 1 13
y
y
y
14
14
10
10
6
6
2
2
30
18
6
218
26
26
6
18
x
26
5
24. y 5 } x 2 2 15x 1 2
3
22
2
7
25. y 5 } x 2 1 35x 2 4
4
y
22
26
22
6
10 x
215
25
230
6
x
y
30
2
2
2
26. y 5 2} x 2 2 20x 1 5
5
y
6
26
26
10 x
6
250
5
15
x
150
218
290
50
230
2150
2150 250
250
50
150
x
27. f(x) 5 9x 2 2 36
3
28. f (x) 5 2} x 2 1 18x 2 7
4
5
29. f (x) 5 } x 2 2 10x 1 3
4
30. Lamps A lighting company offers two models of small lamps, both of
which contain a reflector in the shape of a parabola. The shape of the
reflector in lamp A can be modeled by the function y 5 20.16x 2 1 25
and the shape of the reflector in lamp B can be modeled by the function
y 5 20.2x 2 1 20 where x and y are measured in millimeters.
a. Find the maximum value of each function, which gives the height
of the reflector.
b. How much taller is the reflector for lamp A than the reflector for lamp B?
31. Window An artist designs a window in a house to be in the shape of a
parabola as shown. The top part of the window can be modeled by the
function y 5 21.875x 2 1 7.5x and the bottom part of the window can
be modeled by the function y 5 1.5 where x represents the width of the
window (in feet) and y represents the height of the window (in feet) above
the ground. How tall is the window? Explain how you got your answer.
194
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 10.2
Tell whether the function has a minimum value or a maximum value.
Then find the minimum or maximum value.
Name ———————————————————————
LESSON
10.2
Date ————————————
Review for Mastery
For use with pages 657–662
GOAL
Graph general quadratic functions.
Vocabulary
For y 5 ax 2 1 bx 1 c, the y-coordinate of the vertex is the minimum
value of the function if a > 0 and the maximum value of the function
if a < 0.
EXAMPLE 1
Find the axis of symmetry and the vertex
Consider the function y 5 3x2 2 18x 1 11.
a. Find the axis of symmetry of the graph of the function.
b. Find the vertex of the graph of the function.
Solution
a. For the function y 5 3x 2 2 18x 1 11, a 5 3 and b 5 218.
(218)
b
5 2}
53
x 5 2}
2a
2(3)
Substitute 3 for a and 218 for b.
Then simplify.
The axis of symmetry is x 5 3.
b
b. The x-coordinate of the vertex is 2}, or 3. To find the y-coordinate,
2a
substitute 3 for x in the function and find y.
Copyright © Holt McDougal. All rights reserved.
Substitute 3 for x. Then simplify.
The vertex is (3, 216).
EXAMPLE 2
Find the minimum or maximum value
LESSON 10.2
y 5 3(3)2 218(3) 1 11 5 216
Tell whether the function f(x) 5 x 2 1 14x 2 3 has a minimum value or
a maximum value. Then find the minimum or maximum value.
Solution
Because a 5 1 and 1 > 0, the parabola opens up and the function has a minimum
value. To find the minimum value, find the vertex.
b
14
b
5 2}
5 27
x 5 2}
2a
2(1)
The x-coordinate is 2}
.
2a
f (27) 5 (27)2 1 14(27) 2 3 5 252
Substitute 27 for x. Then simplify.
The minimum value of the function is f(x) 5 252.
Algebra 1
Chapter 10 Resource Book
195
Name ———————————————————————
LESSON
10.2
Review for Mastery
For use with pages 657–662
Date ————————————
continued
Exercises for Examples 1 and 2
Find the axis of symmetry and the vertex of the graph of the function.
1. y 5 5x 2 1 20x 1 9
1
2. y 5 } x 2 2 4x 2 19
3
1
3. Tell whether the function f (x) 5 } x 2 2 8x 1 13 has a minimum value or a
2
maximum value. Then find the minimum value or maximum value.
EXAMPLE 3
Graph y 5 ax 2 1 bx 1 c
1 2
Graph y 5 }
x 2 2x 1 3.
5
Solution
STEP 1
Determine whether the parabola opens up or down.
Because a > 0, the parabola opens up.
STEP 2
Find and draw the axis of symmetry:
(22)
b
5 2}
5 5.
x 5 2}
2a
1
y
21 }5 2
STEP 3
14
Find and plot the vertex.
x55
10
b
or 5. To find the y-coordinate, substitute
5 for x in the function and simplify.
1
y 5 }5 (5)2 2 2(5) 1 3 5 22
(0, 3)
(10, 3)
(1, 1.2)
22
(9, 1.2)
6
10
14 x
(5, 22)
So, the vertex is (5, 22).
STEP 4
Plot two points. Choose two x-values less than the x-coordinate of the vertex.
Then find the corresponding y-values.
x
0
1
y
3
1.2
STEP 5
Reflect the points plotted in Step 4 in the axis of symmetry.
STEP 6
Draw a parabola through the plotted points.
Exercise for Example 3
4. Graph the function f(x) 5 x 2 2 4x 1 7. Label the vertex and axis of symmetry.
196
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 10.2
The x-coordinate of the vertex is 2}
,
2a
Name ———————————————————————
LESSON
10.2
Date ————————————
Problem Solving Workshop:
Worked Out Example
For use with pages 657–662
PROBLEM
Basketball You throw a basketball whose path can be modeled by the graph of
y 5 216x 2 1 19x 1 6 where x is the time (in seconds) and y is the height (in feet) of
the basketball. What is the maximum height of the basketball?
STEP 1
Read and Understand
What do you know?
The equation that models the path of a basketball
What do you want to find out?
The maximum height of the basketball
STEP 2
Make a Plan Use what you know to find the vertex of the parabola.
STEP 3
Solve the Problem The highest point of the basketball is at the vertex of the
parabola. Find the x-coordinate of the vertex. Use a 5 216 and b 5 19.
b
19
5 2}
ø 0.59
x 5 2}
2a
2(216)
Use a calculator.
Substitute 0.59 for x in the equation to find the y-coordinate of the vertex.
y ø 216(0.59)2 1 19(0.59) 1 6 ø 11.64
PRACTICE
Look Back By graphing the function, it
appears that the maximum occurs after
about 0.6 second and at a height between
11 and 12 feet. The answer seems reasonable.
1. Suspension Bridges The cables
between the two towers of the Golden
Gate Bridge in California form a
parabola that can be modeled by the
graph of y 5 0.00012x 2 2 0.505x 1 746
where x and y are measured in feet. What
is the height of the cable above the water
at its lowest point?
2. Baseball You hit a baseball whose path
can be modeled by the graph of
y 5 216x 2 1 40x 1 3 where x is the
time (in seconds) since the ball was
hit and y is the height (in feet) of the
baseball. What is the maximum height
of the baseball?
y
12
10
8
6
4
2
0
0
0.4
0.8
1.2
Time (seconds)
x
LESSON 10.2
Copyright © Holt McDougal. All rights reserved.
STEP 4
Height (feet)
The basketball reaches a maximum height of about 11.64 feet.
3. Tunnel The shape of a tunnel for cars
can be modeled by the graph of the
equation y 5 20.5x 2 1 4x where x and
y are measured in feet. On a coordinate
plane, the ground is represented by
the x-axis. How wide is the tunnel at
its base?
4. Sprinkler A sprinkler ejects water at an
angle of 35° with the ground. The path
of the water can be modeled by the
equation y 5 20.06x2 1 0.7x 1 0.5
where x and y are measured in feet.
What is the maximum height of
the water?
Algebra 1
Chapter 10 Resource Book
197
Name ———————————————————————
LESSON
10.2
Date ————————————
Challenge Practice
For use with pages 657–662
In Exercises 1–5, write the function of the form y 5 ax 2 1 bx 1 c whose
graph passes through the three given points.
1. (0, 1), (1, 0), (2, 3)
2. (1, 2), (0, 4), (21, 4)
3. (21, 6), (1, 2), (3, 6)
4. (2, 0), (1, 1), (0, 4)
5. (1, 12), (2, 9), (3, 0)
In Exercises 6–10, use the given information to write a function of the
form f (x) 5 ax 2 1 bx 1 c.
3
6. f(x) has an axis of symmetry at x 5 }, x-intercepts at x 5 1 and x 5 2, and a
2
y-intercept at y 5 2.
3
5
7. f(x) has an axis of symmetry at x 5 }, x-intercepts at x 5 21 and x 5 },
4
2
and a y-intercept at y 5 5.
5
7
8. f(x) has an axis of symmetry at x 5 2} , x-intercepts at x 5 2} and x 5 1,
4
2
198
5
1
1
9. f(x) has an axis of symmetry at x 5 }, x-intercepts at x 5 } and x 5 },
12
3
2
and a y-intercept at y 5 21.
19
1
10. f(x) has an axis of symmetry at x 5 }, x-intercepts at x 5 } and x 5 6,
6
3
and a y-intercept at y 5 6.
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 10.2
and a y-intercept at y 5 27.
Name ———————————————————————
FOCUS ON
10.2
Date ————————————
Practice
For use with pages 663–664
Graph the quadratic function. Label the vertex, axis of symmetry, and
x-intercepts. Identify the domain and range of the function.
1. y 5 (x 1 2)(x 2 4)
2. y 5 2(x 1 1)(x 2 3)
Y
/
X
3. y 5 2(x 1 5)(x 1 1)
Y
Y
X
/
/
4. y 5 24(x 2 1)(x 2 3)
5. y 5 (x 1 4)2
Y
X
6. y 5 5x2 2 45
Y
Y
/
X
X
/
/
7. y 5 2x2 + 4x 2 4
8. y 5 2x2 + 6x 2 8
Y
Copyright © Holt McDougal. All rights reserved.
/
9. y 5 25x2 + 10x 1 40
Y
Y
X
X
/
10. Follow the steps below to write an equation
X
FOCUS ON 10.2
/
X
Y
of the parabola shown.
a. Find the x-intercepts.
b. Use the values of p and q and the coordinates of
the vertex to find the value of a in the equation
y 5 a(x 2 p)(x 2 q).
c. Write a quadratic equation in intercept form.
/
X
11. Challenge A baseball is thrown into the air. The path of a baseball is parabolic.
The ball reaches a height of 25 feet before it starts to descend and lands 50 feet
from the point where it was thrown. What is the equation, in intercept form, which
models the path of the baseball? Assume the baseball was thrown at (0, 0).
Algebra 1
Chapter 10 Resource Book
199
Name ———————————————————————
FOCUS ON
10.2
Date ————————————
Review for Mastery
For use with pages 663–664
GOAL
Graph quadratic functions in intercept form.
Vocabulary
The intercept form of a quadratic function is y 5 a(x 2 p)(x 2 q)
where a Þ 0 and p and q are the x-intercepts. The axis of symmetry is
halfway between (p, 0) and (q, 0). The parabola opens up if a . 0 and
opens down if a , 0.
EXAMPLE 1
Graph a quadratic function in intercept form
Graph y 5 2(x 2 1)(x 1 3).
Solution
STEP 1
Identify and plot the x-intercepts. Because p 5 1 and q 5 23,
the x-intercepts occur at the points (1, 0) and (23, 0).
STEP 2
Find and draw the axis of symmetry:
p1q
y
( 3, 0)
1 1 (23)
O
1 (1, 0)
x
5}
5 21
x5}
2
2
FOCUS ON 10.2
STEP 3
Find and plot the vertex.
To find the y-coordinate of the vertex,
substitute 21 for x and simplify.
( 1, 8)
y 5 2(21 2 1)(21 1 3) 5 28
So, the vertex is (21, 28).
STEP 4
Draw a parabola through the vertex and the points where
the x-intercepts occur.
Exercises for Example 1
Graph the quadratic function. Label the vertex, axis of symmetry, and
x-intercepts. Identify the domain and range of the function.
1. y 5 22(x 1 3)(x 2 3)
200
Algebra 1
Chapter 10 Resource Book
2. y = 4(x 1 2)(x 2 4)
Copyright © Holt McDougal. All rights reserved.
The x-coordinate of the vertex is 21.
Name ———————————————————————
FOCUS ON
10.2
Review for Mastery
For use with pages 663–664
EXAMPLE 2
Date ————————————
continued
Graph a quadratic function
Graph y 5 3x 2 2 3.
Solution
STEP 1
Rewrite the quadratic function in intercept form.
y 5 3x2 2 3
Write original function.
5 3(x2 2 1)
Factor out common factor.
5 3(x 1 1)(x 2 1)
Difference of two squares pattern
STEP 2
Identify and plot the x-intercepts. Because p 5 21 and q 5 1,
the x-intercepts occur at the points (21, 0) and (1, 0).
STEP 3
Find and draw the axis of symmetry.
p1q
Y
21 1 1
5}
50
x5}
2
2
/
X
STEP 4
FOCUS ON 10.2
Copyright © Holt McDougal. All rights reserved.
Find and plot the vertex.
The x-coordinate of the vertex is 0.
The y-coordinate of the vertex is:
y 5 3(0)2 2 3 5 23
So, the vertex is (0, 23).
STEP 5
Draw a parabola through the vertex and the points where
the x-intercepts occur.
Exercises for Example 2
Graph the quadratic function. Label the vertex, axis of symmetry, and
x-intercepts. Identify the domain and range of the function.
3. y 5 3x2 2 12
4. y 5 23x2 1 12x
Algebra 1
Chapter 10 Resource Book
201
Name ———————————————————————
Graphing Calculator Activity Keystrokes
LESSON
10.3
For use with pages 672 and 673
TI-83 Plus
Casio CFX-9850GC Plus
Example 1
Y=
�
Example 1
2
(�)
7
x
X,T,�,n
6
(�)
WINDOW
2
6
�
X,T,�,n
From the main menu, choose GRAPH.
(�)
ENTER
2
X,,T
10
ENTER
1
ENTER
(�)
20
ENTER
EXE
SHIFT
20
ENTER
2
ENTER
2nd
[CALC] 4
EXE
(�)
(�)
4
ENTER
1
(�)
ENTER
1
ENTER
Y=
5
ENTER
10
2
ENTER
WINDOW
1
ENTER
2nd
x2
X,T,�,n
ENTER
ENTER
.5
F6
EXIT
6
�
F3
(�)
6
EXE
20
EXE
F5
SHIFT
X,,T
�
10
EXE
EXE
2
7
1
EXE
F2
(�)
(�)
ENTER
1
1
2nd
3
5
(�)
10
[CALC] 2
ENTER
[CALC] 2 0
From the main menu, choose GRAPH.
2
�
(�)
ENTER
1.5
x2
X,,T
SHIFT
EXE
EXIT
(�)
F3
(�)
F6
2
�
10
5
EXE
SHIFT
X,,T
EXE
10
F5
5
EXE
4
�
EXE
1
EXE
1
EXE
F1
2
ENTER
LESSON 10.3
Copyright © Holt McDougal. All rights reserved.
ENTER
4
�
ENTER
(�)
3
CLEAR
X,T,�,n
20
x2
Example 2
Example 2
202
Date ————————————
Algebra 1
Chapter 10 Resource Book
Name ———————————————————————
LESSON
10.3
Date ————————————
Practice A
For use with pages 665–673
Write the equation in standard form.
1. x 2 1 3x 5 212
2. x 2 2 8x 5 14
3. x 2 5 9x 2 1
4. x 2 5 6 2 10x
5. 14 2 x 2 5 3x
1
6. } x 2 5 23x 2 7
2
Determine whether the given value is a solution of the equation.
7. x 2 1 36 5 0; 26
8. 100 2 x 2 5 0; 210
10. x 2 2 5x 1 6 5 0; 2
11. 2x 2 1 4x 2 4 5 0; 4
9. 0 5 x 2 1 6x 1 5; 21
12. 0 5 2x 2 1 8x 1 3; 8
Use the graph to find the solutions of the given equation.
13. x 2 1 5 5 0
14. 2x 2 1 4 5 0
y
15. x 2 1 4x 1 3 5 0
y
y
5
3
3
3
1
1
23
23
21
1
25
21
1
1 x
21
3 x
3 x
16. x 2 2 16 5 0
17. x 2 2 2 5 0
y
18. x 2 1 2x 2 8 5 0
y
y
1
23
12 x
212
21
21
1
2
3 x
26
x
22
22
26
25
Solve the equation by graphing.
19. 8x 2 1 2x 1 3 5 0
y
23
1
21. } x 2 1 4x 1 6 5 0
2
20. 2x 2 1 3x 1 1 5 0
y
y
15
5
6
9
3
2
3
1
21
1
3 x
23
21
26
1
22
22
2
x
LESSON 10.3
Copyright © Holt McDougal. All rights reserved.
4
3 x
Algebra 1
Chapter 10 Resource Book
203
Name ———————————————————————
Date ————————————
Practice A
LESSON
10.3
For use with pages 665–673
22. x 2 2 2x 2 15 5 0
23. 22x 2 1 x 2 3 5 0
y
y
3
y
3
23
23
29
24. 2x 2 2 2x 1 3 5 0
9 x
3
21
23
23
3
3 x
1
1
29
29
215
215
23
21
21
1
x
Find the zeros of the function by graphing the function.
26. f (x) 5 2x 2 1 9
y
y
5
215
25
25
15 x
5
29
215
y
9
3
3
1
23
23
9 x
3
23
21
21
29
225
28. f(x) 5 x 2 2 4x 2 12
y
6
x
3 x
30. f (x) 5 3x 2 2 30x
y
y
50
2
1
23
29. f (x) 5 2x 2 2 3x 1 40
4
22
24
27. f (x) 5 2x 2 1 4x
22
212
2
6
10 x
30
236
212
220
10
26
22
210
2
x
260
204
in the shape of a parabola. The cross section can be modeled by
the function y 5 20.1875x 2 1 3x where x is the width of the
cover (in inches) and y is the height of the cover (in inches).
a. Graph the function.
b. Find the domain and range of the function in this situation.
c. How wide is the cover?
d. How tall is the cover?
Algebra 1
Chapter 10 Resource Book
Height (inches)
LESSON 10.3
31. Plate Cover A plate cover made of netting has a cross section
y
12
10
8
6
4
2
0
0 2 4 6 8 10 12 14 16 x
Width (inches)
Copyright © Holt McDougal. All rights reserved.
25. f(x) 5 x 2 2 25
Name ———————————————————————
Date ————————————
Practice B
LESSON
10.3
For use with pages 665–673
Determine whether the given value is a solution of the equation.
1. x 2 2 2x 1 15 5 0; 3
2. x 2 2 4x 2 12 5 0; 2
3. 2x 2 2 5x 2 6 5 0; 3
4. x 2 1 3x 2 4 5 0; 1
5. 2x 2 1 9x 2 5 5 0; 22
6. 3x 2 2 5x 2 2 5 0; 2
Use the graph to find the solutions of the given equation.
7. x 2 1 8x 1 16 5 0
8. 2x 2 1 36 5 0
y
20
9. x 2 1 5x 2 24 5 0
y
y
6
26
12
22
26
2
x
218
24
24
212
6
x
4
18 x
218
10. x 2 1 11x 1 30 5 0
11. x 2 2 25 5 0
y
12. x 2 1 7 5 0
y
y
5
10
15 x
215
6
6
2
26
2 x
22
26
22
6 x
2
Solve the equation by graphing.
13. 2x 2 2 6x 5 0
14. 2x 2 5 2
15. x 2 2 7x 1 10 5 0
y
y
y
3
10
1
6
3 x
23
22
22
2
2
x
22
22
23
16. x 2 5 10x
17. x 2 2 6x 1 9 5 0
y
15
x
3
29
6
215
225
y
10
5
22
22
23
23
9 x
29
2
26
x
18. 2x 2 1 9x 5 18
y
5
25
25
6
LESSON 10.3
Copyright © Holt McDougal. All rights reserved.
210
2
225
2
6 x
215
Algebra 1
Chapter 10 Resource Book
205
Name ———————————————————————
LESSON
10.3
Date ————————————
Practice B
For use with pages 665–673
Find the zeros of the function by graphing.
19. f(x) 5 2x 2 2 5x 2 10
20. f (x) 5 x 2 1 12x 1 36
y
y
y
2
26
22
22
2
218
22. f(x) 5 x 2 2 49
218
30
6 x
210
26
6
y
26
212
y
230
y
26
22
22
6 x
23
21
18 x
24. f (x) 5 3x 2 1 12x
3
2
6
18 x
23. f (x) 5 2x 2 1 1
10
22
210
21. f (x) 5 2x 2 1 24x
2
6 x
3 x
building 50 feet above the ground onto a pad on the ground
below. The stunt double jumps with an initial vertical velocity
of 10 feet per second.
a. Write and graph a function that models the height h (in feet)
of the stunt double t seconds after she jumps.
b. How long does it take the stunt double to reach the ground?
Height (feet)
25. Stunt Double A movie stunt double jumps from the top of a
LESSON 10.3
206
wastebasket from a height of about 1.3 feet above the
floor with an initial vertical velocity of 3 feet per second.
a. Write and graph a function that models the height h (in feet)
of the paper t seconds after it is thrown.
b. If you miss the wastebasket and the paper hits the floor, how
long does it take for the ball of paper to reach the floor?
c. If the ball of paper hits the rim of the wastebasket one-half
foot above the ground, how long was the ball in the air?
Algebra 1
Chapter 10 Resource Book
Height (feet)
26. Wastebasket You throw a wad of used paper towards a
h
50
40
30
20
10
0
h
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
0
0.5
1.0
1.5
Time (seconds)
2.0 t
0
0.1
0.2
0.3
Time (seconds)
0.4 t
Copyright © Holt McDougal. All rights reserved.
23
Name ———————————————————————
Date ————————————
Practice C
LESSON
10.3
For use with pages 665–673
Solve the equation by graphing.
1. x 2 5 4
2. x 2 1 3x 5 4
y
3. 2x 2 2 14x 2 49 5 0
y
y
5
6
1
23
21
21
210
2
3 x
1
26
22
22
22
25
26
6 x
2
x
215
23
26
4. 2x 2 1 6x 1 16 5 0
225
5. x 2 1 10x 1 25 5 0
y
6. x 2 1 8x 1 15 5 0
y
y
25
15
15
9
5
3
20
12
4
215
22
2
6
x
7. x 2 1 2 5 0
25
25
15 x
5
8. x 2 5 4x 1 12
y
25
23
21
23
x
9. 2x 2 1 25 5 0
y
y
25
3
4
23
21
21
26
1
22
24
3 x
15
6 x
2
5
212
215
23
25
25
5
15 x
Find the zeros of the function by graphing.
10. f (x) 5 2x 2 2 8x 2 10
11. f (x) 5 23x 2 2 6x 1 24
y
12. f (x) 5 4x 2 2 4x 2 8
y
y
30
4
21
24
2
1
3
18
5 x
23
21
22
1
3 x
6
212
23
21
26
1
x
26
Algebra 1
Chapter 10 Resource Book
LESSON 10.3
Copyright © Holt McDougal. All rights reserved.
1
207
Name ———————————————————————
LESSON
10.3
Date ————————————
Practice C
For use with pages 665–673
Approximate the zeros of the function to the nearest tenth by graphing.
13. f(x) 5 22x 2 1 5x 1 1
14. f (x) 5 3x 2 2 5
y
15. f (x) 5 4x 2 2 3x 2 4
y
y
1
3
23
21
21
1
1
3 x
23
1
23
21
21
1
3 x
21
21
3 x
1
23
23
25
Use the given surface area S of the cylinder to find the radius r to the nearest tenth.
(Use 3.14 for π.)
17. S 5 58 ft2
18. S 5 1356 cm2
r
r
6 in.
r
3 ft
12 cm
It jumps with an initial vertical velocity of 5 feet per second.
a. Write and graph a function that models the height h (in feet) of
the cat t seconds after it jumps. Explain how you got your model.
b. How far above the ground is the cat after one half of a second?
c. How long does it take the cat to reach the ground?
Height (feet)
19. Jumping A cat jumps from a countertop 30 inches above the floor.
208
height of 6 feet with an initial vertical velocity of 50 feet per second.
a. Write and graph a function that models the height h (in feet) of
the ball t seconds after it is thrown.
b. If the player misses the hoop completely and the ball lands on
the ground, how long was the ball in the air?
c. If an opposing player catches the ball at a height of 5 feet, how
long was the ball in the air? Explain your reasoning.
Algebra 1
Chapter 10 Resource Book
Height (feet)
LESSON 10.3
20. Basketball A basketball player throws a ball towards a hoop at a
h
2.5
2.0
1.5
1.0
0.5
0
0
h
40
30
20
10
0
0
0.2
0.4
0.6 t
Time (seconds)
1
2
3 t
Time (seconds)
Copyright © Holt McDougal. All rights reserved.
16. S 5 301 in.2
Name ———————————————————————
LESSON
10.3
Date ————————————
Review for Mastery
For use with pages 665–673
GOAL
Solve quadratic equations by graphing.
Vocabulary
A quadratic equation is an equation that can be written in the
standard form ax 2 1 bx 1 c 5 0 where a Þ 0 and a is called
the leading coefficient.
EXAMPLE 1
Solve a quadratic equation having two solutions
Solve x 2 1 5x 5 14 by graphing.
x 5 27 y
Solution
STEP 1
x 2 1 5x 5 14
x 2 1 5x 2 14 5 0
STEP 2
22
22
Write the equation in standard form.
x52
x
26
Write original equation.
Subtract 14 from
each side.
210
214
Graph the function y 5 x 2 1 5x 2 14.
The x-intercepts are 27 and 2.
The solutions of the equation x 2 1 5x 5 14
are 27 and 2.
You can check 27 and 2 in the original equation.
x 2 1 5x 5 14
x 2 1 5x 5 14
(27)2 1 5(27) 0 14
(2)2 1 5(2) 0 14
14 5 14 ✓
EXAMPLE 2
14 5 14 ✓
Write original equation.
Substitute for x.
Simplify. Each solution checks.
Solve a quadratic equation having one solution
Solve x 2 1 25 5 10x by graphing.
Solution
y
STEP 1
Write the equation in standard form.
x 2 1 25 5 10x
2
x 2 10x 1 25 5 0
STEP 2
Write original equation.
Subtract 10x from
each side.
Graph the function y 5 x 2 2 10x 1 25.
The x-intercept is 5.
LESSON 10.3
Copyright © Holt McDougal. All rights reserved.
CHECK
10
6
x55
2
2
6
10
x
The solution of the equation x 2 1 25 5 10x is 5.
Algebra 1
Chapter 10 Resource Book
209
Name ———————————————————————
LESSON
10.3
Review for Mastery
For use with pages 665–673
EXAMPLE 3
Date ————————————
continued
Solve a quadratic equation having no solution
Solve x2 1 11 5 5x by graphing.
Solution
STEP 1
x 2 1 11 5 5x
x2 2 5x 1 11 5 0
STEP 2
y
Write the equation in standard form.
Write original equation.
10
Subtract 5x from each side.
6
Graph the function y 5 x 2 2 5x 1 11.
2
The graph has no x-intercepts.
22
The equation x2 1 11 5 5x has no solution.
2
6
x
Exercises for Examples 1, 2, and 3
Solve the equation by graphing.
1. x 2 5 2x 1 15
2. x 2 1 4 5 24x
3. x 2 1 6x 5 24
Find the zeros of a quadratic function
Find the zeros of f(x) 5 x 2 2 10x 1 24.
Solution
Graph the function f(x) 5 x 2 2 10x 1 24.
The x-intercepts are 4 and 6.
y
10
The zeros of the function are 4 and 6.
6
x56
2
LESSON 10.3
x54
Exercises for Example 4
Find the zeros of the function.
4. f (x) 5 x 2 2 4
5. f (x) 5 x 2 1 5x 2 14
210
Algebra 1
Chapter 10 Resource Book
10
x
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 4
Name ———————————————————————
LESSON
10.3
Date ————————————
Challenge Practice
For use with pages 665–673
In Exercises 1–5, graph each quadratic function on the same coordinate
system and use the graph to identify the points of intersection.
1. y 5 3x 2 1 1
y 5 2x 2 1 5
1
2. y 5 } x 2 2 1
2
1
y 5 2} x 2 1 8
2
1
3. y 5 2x 2 2 }
2
7
y 5 x2 1 }
2
4. y 5 2x 2 1 4x 1 3
y 5 x2 1 x 1 3
5. y 5 2x 2 1 3x 1 1
y 5 22x 2 2 3x 1 1
In Exercises 6–8, use the following information.
A batter hits a baseball in such a way that its path is described by the quadratic function
A fence of varying height surrounds the baseball field. Given the information in the
exercise, determine whether the ball goes over the fence, hits the fence, or hits the
ground before reaching the fence.
6. The fence is 380 feet away from the batter, and the fence is 10 feet high.
7. The fence is 410 feet away from the batter, and the fence is 5 feet high.
8. The fence is 360 feet away from the batter, and the fence is 15 feet high.
LESSON 10.3
Copyright © Holt McDougal. All rights reserved.
y 5 20.00126875x 2 1 0.5x 1 3.
Algebra 1
Chapter 10 Resource Book
211
Name ———————————————————————
Practice A
LESSON
10.4
LESSON 10.4
Date ————————————
For use with pages 674–680
Evaluate the expression.
}
}
1. Ï 49
}
2. Ï 225
3. Ï 100
Isolate the variable in the equation.
4. 9x 2 2 18 5 0
5. 4x 2 2 12 5 0
6. 10x 2 2 40 5 0
8. x 2 2 9 5 0
9. 5x 2 5 20
Solve the equation.
7. x 2 5 36
10. 5x 2 2 45 5 0
11. 2x 2 2 18 5 0
12. 3x 2 2 12x 5 0
Evaluate the expression. Round your answer to the nearest hundredth.
}
}
}
14. Ï 10
13. Ï 5
15. Ï 12
Solve the equation. Round the solutions to the nearest hundredth.
16. x 2 5 8
17. x 2 2 3 5 0
18. 7x 2 2 14 5 0
Use the given area A of the circle to find the radius r or the diameter d of
the circle. Round the answer to the nearest hundredth, if necessary.
20. A = 121π in.2
r
21. A = 23π cm2
d
r
22. Boat Racing The maximum speed s (in knots or nautical miles per hour) that some
16
kinds of boats can travel can be modeled by s 2 5 }
x where x is the length of the
9
water line in feet. Find the maximum speed of a sailboat with a 20-foot water line.
Round your answer to the nearest hundredth.
23. Stockpile You can find the radius r (in inches) of a cylindrical air compressor
1
receiver tank by using the formula c 5 }
hr 2 where h is the height of the tank
73.53
(in inches) and c is the capacity of the tank (in gallons). Find the tank radius of each
tank in the table. Round your answers to the nearest inch.
212
Tank
Height (in.)
Radius (in.)
Capacity (in.3)
A
24
?
12
B
36
?
24
C
48
?
65
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
19. A = 25π m2
Name ———————————————————————
LESSON
10.4
Date ————————————
Practice B
For use with pages 674–680
1. 6x 2 2 24 5 0
2. 8x 2 2 128 5 0
3. x 2 2 13 5 23
4. 3x 2 2 60 5 87
5. 2x 2 2 33 5 17
6. 5x 2 2 200 5 205
7. 4x 2 2 125 5 225
8. 7x 2 2 50 5 13
1
1
9. } x 2 2 } 5 0
2
2
LESSON 10.4
Solve the equation.
Solve the equation. Round the solutions to the nearest hundredth.
10. x 2 1 15 5 23
11. x 2 2 16 5 213
12. 12 2 x 2 5 17
13. 3x 2 2 8 5 7
14. 9 2 x 2 5 9
15. 4 1 5x 2 5 34
16. 48 5 14 1 2x 2
17. 8x 2 5 50
18. 3x 2 1 23 5 18
19. (x 2 3)2 5 5
20. (x 1 2)2 5 10
21. 3(x 2 4)2 5 18
Use the given area A of the circle to find the radius r or the diameter d of
the circle. Round the answer to the nearest hundredth, if necessary.
22. A 5 169π m2
23. A 5 38π in.2
r
24. A 5 45π cm2
d
r
Copyright © Holt McDougal. All rights reserved.
25. Flower Seed A manufacturer is making a cylindrical can that will hold
and dispense flower seeds through small holes in the top of the can.
The manufacturer wants the can to have a volume of 42 cubic inches
and be 6 inches tall. What should the diameter of the can be? (Hint: Use
the formula for volume, V = πr 2h, where V is the volume, r is the radius,
and h is the height.) Round your answer to the nearest inch.
6 in.
26. Stockpile You can find the diameter D (in feet) of a conical pile of sand,
dirt, etc. by using the formula V 5 0.2618hD 2 where h is the height of the
pile (in feet) and V is the volume of the pile (in cubic feet). Find the diameter
of each stockpile in the table. Round your answers to the nearest foot.
Stockpile
Height (ft)
Diameter (ft)
Volume (ft3)
A
10
?
68
B
15
?
230
C
20
?
545
Algebra 1
Chapter 10 Resource Book
213
Name ———————————————————————
LESSON
LESSON 10.4
10.4
Date ————————————
Practice C
For use with pages 674–680
Solve the equation.
1. 4x 2 2 29 5 7
2. 2x 2 2 50 5 48
3. 5x 2 2 120 5 240
1
4. } x 2 2 2 5 0
2
1
5. } x 2 2 8 5 4
3
6. 0.1x 2 2 6.4 5 0
Solve the equation. Round the solutions to the nearest hundredth.
7. 4x 2 2 8 5 12
8. 7x 2 2 43 5 34
9. 2x 2 1 7 5 1
10. 3x 2 1 23 5 74
11. 6x 2 2 27 5 9
12. 5(x 2 8)2 5 15
13. 4(x 1 9)2 5 24
1
14. } (x 2 4)2 5 7
2
3
15. } (x 1 7)2 5 9
4
2
16. } (x 2 4)2 5 16
5
17. 7x 2 2 34 5 2x 2 1 16
18. 24 5 3(x 2 1 7)
19. 9x 2 1 3 5 4(3x 2 2 6)
20.
2
x24
1}
5 2
5 36
21.
(16x 2 2 8)2 5 81
Solve the equation without graphing.
22. x 2 1 6x 1 9 5 16
23. x 2 2 4x 1 4 5 100
24. x 2 2 10x 1 25 5 121
25. 2x 2 2 28x 1 98 5 72
26. 23x 2 1 6x 2 3 5 227
1
27. } x 2 1 4x 1 8 5 8
2
hold granulated plant food. The manufacturer wants the canister to
have a volume of 2036 cubic centimeters and be 18 centimeters tall.
What should the diameter of the canister be? (Hint: Use the formula
for volume, V = πr 2h, where V is the volume, r is the radius, and h is
the height.) Round your answer to the nearest centimeter.
18 cm
29. Speed To estimate the speed s (in feet per second) of a car involved in an accident,
}
11 3
Ï
} l where l represents the length (in feet)
investigators use the formula s 5 }
2 4
of tire skid marks on the pavement. After an accident, an investigator measures skid
marks that are 180 feet long. Approximately how fast was the car traveling?
214
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
28. Plant Food A manufacturer is making a cylindrical canister that will
Name ———————————————————————
LESSON
10.4
Date ————————————
Review for Mastery
For use with pages 674–680
EXAMPLE 1
Solve a quadratic equation by finding square roots.
LESSON 10.4
GOAL
Solve quadratic equations
Solve the equation.
a. x 2 2 7 5 9
b.
11y 2 5 11
c. z 2 1 13 5 5
Solution
a. x 2 2 7 5 9
Write original equation.
x 2 5 16
Add 7 to each side.
}
x 5 6Ï 16
Take square roots of each side.
Simplify.
5 64
The solutions are 24 and 4.
b. 11y 2 5 11
Write original equation.
y2 5 1
Divide each side by 11.
}
y 5 6Ï 1
Take square roots of each side.
Simplify.
5 61
The solutions are 21 and 1.
c. z 2 1 13 5 5
Write original equation.
Subtract 13 from each side.
z 5 28
Negative real numbers do not have real square roots. So, there is
no solution.
Copyright © Holt McDougal. All rights reserved.
2
EXAMPLE 2
Take square roots of a fraction
Solve 9m 2 5 169.
Solution
9m2 5 169
Write original equation.
169
m2 5 }
9
Divide each side by 9.
}
Ï
169
Take square roots of each side.
m56 }
9
13
Simplify.
m 5 6}
3
13
13
The solutions are 2}
and }
.
3
3
Algebra 1
Chapter 10 Resource Book
215
Name ———————————————————————
LESSON
LESSON 10.4
10.4
Review for Mastery
For use with pages 674–680
EXAMPLE 3
Date ————————————
continued
Approximate solutions of a quadratic equation
Solve 2x 2 1 5 5 15. Round the solutions to the nearest hundredth.
Solution
2x 2 1 5 5 15
Write original equation.
2x 2 5 10
Subtract 5 from each side.
x2 5 5
Divide each side by 2.
}
x 5 ± Ï5
Take square roots of each side.
x ≈ ±2.24
Use a calculator. Round to the nearest hundredth.
The solutions are about 22.24 and about 2.24.
Exercises for Examples 1, 2, and 3
Solve the equation.
1. w 2 2 9 5 0
2.
4r 2 2 7 5 9
3. 5s 2 1 13 5 9
4. 36x 2 5 121
5.
16m2 1 81 5 81
6. 4q2 2 225 5 0
Solve the equation. Round the solutions to the nearest hundredth.
7. 7x 2 2 8 5 13
26y 2 1 15 5 215
9. 4z 2 1 7 5 12
Solve a quadratic equation
Solve 3(x 1 3)2 5 39. Round the solutions to the nearest hundredth.
Solution
3(x 1 3)2 5 39
Write original equation.
2
(x 1 3) 5 13
Divide each side by 3.
}
x 1 3 5 ± Ï 13
Take square roots of each side.
}
x 5 23 ± Ï 13
Subtract 3 from each side.
}
}
The solutions are 23 1 Ï 13 ø 0.61 and 23 2 Ï 13 ø 26.61.
Exercises for Example 4
Solve the equation.
10. 5(x 2 1)2 5 40
216
Algebra 1
Chapter 10 Resource Book
11.
2( y 1 4)2 5 18
12. 4(z 2 5)2 5 32
Copyright © Holt McDougal. All rights reserved.
EXAMPLE 4
8.
Name ———————————————————————
LESSONS
10.1–10.4
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 648–680
development expenditures for a company
from 1991 to 2003 can be modeled by the
function y 5 2x2 2 12x 1 3600 where y is
the expenditure (in thousands of dollars) and
x is the number of years since 1991.
a. In what year was the research and
development expenditure the least?
b. What was the lowest research and
development expenditure?
2. Multi-Step Problem Use the rectangle
below.
(8 2 x) in.
3x in.
a. Find the value of x that gives the
greatest possible area of the rectangle.
b. What is the greatest possible area of
the rectangle?
Copyright © Holt McDougal. All rights reserved.
3. Short Response For the period 1998–
2001, the number of oil spills O in U.S.
water can be modeled by the function
O 5 2256t 2 1 519t 1 8305 where t is the
number of years since 1998. Did the greatest
number of oil spills occur in 1999? Explain.
4. Open-Ended Write an equation that
models the height of an object being
dropped as a function of time. Use the
equation to determine the time it takes the
object to hit the ground.
5. Gridded Response The skid distance D
(in feet) a car travels after applying the
S2
30f
brakes is given by D 5 } where S is the
speed of the car (in miles per hour) at the
time of applying the brakes and f is the drag
factor of the road surface. A car skids for
75 feet on a road surface that has a drag
factor of 0.9. Find the speed (in miles per
hour) when the brakes were applied.
6. Extended Response You throw a
football twice into the air.
a. For your first throw, the ball is
released 6 feet above the ground with
an initial vertical velocity of 25 feet
per second. Use the vertical motion
model to write an equation for the
height h (in feet) of the football as a
function of time t (in seconds).
b. For your second throw, the ball is
released 5.5 feet above the ground
with an initial vertical velocity of
30 feet per second. Use the vertical
motion model to write an equation for
the height h (in feet) of the football as
a function of time t (in seconds).
c. If no one catches either throw, for
which of your throws is the ball in the
air longer? Explain.
7. Gridded Response
LESSON 10.4
1. Multi-Step Problem The research and
r ft
The volume of the cylinder
is 144π cubic feet. What is
the radius of the cylinder,
in feet?
9 ft
8. Extended Response Students are selling
T-shirts to raise money for a class trip. Last
year, when the students charged $8 per
T-shirt, they sold 100 T-shirts. The students
want to increase the cost per T-shirt. They
estimate that they will lose 5 sales for each
$1 increase in the cost per T-shirt. The
revenue R (in dollars) generated by selling
the T-shirts is given by the function
R 5 (8 1 n)(100 2 5n) where n is the
number of $1 increases.
a. Write the function in standard form.
b. Find the maximum value of the
function.
c. At what price should the T-shirts be
sold to generate the most revenue?
Explain your reasoning.
Algebra 1
Chapter 10 Resource Book
217
Name ———————————————————————
LESSON
LESSON 10.4
10.4
Date ————————————
Challenge Practice
For use with pages 674–680
In Exercises 1–5, solve the equation by writing the left side of the
equation as a perfect square, then use square roots to solve the problem.
1. x 2 1 6x 1 9 5 81
2. 4x 2 1 20x 1 25 5 16
1
3. } x 2 1 2x 1 4 5 0
4
4. 36x 2 1 12x 1 1 5 4
5. 49x 2 1 112x 1 64 5 25
In Exercises 6–8, use the following information.
A NASA mission plans to send a probe to a moon of a distant planet in our solar system.
The probe will orbit the moon at a height of 100 kilometers above the moon’s surface, then
fall out of orbit to the surface of the moon. Once the probe begins to fall to the surface of
1
the moon, its height is modeled by the equation h 5 2}4 t 2 1 100, where t is the time in
minutes and h is the height in kliometers.
6. Once the probe begins to fall, how many minutes pass until the probe hits the
surface of the moon?
7. A NASA scientist needs to know how many minutes pass between the time the probe
falls out of orbit until the probe is 64 kilometers above the surface of the moon. Find
the number of minutes to answer the scientist’s question.
of the moon it fires a rocket to temporarily stop the descent and then releases a
parachute. Once the parachute is released, the height of the probe is modeled by the
1
equation h 5 2}
t 2 1 64. Find the number of minutes between the release of the
16
parachute and the probe striking the surface of the moon.
218
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
8. Suppose that once the probe reaches a height of 64 kilometers above the surface
Name ———————————————————————
LESSON
10.5
Date ————————————
Practice A
For use with pages 685–690
Match the expression with the value of c that makes the expression
a perfect square trinomial.
1. x 2 1 8x 1 c
2. x 2 1 16x 1 c
3. x 2 1 4x 1 c
A. 4
B. 16
C. 64
Write the expression as a square of a binomial.
4. x 2 1 2x 1 1
5. x 2 2 14x 1 49
6. x 2 1 18x 1 81
7. x 2 2 4x 1 4
8. x 2 1 22x 1 121
9. x 2 2 24x 1 144
Find the value of c that makes the expression a perfect square trinomial.
Then write the expression as a square of a binomial.
11. x 2 2 8x 1 c
12. x 2 2 6x 1 c
13. x 2 1 22x 1 c
14. x 2 2 12x 1 c
15. x 2 1 20x 1 c
16. x 2 2 30x 1 c
17. x 2 1 26x 1 c
18. x 2 1 40x 1 c
19. x 2 1 3x 1 c
20. x 2 1 11x 1 c
21. x 2 2 7x 1 c
LESSON 10.5
10. x 2 2 10x 1 c
Solve the equation by completing the square. Round your solutions to the
nearest hundredth, if necessary.
22. x 2 1 6x 5 2
23. x 2 1 10x 5 1
24. x 2 2 4x 5 3
Copyright © Holt McDougal. All rights reserved.
25. Flight of an Arrow An arrow is shot into the air with an upward velocity of 64 feet
per second from a hill 32 feet high. The height h of the arrow (in feet) can be found
by using the model h 5 216t 2 1 64t 1 32 where t is the time (in seconds).
a. Write an equation that you can use to find when the arrow will be 64 feet above
the ground.
b. When will the arrow be 64 feet above the ground? Round your answer(s) to the
nearest hundredth.
c. Write and solve an equation that you can use to find when the arrow will be
32 feet above the ground.
26. Tile Floor You are tiling a floor so that it has marble in the center
and ceramic tile around the border as shown. The ceramic tile
border has a uniform width x (in feet). You have enough money
in your budget to purchase marble to cover 28 square feet.
a. Solve the equation 28 5 (12 2 2x)(15 2 2x) to find the width
of the border.
b. How many square feet of ceramic tile will you need for the
project? Explain how you found your answer.
x
x
x
12 ft
x
15 ft
Algebra 1
Chapter 10 Resource Book
219
Name ———————————————————————
LESSON
10.5
Date ————————————
Practice B
For use with pages 685–690
Find the value of c that makes the expression a perfect square trinomial.
Then write the expression as a square of a binomial.
1. x 2 1 12x 1 c
2. x 2 1 50x 1 c
3. x 2 2 26x 1 c
4. x 2 2 18x 1 c
5. x 2 1 13x 1 c
6. x 2 2 9x 1 c
7. x 2 2 11x 1 c
1
8. x 2 1 } x 1 c
2
6
9. x 2 2 } x 1 c
5
10. x 2 1 6x 5 1
11. x 2 1 4x 5 13
12. x 2 2 10x 5 15
13. x 2 1 8x 5 10
14. x 2 2 2x 2 7 5 0
15. x 2 2 12x 2 21 5 0
16. x 2 1 3x 2 2 5 0
17. x 2 1 5x 2 3 5 0
18. x 2 2 x 5 1
Find the value of x. Round your answer to the nearest hundredth,
if necessary.
19. Area of triangle 5 30 ft2
20. Area of rectangle 5 140 in.2
2x in.
x ft
(x 1 4) ft
(3x 2 1) in.
Copyright © Holt McDougal. All rights reserved.
LESSON 10.5
Solve the equation by completing the square. Round your solutions to the
nearest hundredth, if necessary.
21. Colorado The state of Colorado is almost perfectly rectangular, with its north
border 111 miles longer than its west border. If the state encompasses
104,000 square miles, estimate the dimensions of Colorado. Round your
answer to the nearest mile.
22. Baseball After a baseball is hit, the height h (in feet) of the ball above the ground
t seconds after it is hit can be approximated by the equation h 5 216t 2 1 64t 1 3.
Determine how long it will take for the ball to hit the ground. Round your answer
to the nearest hundredth.
23. Fenced-In Yard You have 60 feet of fencing to fence in part of
your backyard for your dog. You want to make sure that your dog
has 400 square feet of space to run around in. The back of your
house will be used as one side of the enclosure as shown.
a. Write equations in terms of l and w for the amount of fencing
and the area of the enclosure.
b. Use substitution to solve the system of equations from part (a).
What are the possible lengths and widths of the enclosure?
220
Algebra 1
Chapter 10 Resource Book
House
w
w
l
Name ———————————————————————
LESSON
10.5
Date ————————————
Practice C
For use with pages 685–690
Find the value of c that makes the expression a perfect square trinomial.
Then write the expression as a square of a binomial.
1
2. x 2 2 } x 1 c
4
1. x 2 1 3.6x 1 c
2
3. x 2 1 } x 1 c
3
Solve the equation by completing the square. Round your solutions to the
nearest hundredth, if necessary.
7
4. x 2 2 3x 5 }
4
15
5. x 2 1 11x 5 2 }
4
1
6. x 2 2 } x 5 8
3
7. x 2 2 9x 2 8 5 0
8. x 2 2 5x 1 1 5 0
3
9. x 2 1 7x 1 } 5 0
4
11. 2x 2 1 36x 1 12 5 0
12. 3x 2 2 42x 1 30 5 0
13. 2x 2 1 18x 1 5 5 3
14. 3x 2 2 15x 2 10 5 9
15. 4x 2 1 4x 2 9 5 0
LESSON 10.5
10. 2x 2 2 10x 2 16 5 0
Find the value of x. Round your answer to the nearest hundredth,
if necessary.
16. Area of triangle 5 52 ft2
17. Area of rectangle 5 180 in.2
(x 1 5) ft
Copyright © Holt McDougal. All rights reserved.
(x 1 6) ft
2x in.
(2x 1 3) in.
18. The product of two consecutive negative integers is 240. Find the integers.
19. Stopping Distance A car with good tire tread can stop in less distance than a car
with poor tire tread. The formula for the stopping distance d (in feet) of a car with
good tread on dry cement is approximated by d 5 0.04v 2 1 0.5v where v is the
speed of the car (in miles per hour). If the driver must be able to stop within 80 feet,
what is the maximum safe speed of the car? Round your answer to the nearest mile
per hour.
20. Day Care A day care center has 100 feet of fencing to fence in part
of its land for a safe play area for the children. The people that run
the center figure that they will need 1000 square feet of space for
the play area. One side of the day care building will be used as one
side of the play area as shown.
a. Write equations for the length of the fencing and the area
of the play area.
b. Use substitution to solve the system of equations from part (a).
What are the possible lengths and widths of the play area?
Building
w
w
l
Algebra 1
Chapter 10 Resource Book
221
Name ———————————————————————
LESSON
10.5
Date ————————————
Review for Mastery
For use with pages 685–690
GOAL
Solve quadratic equations by completing the square.
Vocabulary
For an expression of the form x 2 1 bx, you can add a constant c to
the expression so that the expression x 2 1 bx 1 c is a perfect square
trinomial. This process is called completing the square.
EXAMPLE 1
Complete the square
LESSON 10.5
Find the value of c that makes the expression x 2 1 7x 1 c a perfect
square trinomial. Then write the expression as the square of a binomial.
Solution
Find the value of c. For the expression to be a perfect square trinomial,
c needs to be the square of half the coefficient of x.
7 2
49
c 5 1 }2 2 5 }
4
STEP 2
Find the square of half the coefficient of x.
Write the expression as a perfect square trinomial. Then write the expression
as the square of a binomial.
49
x 2 1 7x 1 c 5 x 2 1 7x 1 }
4
7 2
5 1 x 1 }2 2
EXAMPLE 2
49
Substitute }
for c.
4
Square of a binomial.
Solve a quadratic equation
Solve x 2 1 14x 5 213 by completing the square.
Solution
x 2 1 14x 5 213
x 2 1 14x 1 (7)2 5 213 1 72
(x 1 7)2 5 213 1 72
2
(x 1 7) 5 36
x17566
x 5 27 6 6
Write original equation.
14 2
Add 1 }
, or 72, to each side.
22
Write left side as the square of a binomial.
Simplify the right side.
Take square roots of each side.
Subtract 7 from each side.
The solutions of the equation are 27 1 6 5 21 and 27 2 6 5 213.
222
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
LESSON
10.5
Review for Mastery
For use with pages 685–690
EXAMPLE 3
Date ————————————
continued
Solve a quadratic equation in standard form
Solve 3x2 1 18x 2 9 5 0 by completing the square. Round your
solutions to the nearest hundredth.
Solution
3x2 1 18x 2 9 5 0
Write original equation.
2
Add 9 to each side.
x2 1 6x 5 3
Divide each side by 3.
3x 1 18x 5 9
6 2
Add 1 }2 2 , or 32, to each side.
x2 1 6x 1 32 5 3 1 32
Write left side as the square of a binomial.
}
x 1 3 5 Ï 12
Take square roots of each side.
}
x 5 23 6 Ï 12
}
Subtract 3 from each side.
}
The solutions are 23 1 Ï 12 ø 0.46 and 23 2 Ï 12 ø 26.46.
LESSON 10.5
(x 1 3)2 5 12
Exercises for Examples 1, 2, and 3
Find the value of c that makes the expression a perfect square
trinomial. Then write the expression as the square of a binomial.
1. x 2 2 9x 1 c
Copyright © Holt McDougal. All rights reserved.
2. x 2 1 11x 1 c
3. x 2 2 16x 1 c
Solve the equation by completing the square. Round your solutions to
the nearest hundredth if necessary.
4. q2 2 8q 5 7
5. r 2 1 12r 5 23
6. 2s 2 2 28s 1 8 5 0
Algebra 1
Chapter 10 Resource Book
223
Name ———————————————————————
LESSON
10.5
Date ————————————
Challenge Practice
For use with pages 685–690
1. The product of two consecutive positive even integers is 224. Find the integers.
2. The product of two consecutive positive odd integers is 143. Find the integers.
3. The product of two consecutive positive integers is equal to eleven times the sum of
the two integers plus 35. Find the integers.
4. The sum of the squares of two consecutive positive integers is 421. Find the integers.
5. The sum of the squares of a positive integer and five more than twice the integer is
equal to 1810. Find the integer.
In Exercises 6–9, complete the square to solve for x.
LESSON 10.5
6. x 2 1 bx 1 5 5 12
7. x 2 2 5x 1 c 5 3
8.
x 2 1 bx 1 c 5 0
9. ax 2 1 bx 1 c 5 0
10. You are planning a vegetable garden and you lay out a rectangular design 10 feet
11.
224
The path of a rocket shot into the air is modeled by the equation h 5 225t2 1 50t 1 4
where h is the height (in feet) of the rocket above the ground t seconds after it is
launched. Find the number of seconds after launch it takes for the rocket to touch
back down to the ground. Round your answer to the nearest hundredth second.
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
wide by 20 feet long. After laying out the design you decide you want a larger
garden and decide to increase the length of the garden by a length of 2x feet and
increase the width by a length of x feet. You have enough dirt to cover an area of
600 square feet, and you want to make the garden as large as possible. What are the
dimensions of the finished garden? Round your answer to the nearest foot.
Name ———————————————————————
Date ————————————
Practice
FOCUS ON
10.5
For use with pages 691–692
Graph the quadratic function. Label the vertex and axis of symmetry.
1. y 5 (x 1 1)2 2 3
2. y 5 3(x 1 2)2 2 1
3. y 5 2(x 2 2)2 1 4
Y
Y
Y
X
/
/
X
/
Y
Y
X
X
X
/
/
X
FOCUS ON 10.5
Y
3
6. y 5 2} (x 2 2)2 1 2
2
1
5. y 5 } (x 2 2)2 2 3
2
4. y 5 22(x 1 4)2 1 2
/
Write the function in vertex form, then graph the function. Label the vertex
and axis of symmetry.
1
7. y 5 2x2 2 12x 1 2
8. y 5 24x2 2 2x 1 16
9. y 5 } x2 2 2x 2 1
2
Copyright © Holt McDougal. All rights reserved.
Y
Y
y
/
X
/
4
O
1
X
x
10. Write an equation in vertex form of the parabola
shown. Use the coordinates of the vertex and the
coordinates of a point on the graph to write
the equation.
y
1
1
1
3 1
(2 , 2)
( 2, 2 )
O
1
x
1
( 2 , 1)
11. Challenge The path of a soccer ball is parabolic. The ball reaches a height of
12 feet before it starts to descend and lands 32 feet from the point where it was
kicked. What is the equation, in vertex form, which models the path of the
soccer ball? Assume the ball was kicked at (0, 0).
Algebra 1
Chapter 10 Resource Book
225
Name ———————————————————————
FOCUS ON
10.5
Date ————————————
Review for Mastery
For use with pages 691–692
GOAL
Graph quadratic functions in vertex form.
Vocabulary
The vertex form of a quadratic function is y 5 a(x 2 h)2 1 k where
a Þ 0. The vertex of the graph is (h, k) and the axis of symmetry is
x 5 h. The parabola opens up if a . 0 and opens down if a , 0.
The graph of y 5 a(x 2 h)2 1 k is the graph of y 5 ax2 translated h
units horizontally and k units vertically.
Graph a quadratic function in vertex form
Graph y 5 2(x 2 1)2 2 2.
Solution
STEP 1
Identify the values of a, h, and k: a 5 2, h 5 1, and k 5 22.
Because a . 0, the parabola opens up.
STEP 2
Draw the axis of symmetry, x 5 1.
STEP 3
Plot the vertex (h, k) 5 (1, 22).
STEP 4
Plot four points. Evaluate the function for
two x-values less than the x-coordinate
of the vertex.
1
x 5 0: y 5 2(0 2 1)2 2 2 5 0
O
y
x
x 5 21: y 5 2(21 2 1)2 2 2 5 6
1
1
x
(1, 2)
Plot the points (0, 0) and (–1, 6) and their
reflections (2, 0) and (3, 6), in the axis of symmetry.
STEP 5
Draw a parabola through the plotted points.
Exercises for Example 1
Graph the quadratic function. Label the vertex and axis of symmetry.
1. y 5 3(x 1 1)2 2 5
226
Algebra 1
Chapter 10 Resource Book
2. y = 22(x 2 3)2 1 1
Copyright © Holt McDougal. All rights reserved.
FOCUS ON 10.5
EXAMPLE 1
Name ———————————————————————
FOCUS ON
10.5
Review for Mastery
Date ————————————
continued
For use with pages 691–692
EXAMPLE 2
Graph a quadratic function
Graph y 5 x 2 2 4x 1 1.
Solution
STEP 1
Write the function in vertex form by completing the square.
y 5 x2 2 4x 1 1
y1
5 x2 2 4x 1
Write original function.
11
4 2
Add 1 –}2 2 5 (–2)2 5 4 to each side.
y 1 4 5 (x 2 2)2 1 1
Write x2 2 4x 1 4 as a square of a binomial.
Subtract 4 from each side.
Identify the values of a, h, and k: a 5 1, h 5 2,
k 5 23. Because a . 0, the parabola opens up.
y
x
2
FOCUS ON 10.5
y 1 4 5 (x2 2 4x 1 4) 1 1
y 5 (x 2 2)2 2 3
STEP 2
Prepare to complete the square.
1
STEP 3
Draw the axis of symmetry, x 5 2.
STEP 4
Plot the vertex (h, k) 5 (2, 23).
STEP 5
Plot four more points. Evaluate the function
for two x-values less than the x-coordinate
of the vertex.
O
1
x
(2, 3)
Copyright © Holt McDougal. All rights reserved.
x 5 1: y 5 (1 2 2)2 2 3 5 22
x 5 0: y 5 (0 2 2)2 2 3 5 1
Plot the points (1, 22) and (0, 1) and their reflections (3, 22) and (4, 1),
in the axis of symmetry.
STEP 6
Draw a parabola through the plotted points.
Exercises for Example 2
Write the function in vertex form, then graph the function.
Label the vertex and the axis of symmetry.
3. y 5 22x2 2 8x 2 7
4. y 5 2x2 1 4x 2 1
Algebra 1
Chapter 10 Resource Book
227
Name ———————————————————————
LESSON
10.6
Date ————————————
Practice A
For use with pages 693–698
Identify the values of a, b, and c in the quadratic equation.
1. 5x 2 1 7x 1 1 5 0
2. 2x 2 2 6x 1 11 5 0
3. 2x 2 1 17x 2 23 5 0
4. 10x 2 2 8x 2 13 5 0
5. 23x 2 1 x 2 2 5 0
6. 5x 2 2 18x 2 3 5 0
Match the quadratic equation with the formula that gives its solution(s).
7. 2x 2 1 x 2 4 5 0
}}
24 6 Ï 42 2 4(21)(2)
A. x 5 }}
2(21)
8. 4x 2 2 x 1 2 5 0
}}
21 6 Ï 12 2 4(2)(24)
B. x 5 }}
2(2)
9. 2x 2 1 4x 1 2 5 0
}}
2(21) 6 Ï (21)2 2 4(4)(2)
C. x 5 }}}
2(4)
10. x 2 1 6x 2 10 5 0
11. x 2 2 4x 2 9 5 0
12. 5x 2 1 2x 2 3 5 0
13. x 2 1 8x 1 2 5 0
14. x 2 1 10x 1 1 5 0
15. 2x 2 2 3x 1 5 5 0
16. 3x 2 1 5x 2 2 5 0
17. 6x 2 2 2x 1 5 5 0
18. 2x 2 2 8x 1 3 5 0
19. 2x 2 1 4x 2 16 5 0
20. 23x 2 1 7x 2 2 5 0
21. 5x 2 2 2x 1 1 5 0
22. Nuts For the period 1990–2002, the amount of shelled nuts y (in millions
of pounds) imported into the United States can be modeled by the function
y 5 1.55x 2 2 5.1x 1 197 where x is the number of years since 1990.
a. Write and solve an equation that you can use to approximate the year in which
300 million pounds of nuts were imported.
b. Write and solve an equation that you can use to approximate the year in which
237 million pounds of nuts were imported.
23. Soybeans For the period 1995–2003, the number of acres y (in millions)
of soybeans harvested in the United States can be modeled by the function
y 5 20.31x 2 1 3.8x 1 61.6 where x is the number of years since 1995.
a. Write and solve an equation that you can use to approximate the year(s) in which
73 million acres of soybeans were harvested.
b. Graph the function on a graphing calculator. Use the trace feature to find the year
in which 73 million acres of soybeans were harvested. Use the graph to check
your answer from part (a).
228
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 10.6
Use the quadratic formula to solve the equation. Round your solutions to
the nearest hundredth, if necessary.
Name ———————————————————————
LESSON
10.6
Date ————————————
Practice B
For use with pages 693–698
Use the quadratic formula to solve the equation. Round your solutions to
the nearest hundredth, if necessary.
1. x 2 1 7x 2 80 5 0
2. 3x 2 2 x 2 16 5 0
3. 8x 2 2 2x 2 30 5 0
4. x 2 1 4x 1 1 5 0
5. 2x 2 1 x 1 12 5 0
6. 23x 2 2 4x 1 10 5 0
7. 5x 2 1 30x 1 32 5 0
8. x 2 1 6x 2 100 5 0
9. 4x 2 2 x 2 20 5 0
10. 5x 2 1 x 2 9 5 0
11. 6x 2 1 7x 2 3 5 0
12. 10x 2 2 7x 1 5 5 0
Tell which method(s) you would use to solve the quadratic equation.
Explain your choice(s).
13. 6x 2 2 216 5 0
14. 8x 2 5 56
15. 5x 2 2 10x 5 0
16. x 2 1 8x 1 7 5 0
17. x 2 2 6x 1 1 5 0
18. 29x 2 1 10x 5 5
Solve the quadratic equation using any method. Round your solutions
to the nearest hundredth, if necessary.
19. 210x 2 5 250
20. x 2 2 16x 5 264
21. x 2 1 3x 2 8 5 0
22. x 2 5 14x 2 49
23. x 2 1 6x 5 14
24. 25x 2 1 x 5 13
Copyright © Holt McDougal. All rights reserved.
(in thousands of metric tons) imported into the United States can be modeled by
the function y 5 1.36x2 1 27.8x 1 304 where x is the number of years since 1990.
a. Write and solve an equation that you can use to approximate the year in which
500 thousand metric tons of biscuits, pasta, and noodles were imported.
b. Write and solve an equation that you can use to approximate the year in which
575 thousand metric tons of biscuits, pasta, and noodles were imported.
LESSON 10.6
25. Pasta For the period 1990–2003, the amount of biscuits, pasta, and noodles y
26. Eggs For the period 1997–2003, the number of eggs y (in billions) produced in
the United States can be modeled by the function y 5 20.27x 2 1 3.3x 1 77 where
x is the number of years since 1997.
a. Write and solve an equation that you can use to approximate the year(s) in which
80 billion eggs were produced.
b. Graph the function on a graphing calculator. Use the trace feature to find the year
when 80 billion eggs were produced. Use the graph to check your answer from
part (a).
Algebra 1
Chapter 10 Resource Book
229
Name ———————————————————————
LESSON
10.6
Date ————————————
Practice C
For use with pages 693–698
Use the quadratic formula to solve the equation. Round your solutions to
the nearest hundredth, if necessary.
1. 15x 2 1 8x 1 1 5 0
2. 4x 2 2 6x 1 2 5 0
3. 9x 2 1 9x 2 1 5 0
4. x 2 2 6x 5 15
5. 4x 2 2 3 5 10x
6. 2x 2 1 6x 1 5 5 7
7. 8x 2 5 5x 2 1 9x 1 3
8. 212 5 x 2 2 14x 1 30
9. 5x 2 2 10x 2 16 5 4x
11. 6x 2 2 5x 5 3 2 5x 2
10. 10x 2 1 10 5 8 2 6x
12. 22x 2 2 x 1 4 5 2x 1 3
Tell which method(s) you would use to solve the quadratic equation.
Explain your choice(s).
13. 13x 2 2 26x 5 0
14. 2x 2 2 9x 1 5 5 0
15. x 2 2 8x 1 1 5 0
16. 23x 2 5 218
17. x 2 2 5x 1 10 5 0
18. x 2 1 3x 2 1 5 0
19. x 2 5 9x 2 81
20. x 2 1 6x 5 10
21. 25x 2 1 x 5 13
22. 10x 2 2 4 5 6x 2 1 5
23. 2x 2 2 18 5 x 2 1 12x
24. (x 1 9)2 5 64
25. Books For the period 1990–2002, the amount of money y (in billions of dollars)
spent in the United States on books and maps can be modeled by the function
y 5 0.0178x 2 1 1.5x 1 16 where x is the number of years since 1990.
a. Find the year in which 20 billion dollars were spent on books and maps.
b. Find the year in which 32 billion dollars were spent on books and maps.
c. Graph the function on a graphing calculator. Use the trace feature to check
your answers from parts (a) and (b).
26. Spectator Sports For the period 1990–2002, the amount of money y (in billions
of dollars) spent in the United States on admissions to spectator sports can be
modeled by the function y 5 0.0284x 2 1 0.388x 1 5 where x is the number of
years since 1990.
a. Find the year in which 7 billion dollars were spent.
b. Graph the function on a graphing calculator. Use the trace feature to find the
year in which 7 billion dollars were spent. Use the graph to check your answer
from part (a).
230
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 10.6
Solve the quadratic equation using any method. Round your solutions
to the nearest hundredth, if necessary.
Name ———————————————————————
LESSON
10.6
Date ————————————
Review for Mastery
For use with pages 693–698
GOAL
Solve quadratic equations using the quadratic formula.
Vocabulary
By completing the square for the quadratic equation ax2 1 bx 1 c 5 0,
}
2b 6 Ï b2 2 4ac
, that gives the
you can develop a formula, x 5 }}
2a
solutions of any quadratic equation in standard form. This formula
is called the quadratic formula.
EXAMPLE 1
Solve a quadratic equation
Solve 5x 2 2 3 5 4x.
Solution
5x 2 2 3 5 4x
Write original equation.
5x 2 2 4x 2 3 5 0
Write in standard form.
}
2b 6 Ï b2 2 4ac
Quadratic formula
x 5 }}
2a
}}
2(24) 6 Ï (24)2 2 4(5)(23)
5 }}}
2(5)
Substitute values in the quadratic formula:
a 5 5, b 5 24, and c 5 23.
Copyright © Holt McDougal. All rights reserved.
LESSON 10.6
}
4 6 Ï76
Simplify.
5}
10
}
}
4 1 Ï 76
4 2 Ï 76
ø 1.27 and }
ø 20.47.
The solutions are }
10
10
Exercises for Example 1
Use the quadratic formula to solve the equation. Round your solutions
to the nearest hundredth, if necessary.
1. x 2 2 12x 2 14 5 0
2. 5y 2 2 7 5 11y
3. 9z 2 1 3z 5 5
Algebra 1
Chapter 10 Resource Book
231
Name ———————————————————————
LESSON
10.6
Review for Mastery
For use with pages 693–698
EXAMPLE 2
Date ————————————
continued
Use the quadratic formula
Retirement Savings For the period 1995–2005, the amount of dollars
invested in an individual’s retirement account can be modeled by the
function y 5 30x2 2 24x 1 15,500 where x is the number of years since
1995. In what year was $17,000 invested?
Solution
y 5 30x 2 2 24x 1 15,500
Write function.
2
17,000 5 30x 2 24x 1 15,500
Substitute 17,000 for y.
0 5 30x 2 2 24x 2 1500
Write in standard form.
}}
2(224) 6 Ï (224)2 2 4(30)(21500)
2(30)
x 5 }}}
Substitute values in the quadratic formula:
a 5 30, b 5 224, and c 5 21500.
}
24 6 Ï180,576
5 }}
60
Simplify.
}
}
24 1 Ï 180,576
60
24 2 Ï 180,576
60
The solutions are }} ø 7 and }} ø 27.
The year when $17,000 is invested is about 7 years after 1995, or 2002.
Choose a solution method
Tell what method you would use to solve the quadratic equation.
Explain your choice(s).
a. 3x 2 1 13x 5 11
b.
x 2 1 8x 5 7
c. 4x 2 2 25 5 0
Solution
a. The quadratic equation cannot be factored easily, and completing
the square will result in many fractions. So, the equation can be
solved using the quadratic formula.
b. The quadratic equation can be solved by completing the square
because the equation can be rewritten in the form ax 2 1 bx 1 c 5 0
where a 5 1 and b is an even number.
c. The quadratic equation can be solved using square roots because
the equation can be written in the form x 2 5 d.
Exercises for Examples 2 and 3
4. In Example 2, find the year when $18,000 was invested.
Tell what method you would use to solve the quadratic equation.
Explain your choice(s).
5. x 2 1 11x 5 0
232
Algebra 1
Chapter 10 Resource Book
6.
23x 2 1 19x 5 27
7. 4x 2 1 16x 5 12
Copyright © Holt McDougal. All rights reserved.
LESSON 10.6
EXAMPLE 3
Name ———————————————————————
LESSON
10.6
Date ————————————
Problem Solving Workshop:
Using Alternative Methods
For use with pages 693–698
Another Way to Solve Example 3 on page 694
Multiple Representations In Example 3 on page 694, you saw how to solve a
problem about films produced in the world from 1971–2001 by using the quadratic formula.
You can also solve the problem by using a graph.
PROBLEM
Film Production For the period 1971–2001, the number y of films produced in the
world can be modeled by the function y 5 10x2 2 94x 1 3900 where x is the number
of years since 1971. In what year were 4200 films produced?
Using a Graph You can solve the problem by using a graph.
STEP 1
PRACTICE
10,000
8,000
6,000
4,000
2,000
0
Intersection
X=11.91734 Y=4200
0
5 10 15 20 25 30
Years since 1971
Find the intersection of the graphs by using the intersect feature on your
calculator. You only need to consider positive values of x because a negative
solution does not make sense in this situation. The intersection occurs at
(11.91734, 4200). There were 4200 films produced about 12 years after 1971,
or in 1983.
1. Cassettes For the period 1998–2003,
3. Diving Board A person jumps off of a
the number y of cassettes (in millions)
in manufacturers’ shipments can
be modeled by the function
y 5 4.3x2 2 50.4x 1 162 where x is
the number of years since 1998. In
what year were 50 million cassettes
shipped?
6-foot high diving board with an initial
velocity of 13 feet per second. How
many seconds does it take the person
to hit the water? Round your answer to
the nearest tenth of a second.
2. Error Analysis Describe and correct
the error made in Exercise 1.
}}
50.4 6 Ï(50.4)2 2 4(4.3)(162)
x 5 }}}
2(4.3)
}
50.4 6 Ï2246.24
x 5 }}
8.6
LESSON 10.6
Copyright © Holt McDougal. All rights reserved.
STEP 2
Graph the equation for the number of films
produced in the world using a graphing
calculator. Graph y1 5 10x 2 2 94x 1 3900.
Because you are looking for when the
number of films produced is 4200, graph
y2 5 4200 and find the intersection between
the graphs. You only need to consider
x-values between 0 and 30 because that is
the interval for the equation.
Number of films
METHOD
4. Federal Aid For the period
1998–2003, the amount of money y
(in billions of dollars) of federal aid
grants to state and local governments
can be modeled by the function
y 5 1.71x 2 1 19.14x 1 244.92 where
x is the number of years since 1998.
In what year was 290 billion dollars
given to state and local governments?
There was no time from 1998–2003
when 50 million cassettes were shipped.
Algebra 1
Chapter 10 Resource Book
233
Name ———————————————————————
LESSON
10.6
Date ————————————
Challenge Practice
For use with pages 693–698
In Exercises 1–5, the solution to a quadratic equation is given.
Write an equation in standard form that has the solution.
}
22 6 Ï3
Example: x 5 }
5
Solution: The solution to the quadratic equation ax 2 1 bx 1 c 5 0 is given by
}
5
2b 6 Ïb 2 2 4ac
x 5 }}
. Letting 2b 5 22 gives b 5 2, letting 2a 5 5 gives a 5 }2 ,
2a
b2 2 3
and letting b2 2 4ac 5 3 gives c 5 }
. Substituting the values for a and b you get
4a
5
1
1
c5}
. So the equation }2 x 2 1 2x 1 }
5 0 has the desired solutions.
10
10
}
24 6 Ï 10
1. x 5 }
3
}
26 6 Ï25
2. x 5 }
7
}
1 6 Ï0
3. x 5 }
3
}
217 6 Ï 21
4. x 5 }
15
}
In Exercises 6–8, use the following information.
If the graph of a parabola has x-intercepts, then the axis of symmetry of the parabola can
be found at the position that is the average of the two x-intercepts. Use this concept to find
the axis of symmetry for the parabola modeled by the equation.
6. y 5 3x 2 1 5x 1 2
7. y 5 2x2 2 4x 1 1
8. y 5 6x2 1 x 2 1
234
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 10.6
11 6 Ï11
5. x 5 }
11
Name ———————————————————————
LESSON
10.7
Date ————————————
Practice A
For use with pages 699–705
Identify the values of a, b, and c in the quadratic equation.
1. 2x 2 1 x 2 10 5 0
2. 4x 2 2 5x 1 2 5 0
3. x 2 2 8x 1 11 5 0
4. 2x 2 1 6x 2 3 5 0
5. 12 2 3x 2 x 2 5 0
6. 3x 2 2 4x 1 15 5 0
Find the discriminant of the quadratic equation.
7. x 2 1 3x 1 6 5 0
8. x 2 2 5x 1 12 5 0
9. x 2 2 2x 2 10 5 0
10. 3x 2 2 4x 1 1 5 0
11. 5x 2 1 x 1 4 5 0
12. 2x 2 1 8x 2 3 5 0
13. 24x 2 2 6x 1 3 5 0
14. 10x 2 2 3x 1 7 5 0
15. 2x 2 2 9x 2 3 5 0
Tell whether the equation has two solutions, one solution, or no solution.
16. 3x 2 1 x 1 1 5 0
17. 2x 2 1 5x 1 7 5 0
18. x 2 2 10x 1 8 5 0
19. 4x 2 1 x 2 6 5 0
20. 2x 2 2 5x 2 8 5 0
21. 26x 2 2 2x 1 7 5 0
22. 10x 2 1 12x 2 1 5 0
23. 8x 2 2 x 1 15 5 0
24. 3x 2 1 12x 1 12 5 0
Find the number of x-intercepts that the graph of the function has.
25. y 5 x 2 2 5x 2 3
26. y 5 3x 2 2 x 2 1
27. y 5 4x 2 1 6x 1 1
28. y 5 2x 2 2 7x 1 7
29. y 5 8x 2 2 4x 1 1
30. y 5 x 2 1 2x 1 1
10 ft
x
You have blueprints which show that the shed is 15 feet
long and 10 feet wide. You want to change the dimensions
as shown. The new area can be modeled by the function
y 5 2x 2 1 5x 1 150.
15 ft
a. Write an equation that you can use to determine if there
is a value of x that gives an area of 155 square feet.
b. Use the discriminant of your equation from part (a) to
show that it is possible to find a value of x for which the
area is 155 square feet.
c. Find the value(s) of x for which the area is 155 square feet. Round your answer(s)
to the nearest tenth.
15 2 x
32. House Painting You are painting a house. While standing on a ladder that is
15 feet above the ground, you ask your friend to toss you a paintbrush. The starting
height of the paintbrush is 5.5 feet and its initial vertical velocity is 20 feet per
second. Write an equation that you can use to determine whether or not the paintbrush reaches you. Then use the discriminant to determine whether the paintbrush
reaches you.
Algebra 1
Chapter 10 Resource Book
LESSON 10.7
Copyright © Holt McDougal. All rights reserved.
31. Blueprints You want to build a shed in your backyard.
235
Name ———————————————————————
LESSON
10.7
Date ————————————
Practice B
For use with pages 699–705
Tell whether the equation has two solutions, one solution, or no solution.
1. x 2 1 x 1 3 5 0
2. 2x 2 2 4x 2 5 5 0
3. 22x 2 1 10x 2 5 5 0
4. 3x 2 2 9x 1 8 5 0
5. 10x 2 2 8x 1 1 5 0
6. 24x 2 1 9 5 0
7. 36x 2 2 9x 5 0
8. 3x 2 1 2 5 4x
9. 12 5 x 2 2 6x
1
10. } x 2 1 3 5 x
6
2
11. 28x 2 2 9x 5 }
3
12. 8x 2 1 12x 1 2 5 4x
Find the number of x-intercepts that the graph of the function has.
13. y 5 x 2 2 6x 2 3
14. y 5 5x 2 2 x 2 1
15. y 5 6x 2 2 6x 1 1
16. y 5 x 2 1 x 1 6
17. y 5 24x 2 1 x 1 1
18. y 5 4x 2 1 5x 2 1
19. y 5 2x 2 2 4x 1 2
20. y 5 10x 2 2 5x 1 1
21. y 5 8x 2 1 x 1 4
22. y 5 215x 2 1 3x 1 5
1
23. y 5 } x 2 2 4x 1 8
2
2
24. y 5 } x 2 2 5x 1 2
3
Give a value of c for which the equation has (a) two solutions,
(b) one solution, and (c) no solution.
26. x 2 2 4x 1 c 5 0
27. 25x 2 1 10x 1 c 5 0
28. 49x 2 2 14x 1 c 5 0
29. 2x 2 1 4x 1 c 5 0
30. 3x 2 2 18x 1 c 5 0
31. Playhouse You want to build a playhouse for your sister in your
12 ft
LESSON 10.7
backyard. You have blueprints which show that the playhouse is
12 feet long and 13 feet wide. You want to change the dimensions
as shown. The new area can be modeled by the function
13 ft
y 5 2x 2 1 x 1 156.
a. Write an equation that you can use to determine if there is a
value of x that gives an area of 150 square feet.
b. Use the discriminant of your equation from part (a) to show
that it is possible to find a value of x for which the area is
150 square feet.
c. Find the value(s) of x for which the area is 150 square feet.
236
32. Tennis You and your friend are walking around the exterior of a tennis court that
has a 12-foot high fence around it. You pick up a ball and try to throw it from a
height of 5 feet over the fence. You throw it with an initial vertical velocity of 20 feet
per second. Did the ball make it over the fence?
Algebra 1
Chapter 10 Resource Book
x
13 2 x
Copyright © Holt McDougal. All rights reserved.
25. x 2 1 10x 1 c 5 0
Name ———————————————————————
LESSON
10.7
Date ————————————
Practice C
For use with pages 699–705
Tell whether the equation has two solutions, one solution, or no solution.
1. x 2 1 x 1 5 5 0
2. 100x 2 2 36x 5 0
3. 5x 2 1 4 5 6x
4. 14 5 x 2 2 7x
1
5. } x 2 1 6 5 x
3
3
6. 24x 2 2 5x 5 }
4
7. 9x 2 1 11x 1 1 5 5x
8. 6x 2 1 10 5 3x 2 2 3x 1 4
9. 4x 2 1 4 5 12x 2 4x 2
Find the number of x-intercepts that the graph of the function has.
10. y 5 5x 2 1 4x 2 1
11. y 5 3x 2 2 15x 1 5
12. y 5 4x 2 1 x 1 8
13. y 5 x 2 2 4x 2 2
14. y 5 5x 2 2 10x 1 5
15. y 5 26x 2 1 5x 1 3
16. y 5 6x 2 1 9x 1 1
1
17. y 5 } x 2 2 4x 2 3
5
3
18. y 5 } x 2 2 4x 1 3
4
Give a value of c for which the equation has (a) two solutions,
(b) one solution, and (c) no solution.
19. x 2 1 12x 1 c 5 0
20. x 2 2 8x 1 c 5 0
21. 81x 2 1 18x 1 c 5 0
22. 36x 2 2 12x 1 c 5 0
23. 4x 2 1 24x 1 c 5 0
24. 5x 2 2 45x 1 c 5 0
Tell whether the vertex of the graph of the function lies above, below,
or on the x-axis. Explain your reasoning.
25. y 5 x 2 2 9x 1 20
26. y 5 4x 2 2 24x 1 36
27. y 5 8x 2 2 3x 1 5
from the ground.
a. Use the vertical motion model to write a function that models the height h
(in feet) of the ball after t seconds.
b. Does the ball reach a height of 25 feet? If so, when?
29. Deck Box You want to build a deck box for the deck
off the back of your house. You have blueprints which
show that the base of the deck box is 18 inches wide
and 48 inches long. You want to change the dimensions 18 2 x
as shown. The area can be modeled by the function
x
48 in.
y 5 2x 2 2 30x 1 864.
a. Can you change the dimensions so that the area is 700 square inches?
b. Can you change the dimensions so that the area is 5 square feet? Explain how
you got your answer.
18 in.
Algebra 1
Chapter 10 Resource Book
LESSON 10.7
Copyright © Holt McDougal. All rights reserved.
28. Football You kick a football with an initial upward velocity of 42 feet per second
237
Name ———————————————————————
LESSON
10.7
Date ————————————
Review for Mastery
For use with pages 699–705
GOAL
Use the value of the discriminant.
Vocabulary
In the quadratic formula, the expression b 2 2 4ac is called the
discriminant of the associated equation ax 2 1 bx 1 c 5 0.
EXAMPLE 1
EXAMPLE 2
Use the discriminant
Equation
ax 2 1 bx 1 c 5 0
Discriminant
b 2 2 4ac
Number of
solutions
a.
9x 2 1 30x 1 25 5 0
30 2 2 4(9)(25) 5 0
One solution
b.
7x 2 2 4x 1 6 5 0
(24)2 2 4(7)(6) 5 2152
No solution
c.
4x 2 2 8x 1 3 5 0
(28)2 2 4(4)(3) 5 16
Two solutions
Find the number of solutions
Tell whether the equation 16x2 1 49 5 56x has two solutions, one
solution, or no solution.
Solution
Write the equation in standard form.
16x 2 1 49 5 56x
16x 2 2 56x 1 49 5 0
STEP 2
Write equation.
Subtract 56x from each side.
Find the value of the discriminant.
b2 2 4ac 5 (256)2 2 4(16)(49)
50
Substitute 16 for a, 256 for b,
and 49 for c.
Simplify.
LESSON 10.7
The discriminant is zero, so the equation has one solution.
Exercises for Examples 1 and 2
Tell whether the equation has two solutions, one solution, or
no solution.
1. 2x 2 1 x 5 21
2. 4x 2 1 5x 1 2 5 0
3. 25x 2 1 4 5 20x
238
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
LESSON
10.7
Review for Mastery
For use with pages 699–705
EXAMPLE 3
Date ————————————
continued
Find the number of x-intercepts
Find the number of x-intercepts of the graph of y 5 x2 2 12x 1 36.
Solution
Find the number of solutions of the equation 0 5 x 2 2 12x 1 36.
b2 2 4ac 5 (212)2 2 4(1)(36)
50
Substitute 1 for a, 212 for b, and 36 for c.
Simplify.
The discriminant is zero, so the equation has one solution. This means that the graph
of y 5 x 2 2 12x 1 36 has one x-intercept.
CHECK
You can use a graphing calculator to check your answer. Notice that the
graph of y 5 x 2 2 12x 1 36 intercepts the x-axis once.
Exercises for Example 3
Find the number of x-intercepts of the graph.
5. y 5 x2 1 7x 1 13
6. y 5 4x2 2 12x 1 9
LESSON 10.7
Copyright © Holt McDougal. All rights reserved.
4. y 5 7x2 2 14x
Algebra 1
Chapter 10 Resource Book
239
Name ———————————————————————
LESSON
10.7
Date ————————————
Challenge Practice
For use with pages 699–705
In Exercises 1–5, find the value(s) of k for which the equation has exactly
one solution.
1. x 2 1 kx 1 1 5 0
2. 4x 2 1 2x 1 k 5 0
3. 5kx 2 1 40x 1 6 5 0
4. k 2x 2 1 kx 1 k 5 0
5. kx 2 1 3k 2x 1 2k 4 5 0
In Exercises 6–8, find the value(s) of k for which the equation has
no solution. Write your answer as an inequality.
6. 3x 2 1 2x 1 k 5 0
7. kx 2 1 21x 2 3 5 0
12
8. } x 2 2 6x 1 k 2 5 0
k
In Exercises 9 and 10, find the values of k for which the equation has
exactly two solutions. Write your answer as an inequality.
9. 7kx 2 2 2x 1 3 5 0
10. k 2x 2 1 kx 1 2 5 0
LESSON 10.7
for the sale of x number of snowboards is given by the equation
P(x) 5 2400x 2 1 12,000x 2 80,000. How many snowboards would the
manufacturer have to sell in order to earn a profit? Write your answer
as an inequality.
240
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
11. Suppose a recreation equipment manufacturer determines that the profit
Name ———————————————————————
Date ————————————
Graphing Calculator Activity Keystrokes
LESSON
10.8
For use with pages 714 and 715
Example 1
Example 2
From the home screen, enter the following to
clear lists L1 and L2.
From the home screen, enter the following to
clear lists L1 and L2.
STAT
4
STAT
1
2nd
,
[L1]
[L2]
2nd
ENTER
Move cursor to list L1.
0
1
ENTER
ENTER
4
ENTER
30.7
75.7
ENTER
104
2nd
ENTER
2nd
[L2]
(�)
1
20
160
0
ENTER
[L1]
ENTER
17.5
(
X,T,�,n
GRAPH
[L2]
2nd
3
12
6
ENTER
ENTER
15
ENTER
9
ENTER
WINDOW
1
ENTER
20
GRAPH
,
1.6
31000
ENTER
ENTER
135500
ENTER
ENTER
360000
ENTER
[STAT PLOT] 1
2nd
ENTER
(�)
2nd
[L2]
)
^
2nd
[L2]
(�)
3
(�)
50000
50000
ENTER
ENTER
[L2]
x2
�
1010
ENTER
WINDOW
ENTER
ENTER
3
400000
STAT
ENTER
201500
ENTER
[L1]
ENTER
GRAPH
2nd
18
76000
ENTER
ENTER
2nd
ENTER
ENTER
[L1]
Y=
ENTER
500
ENTER
ENTER
2nd
ENTER
46
ENTER
9
ENTER
ENTER
STAT
Copyright © Holt McDougal. All rights reserved.
ENTER
,
[L1]
Move cursor to list L2.
ENTER
[STAT PLOT] 1
2nd
1
2nd
ENTER
Move cursor to list L2.
ENTER
STAT
0
3
ENTER
15.8
4
Move cursor to list L1.
2
ENTER
STAT
LESSON 10.8
TI-83 Plus
Y=
X,T,�,n
ENTER
5
2nd
1440
�
ENTER
[L1]
,
X,T,�,n
8000
GRAPH
Algebra 1
Chapter 10 Resource Book
241
Name ———————————————————————
Practice A
LESSON
10.8
LESSON 10.8
Date ————————————
For use with pages 706–713
Match the function with the graph it represents.
1. Linear function
A.
y
5
25
23
2. Exponential function
3. Quadratic function
B.
C.
y
5
y
5
3
3
3
1
1
1
21
21
3 x
1
25
23
21
21
3 x
1
23
21
21
1
3
x
2
6
x
5
x
Use the graph to tell whether the points represent a linear function, an
exponential function, or a quadratic function.
4.
5.
y
6.
y
10
3
6
1
2
21
21
y
6
2
22
7.
2
6
1
3
22
22
x
26
8.
y
21
21
5
x
1
23
3
x
9.
y
5
y
1
3
23
1
25
21
21
21
21
1
3
5
23
7 x
Use a graph to tell whether the ordered pairs represent a linear function,
an exponential function, or a quadratic function.
10. (24, 27), (22, 24), (0, 21), (2, 2), (4, 5)
1
1
11. (22, 8), (21, 4), (0, 2), (1, 1), 2, }
2
y
26
y
6
10
2
6
22
22
2
26
242
Algebra 1
Chapter 10 Resource Book
6 x
2
23
21
22
1
3 x
2
Copyright © Holt McDougal. All rights reserved.
26
26
1
Name ———————————————————————
Practice A
LESSON
10.8
For use with pages 706–713
Date ————————————
continued
13. (0, 25), (1, 1), (2, 7), (3, 13), (4, 19)
y
3
15
1
5
21
21
23
y
x
1
21
25
14. (0, 1), (1, 2), (2, 4), (3, 8), (4, 16)
5 x
3
15. (1, 2), (2, 21), (3, 22), (4, 21), (5, 2)
y
y
20
3
12
1
4
21
21
21
24
1
LESSON 10.8
12. (23, 0), (22, 22), (21, 22), (0, 0), (1, 4)
1
3
5 x
1
5 x
3
23
Tell whether the table of values represents a linear function,
an exponential function, or a quadratic function.
28
24
0
4
8
y
21
0
1
2
3
x
24
23
22
21
0
y
7
4
3
4
7
19.
x
23
22
21
0
1
y
625
125
25
5
1
x
21
0
1
2
3
y
23
0
1
0
23
20. Baseball Salaries The graph shows a model for the salaries (in
thousands of dollars) of baseball players for the period 199922003.
a. Is the model a linear function, a quadratic function, or an
exponential function?
b. Is this model good for predicting the salaries of players
after 2003? Explain your reasoning.
21. Consumer Spending The graph shows the amount of money spent
(in billions of dollars) in the United States on video and audio
products, computer equipment, and musical instruments for the
period 199022002. Tell whether the data should be modeled by
a linear function, an exponential function, or a quadratic function.
Salary (thousands
of dollars)
Copyright © Holt McDougal. All rights reserved.
18.
17.
x
Amount spent
(billions of dollars)
16.
y
2500
2000
1500
1000
500
0
y
120
100
80
60
40
0
0 1 2 3 4 5 x
Years since 1999
0 2 4 6 8 10 12 x
Years since 1990
Algebra 1
Chapter 10 Resource Book
243
Name ———————————————————————
Practice B
LESSON
10.8
LESSON 10.8
Date ————————————
For use with pages 706–713
Match the function with the graph it represents.
1. Linear function
2. Exponential function
3. Quadratic function
A.
B.
C.
y
3
y
y
3
3
1
1
1
23
21
21
1
x
21
21
1
x
3
23
21
21
1
x
23
Use a graph to tell whether the ordered pairs represent a linear function,
an exponential function, or a quadratic function.
4. (22, 16), (21, 8), (0, 4), (1, 2), (2, 1)
5. (23, 4), (22, 0), (21, 22), (0, 22), (1, 0)
y
y
20
3
12
1
4
23
21
24
1
3 x
6. (24, 17), (22, 11), (0, 5), (2, 21), (4, 27)
21
21
1
x
7. (29, 21), (26, 22), (23, 23), (0, 24), (3, 25)
y
y
1
12
29
23
21
3
x
4
26
6 x
22
24
25
8.
1 22, }19 2, 1 21, }13 2, (0, 1), (1, 3), (2, 9)
9. (2, 5), (3, 2), (4, 1), (5, 2), (6, 5)
y
23
244
y
10
5
6
3
2
1
21
22
1
Algebra 1
Chapter 10 Resource Book
3 x
1
3
5
x
Copyright © Holt McDougal. All rights reserved.
23
Name ———————————————————————
LESSON
10.8
Practice B
For use with pages 706–713
Date ————————————
continued
10.
12.
14.
16.
x
0
1
2
3
4
y
1
5
25
125
625
x
21
0
1
2
3
y
4
1
0
1
4
x
22
21
0
1
2
y
32
8
2
}
1
2
}
x
22
21
0
1
2
y
1
3
5
7
9
11.
13.
15.
1
8
17.
LESSON 10.8
Tell whether the table of values represents a linear function,
an exponential function, or a quadratic function.
x
22
21
0
1
2
y
210
27
24
21
2
x
210
25
0
5
10
y
4
3.5
3
2.5
2
x
24
23
22
21
0
y
23
0
1
0
23
x
23
22
21
0
1
y
27
9
3
1
}
1
3
18. Use the graph shown.
y (4, 256)
or a quadratic function? Explain your reasoning.
b. Make a table of values for the points on the graph. Then use
differences or ratios to check your answer in part (a).
c. Write an equation for the function that the table of values from
part (b) represents.
224
160
96
(3, 64)
32
(0, 1)
19. Pleasure Boats The graph shows total amount of sales
(in millions of dollars) of pleasure boats in the United States
for the period 1990–2002. Tell whether the data should be
modeled by a linear function, an exponential function, or
a quadratic function. Explain your reasoning.
20. Computer Value The value V of a computer between 1999
and 2003 is given in the table. Tell whether the data should
be modeled by a linear function, an exponential function, or
a quadratic function. Then write an equation for the function.
Years since 1999, t
Value, V (dollars)
0
1
2
3
4
800
725
650
575
500
1
(2, 16)
5 x
(1, 4)
Pleasure Boats
Sales
(millions of dollars)
Copyright © Holt McDougal. All rights reserved.
a. Which function does the graph represent, an exponential function
y
16,000
14,000
12,000
10,000
8,000
6,000
0
0 2 4 6 8 10 12 x
Years since 1990
Algebra 1
Chapter 10 Resource Book
245
Name ———————————————————————
Practice C
LESSON
10.8
LESSON 10.8
Date ————————————
For use with pages 706–713
Match the function with the graph it represents.
1. Linear function
A.
y
2. Exponential function
3. Quadratic function
B.
C.
y
5
y
5
1
1
21
23
3 x
23
3
3
1
1
21
21
1
5 x
3
23
21
21
3
1
x
Use a graph to tell whether the ordered pairs represent a linear function,
an exponential function, or a quadratic function.
4. (25, 5), (23, 23), (21, 23), (0, 0), (1, 5)
5. (24, 222), (22, 212), (0, 22), (2, 8), (4, 18)
y
y
18
6
6
2
25
23
26
1 x
21
22
22
26
6 x
2
218
21
1
21
1
1
1
7. (22, 8), (21, 2), 0, } , 1, } , 2, }
2
8
32
y
2
y
1
22
21
2
6
10 x
6
2
23
23
21
22
1
3 x
25
8. (27, 7), (26, 4), (25, 3), (24, 4), (23, 7)
9. (0, 1), (1, 4), (2, 16), (3, 64), (4, 256)
y
210
246
26
y
10
320
6
192
2
64
22
22
Algebra 1
Chapter 10 Resource Book
2 x
21
264
1
3
5 x
Copyright © Holt McDougal. All rights reserved.
6. (0, 25), (2, 24.5), (4,24), (6, 23.5), (8, 23)
Name ———————————————————————
LESSON
10.8
Practice C
Date ————————————
continued
For use with pages 706–713
10.
12.
14.
16.
11.
x
0
1
2
3
4
y
2
2.1
2.2
2.3
2.4
x
24
23
22
21
0
y
1296
216
36
6
1
x
25
24
23
22
21
y
24
21
0
21
24
x
23
22
21
0
1
y
15
11
7
3
21
13.
15.
17.
x
1
2
3
4
5
y
26
23
22
23
26
x
0
1
2
3
4
y
6
3
0
23
26
x
23
22
21
0
1
y
1024
128
16
2
}
x
2
3
4
5
6
y
2
21
22
21
2
1
4
18. Use the graph shown.
y
(23, 64)
a. Which function does the graph represent, an exponential function
Copyright © Holt McDougal. All rights reserved.
LESSON 10.8
Tell whether the table of values represents a linear function,
an exponential function, or a quadratic function.
or a quadratic function? Explain your reasoning.
b. Make a table of values for the points on the graph. Then use
differences or ratios to check your answer in part (a).
c. Write an equation for the function that the table of values from
part (b) represents.
56
40
(22, 16)
(21, 4)
23
21
24
8
(0, 1)
(1, 0.25)
1
3 x
19. Printer Value The value V of a printer between 1999 and 2003 is given in the
table. Tell whether the data should be modeled by a linear function, an exponential
function, or a quadratic function. Then write an equation for the function.
Years since 1999, t
Value, V (dollars)
0
1
2
3
4
2000
1920
1840
1760
1680
20. Interest The balance B of an account is given in the table. Tell whether the data
should be modeled by a linear function, an exponential function, or a quadratic
function. Then write an equation for the function.
Time, t (years)
Balance, B (dollars)
0
1
2
3
4
1020.20
1040.60
1061.42
1082.64
1104.30
Algebra 1
Chapter 10 Resource Book
247
Name ———————————————————————
LESSON
LESSON 10.8
10.8
Date ————————————
Review for Mastery
For use with pages 706–713
GOAL
EXAMPLE 1
Compare linear, exponential, and quadratic models.
Choose functions using sets of ordered pairs
Use a graph to tell whether the ordered pairs represent a linear
function, an exponential function, or a quadratic function.
a. (22, 216), ( 21, 215), (0, 212), (1, 27), (2, 0)
b. (22, 1), ( 21, 3), (0, 5), (1, 7), (2, 9)
1
1
, 21, }5 2, (0, 1), (1, 5), (2, 25)
1 22, }
25 2 1
c.
Solution
b.
y
y
27
7
21
26
5
15
210
3
9
1
3
x
22
22
21
Quadratic function
EXAMPLE 2
c.
y
9
2
Linear function
1
x
23
21
1
x
Exponential function
Identify functions using differences or ratios
Use differences or ratios to tell whether the table of values represents
a linear function, an exponential function, or a quadratic function.
Solution
x
21
0
1
2
y
1
3
9
27
3
Ratios: }1 5 3
3
3
The table represents an exponential function.
248
Algebra 1
Chapter 10 Resource Book
Copyright © Holt McDougal. All rights reserved.
a.
Name ———————————————————————
LESSON
10.8
Review for Mastery
For use with pages 706–713
Date ————————————
continued
1. Tell whether the ordered pairs represent a linear function, a quadratic
function, or an exponential function: (21, 26), (0, 24), (1, 0), (2, 6).
2. Tell whether the table represents a linear function, a quadratic function, or an
exponential function.
EXAMPLE 3
x
0
1
2
3
y
26
3
12
21
LESSON 10.8
Exercises for Examples 1 and 2
Write an equation for a function
Tell whether the table of values represents a linear function, an
exponential function, or a quadratic function. Then write an equation
for the function.
STEP 1
Determine which type of function the values in the table represent.
x
21
0
1
2
3
y
7
5
3
1
21
Copyright © Holt McDougal. All rights reserved.
First differences: 2 2 2 2 2 2 2 2
The table of values represents a linear function because the first differences
are equal.
STEP 2
Write an equation for the linear function. The equation has the form
y 5 mx 1 b. When x 5 0, y 5 5, so b 5 5. Find m by substituting any
two points into the slope formula.
y
527
0 2 (21)
7
22
5 22
m5}5}
1
5
The equation is y 5 22x 1 5.
CHECK
3
Plot the ordered pairs from the table.
Then graph y 5 22x 1 5 to see that
the graph passes through the plotted
points.
1
21
21
x
1
Exercises for Example 3
Tell whether the table of values represents a linear function, an
exponential function, or a quadratic function. Then write an equation
for the function.
3.
x
21
0
1
2
y
12
6
2
0
4.
x
22
y
0.0625 0.125 0.25 0.5 1
21
0
1
2
Algebra 1
Chapter 10 Resource Book
249
Name ———————————————————————
LESSONS
10.5–10.8
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
1. Multi-Step Problem Different currents
(in amperes) are sent through an electric
circuit. The powers (in volts) that are
recorded from the electric current are shown
in the table.
Current (amperes)
Power (volts)
0.5
5
1
20
1.5
45
2
80
2.5
125
a. Tell whether the data can be modeled
by a linear function, an exponential
function, or a quadratic function.
b. Write an equation for the function.
2. Multi-Step Problem A lacrosse player
throws a ball upward from his playing stick
with an initial height of 6.5 feet above the
ground at initial vertical velocity of 80 feet
per second.
a. Write an equation for the height h
(in feet) of the ball as a function of the
time t (in seconds) after it is thrown.
b. Another player catches the ball when
it is 4 feet above the ground. How
long after the ball is thrown is the ball
caught? Round your answer to the
nearest second.
4. Open-Ended Write a quadratic equation
that has no solution. Use the discriminant
to verify the quadratic equation has no
solution.
5. Short Response For the period
1997–2002, the average monthly basic rate
y (in dollars) for cable television can be
modeled by y 5 0.15x 2 1 0.93x 1 26.55
where x is the number of years since 1997.
a. Use the discriminant to determine the
number of values of x that correspond
to y 5 29.
b. Were there any years during the period
1997–2002 in which the average
monthly basic rate for cable television
reached $29? Explain.
6. Gridded Response The triangle below
has an area of 50 square inches. What is the
value of x? Round your answer to the nearest
tenth.
(x 2 2) in.
(x 2 3) in.
(x 1 8) in.
7. Extended Response You want to place a
walkway around a pool as shown.
x ft
28 ft
40 ft
x ft
3. Multi-Step Problem From the edge of
a ledge directly over a target, you throw a
marker with an initial downward velocity of
230 feet per second from a height of
80 feet.
a. Write an equation for the height h (in
feet) of the marker as a function of the
time t (in seconds) after it is thrown.
b. How long will it take the marker to hit
the target? Round your answer to the
nearest tenth of a second.
250
Algebra 1
Chapter 10 Resource Book
x ft
x ft
a. Write an equation for the area A
(in square inches) of the walkway.
b. You have enough bricks to cover
450 square feet. What should the
width of the walkway be? Round
your answer to the nearest foot.
c. Explain why you could ignore one of
the values of x in part (b).
Copyright © Holt McDougal. All rights reserved.
LESSON 10.8
For use with pages 685–713
Name ———————————————————————
LESSON
10.8
Date ————————————
Challenge Practice
For use with pages 706–713
(0, 3), (2, 7), (3, 9), (5, k)
1. Tell whether the data fits a linear model, quadratic model, or exponential model.
2. Find a value of k that makes the data fit the model selected in Exercise 1.
LESSON 10.8
In Exercises 1–3, use the following data.
3. Write the model for the value of k found in Exercise 2.
In Exercises 4–6, use the following data.
(1, 3), (3, 6.75), (5, 15.1875), (7, k)
4. Tell whether the data fits a linear model, quadratic model, or exponential model.
5. Find a value of k that makes the data fit the model selected in Exercise 4.
6. Write the model for the value of k found in Exercise 5.
In Exercises 7–9, use the following data.
(2, 10), (5, 73), (8, 190), (11, k)
7. Tell whether the data fits a linear model, quadratic model, or exponential model.
8. Find a value of k that makes the data fit the model selected in Exercise 7.
Copyright © Holt McDougal. All rights reserved.
9. Write the model for the value of k found in Exercise 8.
10. The weight of a male African elephant increases during the first year of life
according to the model y 5 10,000 2 9650(k) x where y represents the weight
(in pounds) of the elephant and x represents the number of months after birth.
If a one-year-old male African elephant weights 2000 pounds, how much did the
elephant weigh when it was 4 months old?
Algebra 1
Chapter 10 Resource Book
251
Name ———————————————————————
CHAPTER
10
Date ————————————
Chapter Review Game
For use after Chapter 10
Vertical Motion Puzzle
The vertical motion model h 5 216(t 1 3)(t 2 3) models the height h
(in feet) of an object after t seconds. Use the coordinate plane to plot the
points according to the directions below. Then use the resulting figure to
determine the initial height (in feet) of the object.
Plot the following points and connect them in order.
1. Use the negative solution of x 2 2 2x 2 3 5 0 as the x-coordinate of (x, 2).
2. Use the solution of x 2 1 2x 5 21 as the x-coordinate of (x, 6).
Plot the following points and connect them in order.
3. Use the greatest positive solution of 2x 2 2 11x 1 15 5 0 as the x-coordinate
of (x, 2).
1
4. Use the discriminant of 5x 2 1 4x 1 } 5 0 as the y-coordinate of (3, y).
2
5. Use the positive solution 10x 2 2 10 5 0 as the x-coordinate of (x, 4).
6. Use the positive solution of 3x 2 2 8x 2 3 5 0 as the x-coordinate of (x, 4).
Plot the following points and connect them in order.
7. Use the least positive solution of x 2 2 15x 1 54 5 0 as the x-coordinate of (x, 2).
8. Use the solution of 22x2 1 24x 2 72 5 0 as the x-coordinate of (x, 6).
10. Use the positive solution of 5x 2 2 30x 5 0 as the x-coordinate of (x, 4).
y
7
5
3
CHAPTER REVIEW GAME
1
252
21
Algebra 1
Chapter 10 Resource Book
1
3
5
7 x
Copyright © Holt McDougal. All rights reserved.
9. Use the discriminant of 23x 2 1 8x 5 5 as the y-coordinate of (4, y).
Name ———————————————————————
CHAPTER
11
Date ————————————
Family Letter
For use with Chapter 11
Lesson Title
Lesson Goals
Key Applications
11.1: Graph Square Root
Functions
Graph square root
functions.
• Microphone Sales
• Oceanography
• Long Jump
11.2: Simplify Radical
Expressions
Simplify radical expressions.
• Astronomy
• Finance
• Horizon
Copyright © Holt McDougal. All rights reserved.
Focus on Operations
Perform operations with cube
roots.
11.3: Solve Radical Equations
Solve radical equations.
• Sailing
• Forests
• Biology
11.4: Apply the Pythagorean
Theorem and Its Converse
Use the Pythagorean theorem
and its converse.
• Construction
• Sails
• Screen Sizes
11.5: Apply the Distance and
Midpoint Formulas
Use the distance and midpoint
formulas.
• Sightseeing
• Subway
• Archaeology
CHAPTER SUPPORT
Chapter Overview One way you can help your student succeed in Chapter 11
is by discussing the lesson goals in the chart below. When a lesson is completed, ask your
student the following questions. “What were the goals of the lesson? What new words and
formulas did you learn? How can you apply the ideas of the lesson to your life?”
Key Ideas for Chapter 11
In Chapter 11, you will apply the key ideas listed in the Chapter Opener (see page 733)
and reviewed in the Chapter Summary (see page 778).
1. Graphing square root functions
2. Using properties of radicals in expressions and equations
3. Working with radicals in geometry
Algebra 1
Chapter 11 Resource Book
253
Name ———————————————————————
CHAPTER
CHAPTER SUPPORT
11
Family Letter
Date ————————————
continued
For use with Chapter 11
Key Ideas Your student can demonstrate understanding of key concepts by working through the
following exercises with you.
Lesson
Exercise
}
11.1
Graph the function y 5 2Ï x 2 3 and identify its domain and range.
}
Compare the graph with the graph of y 5 Ïx .
11.2
Simplify the expression.
}
}
(a) Ï220
Focus on
Operations
}
}
(b) 3Ï 5x p Ï 4x
3
2
}
}
(c) (6 1 Ï 7 )(3 2 Ï 7 )
Ï75xy 2
(d) }
}
Ï5x2y 3
Simplify the expression. Assume variables are nonzero.
3}
3}
3}
(b) 9Ï6t 2 13Ï6t
11.3
Solve the equation. Check for extraneous solutions.
}
}
(a) 3Ïx 2 5 2 6 5 9
(b) Ï 2x 1 8 5 x
11.4
A garage door has a height of x feet and a width of (x 1 2.5) feet. If the
diagonal (hypotenuse) of the garage door is 12.5 feet, find the actual height
and width of the garage door.
11.5
Find the distance between the two points. Then find the midpoint of the line
segment connecting the two points.
(a) (25, 1), (7, 3)
(b) (2, 28), (26, 22)
Home Involvement Activity
Directions Create a treasure map, where either the Pythagorean theorem or the
distance formula must be used to get from one point (clue) to another. Have a parent,
sibling, or guardian follow your directions to reach the treasure.
(b) 10 units; (22, 25)
(b) 24Ï 6t 11.3: (a) 30 (b) 4 11.4: 7.5 ft; 10 ft 11.5: (a) 2Ï37 units; (1, 2)
3}
}
11.2: (a) 2Ï55 (b) 6x 2Ï 5x (c)11 2 3Ï 7 (d) }
xy Focus on Operations: (a) 8
}
}
}
Ï15xy
}
1
3
5
x
1
3
11.1:
5
y
Domain: x ≥ 3; Range: y ≥ 0; The graph of y 5 2Ï x 2 3 is a
vertical stretch (by a factor of 2) and a horizontal translation
}
(of 3 units to the right) of the graph of y 5 Ïx .
}
Answers
254
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
3}
(a) Ï4 • Ï 128
Nombre ——————————————————————
CAPÍTULO
11
Fecha ———————————
Carta para la familia
Usar con el Capítulo 11
Copyright © Holt McDougal. All rights reserved.
Título de la lección
Objetivos de la lección
Aplicaciones clave
11.1: Graficar funciones de
raíces cuadradas
Graficar funciones de raíces
cuadradas
• Ventas de micrófonos
• Oceanografía
• Salto largo
11.2: Simplificar expresiones
radicales
Simplificar expresiones
radicales
• Astronomía
• Finanzas
• Horizonte
Enfoque en las
operaciones
Hacer operaciones con raíces
cúbicas
11.3: Resolver ecuaciones
radicales
Resolver ecuaciones radicales
• Navegación
• Bosques
• Biología
11.4: Aplicar el teorema de
Pitágoras y su recíproco
Usar el teorema de Pitágoras
y su recíproco
• Construcción
• Velas
• Tamaños de pantalla
11.5: Aplicar las fórmulas de
distancia y del punto
medio
Usar las fórmulas de distancia
y del punto medio
• Hacer turismo
• Metro
• Arqueología
CHAPTER SUPPORT
Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener éxito en el Capítulo
11 es hablar sobre los objetivos de la lección en la tabla a continuación. Cuando se termina una
lección, pregúntele a su hijo lo siguiente: “¿Cuáles fueron los objetivos de la lección? ¿Qué
palabras y fórmulas nuevas aprendiste? ¿Cómo puedes aplicar a tu vida las ideas de la lección?”
Ideas clave para el Capítulo 11
En el Capítulo 11, aplicarás las ideas clave enumeradas en la Presentación del capítulo
(ver la página 733) y revisadas en el Resumen del capítulo (ver la página 778).
1. Graficar funciones de raíces cuadradas
2. Usar propiedades de radicales en expresiones y ecuaciones
3. Trabajar con radicales en geometría
Algebra 1
Chapter 11 Resource Book
255
Nombre ——————————————————————
CAPÍTULO
CHAPTER
CHAPTER SUPPORT
11
Carta para la familia
Fecha ———————————
continúa
Usar con el Capítulo 11
Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer los siguientes
ejercicios con usted.
Lección
Ejercicio
11.1
Grafica la función y 5 2Ï x 2 3 e identifica su dominio y rango. Compara la gráfica
}
con la gráfica de y 5 Ïx .
11.2
Simplifica la expresión.
}
}
}
(a) Ï220
}
(b) 3Ï 5x p Ï 4x
3
2
}
}
(c) (6 1 Ï 7 )(3 2 Ï 7 )
Ï75xy 2
(d) }
}
Ï5x2y 3
Simplifica la expresión. Asume que las variables no son cero.
3}
3}
(a) Ï4 • Ï 128
3}
3}
(b) 9Ï6t 2 13Ï6t
11.3
Resuelve la ecuación. Busca soluciones extrañas.
}
}
(a) 3Ïx 2 5 2 6 5 9
(b) Ï2x 1 8 5 x
11.4
Una puerta de garaje tiene una altura de x pies y un ancho de (x 1 2.5) pies. Si la
diagonal (hipotenusa) de la puerta es 12.5 pies, halla la altura y el ancho verdaderos
de la puerta.
11.5
Halla la distancia entre dos puntos. Luego halla el punto medio del segmento
conectando los dos puntos.
(a) (25, 1), (7, 3)
(b) (2, 28), (26, 22)
Actividad para la familia
Instrucciones Haz un mapa de tesoro en que se puede usar o el teorema de
Pitágoras o la fórmula de distancia para ir de un punto (pista) a otro. Pida a un padre,
hermano o tutor que siga tus indicaciones para encontrar el tesoro.
(b) 10 units; (22, 25)
(b) 24Ï 6t 11.3: (a) 30 (b) 4 11.4: 7.5 ft; 10 ft 11.5: (a) 2Ï37 units; (1, 2)
3}
}
11.2: (a) 2Ï55 (b) 6x 2Ï 5x (c)11 2 3Ï 7 (d) }
xy Enfoque en las operaciones: (a) 8
}
}
}
Ï15xy
}
1
3
5
x
1
3
11.1:
5
y
Dominio: x ≥ 3; Rango: y ≥ 0; La gráfica de y 5 2Ï x 2 3 es una
extensión vertical (por un factor de 2) y una traslación
}
(de 3 unidades a la derecha) de la gráfica de y 5 Ï x .
}
Respuestas
256
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
Enfoque
en las
operaciones
}
Name ———————————————————————
Graphing Calculator Activity Keystrokes
LESSON
11.1
For use with page 741
TI-83 Plus
Y=
2nd
WINDOW
Date ————————————
Casio CFX-9850GC Plus
}
[Ï ] 2
(�)
5
5
ENTER
(�)
ENTER
GRAPH
X,T,�,n
ENTER
ENTER
�
5
5
3
ENTER
ENTER
From the main menu, choose GRAPH.
)
1
1
}
[Ï ] 2
(
SHIFT
EXE
SHIFT
F3
EXE
(�)
5
EXE
X,,T
(�)
5
5
EXE
�
EXE
1
5
EXE
3
)
EXE
1
EXIT
F6
Copyright © Holt McDougal. All rights reserved.
LESSON 11.1
Algebra 1
Chapter 11 Resource Book
257
Name ———————————————————————
Date ————————————
Practice A
LESSON
11.1
For use with pages 734–741
Match the function with its graph.
}
A.
2. y 5 28Ï x
y
y
B.
5
22
25
1 }
3. y 5 } Ï x
8
}
1. y 5 8Ï x
2
10
6
C.
y
25
0.3
x
15
0.1
215
22
20.1
2
6
10
5
x
22
25
LESSON 11.1
225
10
x
5
x
21
21
1
x
6
10
x
2
6
Graph the function and identify its domain and range. Compare the graph
}
with the graph of y 5 Ï x.
}
}
4. y 5 6Ï x
}
5. y 5 0.4Ï x
y
6. y 5 22Ï x
y
y
10
1.0
1
6
0.6
21
21
2
0.2
23
22
22
2
10
6
x
22
20.2
2
6
10
x
3
1
25
Match the function with its graph.
}
9. y 5 Ï x 2 5 1 2
}
12. y 5 Ï x 1 5 1 2
8. y 5 Ï x 2 2 1 5
}
11. y 5 Ï x 1 2 2 5
10. y 5 Ï x 2 5 2 2
A.
B.
y
}
}
C.
y
y
1
5
21
21
1
1
5 x
3
21
21
25
1
3
x
23
1
23
25
25
D.
F.
y
y
10
6
10
6
2
6
2
22
22
22
22
258
E.
y
23
2
6
Algebra 1
Chapter 11 Resource Book
10
x
26
2
10
x
2
22
22
2
Copyright © Holt McDougal. All rights reserved.
}
7. y 5 Ï x 1 5 2 2
Name ———————————————————————
LESSON
11.1
Practice A
For use with pages 734–741
Date ————————————
continued
Graph the function and identify its domain and range. Compare the graph
}
with the graph of y 5 Ï x.
}
}
13. y 5 Ï x 2 5
}
14. y 5 Ï x 1 3
y
15. y 5 Ï x 2 6
y
y
6
6
3
2
26
2
22
22
2
6
1
x
23
26
21
21
1
3
22
22
x
26
2
6
x
26
23
}
}
17. y 5 Ï x 1 3
y
18. y 5 Ï x 2 5
y
LESSON 11.1
}
16. y 5 Ï x 2 2
y
6
3
6
1
2
2
26
22
22
2
6
x
23
21
21
1
3
x
22
22
2
6
10 x
26
23
26
20. Horizon The distance d (in nautical miles) that a person can see
}
to the horizon is given by the formula d 5 1.17Ï h where h is the
person’s eye level in feet.
a. Graph the function and identify its domain and range.
b. A person can see 20 nautical miles to the horizon.
What is the person’s eye level? Round your answer
to the nearest nautical mile.
f
700
600
500
400
300
200
100
0
0 5 10 15 20 25 30 35 p
Nozzle pressure (lb/in.2 )
Distance
(nautical miles)
2 inches, the}flow rate f (in gallons per minute) can be modeled
by f 5 120Ïp where p is the nozzle pressure in pounds per
square inch.
a. Graph the function and identify its domain and range.
b. If the flow rate is 720 gallons per minute, what is the
nozzle pressure?
Flow rate (gal/min)
Copyright © Holt McDougal. All rights reserved.
19. Fire Hoses For a fire hose with a nozzle that has a diameter of
d
20
15
10
5
0
0
100 200 h
Eye level (feet)
Algebra 1
Chapter 11 Resource Book
259
Name ———————————————————————
Date ————————————
Practice B
LESSON
11.1
For use with pages 734–741
Graph the function and identify its domain and range. Compare the graph
}
with the graph of y 5 Ïx.
}
}
1 }
1. y 5 7Ï x
2. y 5 } Ï x
3. y 5 24Ï x
5
y
y
y
14
3
2
10
26
1
6
23
21
21
1
3
22
22
x
2
6
x
26
2
LESSON 11.1
26
22
2
6
210
23
x
}
Describe how you would graph the function by using the graph of y 5 Ï x.
}
}
4. y 5 Ï x 2 8
}
5. y 5 Ï x 1 3
}
6. y 5 Ï x 1 7
7. y 5 Ï x 2 5
Îx 2 }12
}
}
8. y 5 Ï x 1 3.5
9. y 5
Match the function with its graph.
}
11. y 5 Ï x 2 3 1 4
}
14. y 5 Ï x 1 3 2 4
13. y 5 Ï x 2 4 2 3
A.
}
12. y 5 Ï x 2 4 1 3
}
15. y 5 Ï x 1 3 1 3
B.
y
1
1
5
3
D.
y
x
21
21
Algebra 1
Chapter 11 Resource Book
1
3 x
3
25
21
21
F.
y
23
1
3 x
1
1
3
1
23
E.
y
5
21
21
5
260
C.
y
23
3
23
}
1
5
21
21
}
21
1
3
5
x
1
3
5
x
y
1
x
21
23
23
25
25
Copyright © Holt McDougal. All rights reserved.
10. y 5 Ï x 1 4 2 3
Name ———————————————————————
Practice B
LESSON
11.1
Date ————————————
continued
For use with pages 734–741
Graph the function and identify its domain and range. Compare the graph
}
with the graph of y 5 Ï x.
}
}
}
16. y 5 Ï x 1 4 2 4
18. y 5 Ï x 2 6 1 4
17. y 5 Ï x 1 5 1 1
y
y
y
3
6
1
23
21
21
3
1
1
x
23
25
2
21
21
26
3 x
1
22
22
2
6
10 x
1
3 x
23
23
26
}
20. y 5 Ï x 2 1 1 2
y
LESSON 11.1
}
}
19. y 5 Ï x 2 5 2 7
21. y 5 Ï x 1 5 2 4
y
y
6
3
3
2
26
22
22
1
2
6
1
10 x
23
21
21
25
1
3
23
21
21
x
26
23
23
23. Steel Pipe The inside diameter d of a steel pipe (in inches)
and the weight w of water }
in the pipe (in pounds) are related
by the function d 5 1.71Ïw.
a. Graph the function and identify its domain and range.
b. What does the water weigh in a pipe with an inside
diameter of 17 inches? Round your answer to the
nearest pound.
c. What does the water weigh in a pipe with an inside
diameter of 3.5 inches? Round your answer to the
nearest pound.
d
15
12
9
6
3
0
Side length (inches)
popcorn. The box must be 9 inches tall. The side length x (in inches)
1 }
of the box is given by the function x 5 }3 Ï V where V is the volume
(in cubic inches) of the box.
a. Graph the function and identify its domain and range.
b. What is the volume of a box with a side length of 5 inches?
c. What is the volume of a box with a side length of 8 inches?
Diameter (inches)
Copyright © Holt McDougal. All rights reserved.
22. Box Design You are designing a box with a square base that will hold
x
7
6
5
4
3
2
1
0
0
200 400 V
Volume
(cubic inches)
0 10 20 30 40 50 60 70 80 90 w
Weight (pounds)
Algebra 1
Chapter 11 Resource Book
261
Name ———————————————————————
Date ————————————
Practice C
LESSON
11.1
For use with pages 734–741
Graph the function and identify its domain and range. Compare the graph
}
with the graph of y 5 Ï x.
}
}
3 }
1. y 5 2.5Ï x
2. y 5 2} Ï x
3. y 5 20.25Ï x
5
y
y
y
0.75
3
1.8
1
0.6
0.25
23
21
21
1
3
x
23
23
21
20.6
3
1
21
20.25
x
1
3
x
5
x
20.75
LESSON 11.1
23
21.8
}
Describe how you would graph the function by using the graph of y 5 Ï x.
}
3
5. y 5 Ï x 2 }
2
}
4. y 5 Ï x 1 2.5
Îx 2 }14
}
7. y 5
}
6. y 5 Ï x 1 12
}
3
9. y 5 Ï x 1 }
4
}
8. y 5 Ï x 1 5.5
Match the function with its graph.
}
13. y 5 Ï x 2 3 2 2
A.
}
12. y 5 3Ï x 2 1 1 2
}
15. y 5 Ï x 1 2 1 3
B.
7 x
3
E.
y
5
3
3
1
1
1
5 x
3
262
21
21
1
Algebra 1
Chapter 11 Resource Book
3
21
1
3
y
5
3
3
1
1
1
x
21
F.
y
1
23
y
5
21
5
C.
y
1
D.
}
14. y 5 3Ï x 1 1 2 2
y
1
}
11. y 5 2Ï x 2 1 1 3
1
3
5
x
3
5 x
Copyright © Holt McDougal. All rights reserved.
}
10. y 5 3Ï x 1 2 2 1
Name ———————————————————————
LESSON
11.1
Practice C
For use with pages 734–741
Date ————————————
continued
Graph the function and identify its domain and range. Compare the graph
}
with the graph of y 5 Ï x.
}
}
16. y 5 Ï x 1 6 2 4
y
26
}
18. y 5 Ï x 2 3 2 3
17. y 5 2Ï x 2 1 1 5
y
y
6
6
3
2
2
1
22
22
2
6
x
26
22
22
26
2
x
6
21
21
26
5
x
23
}
}
21. y 5 2Ï x 2 4.5 1 2.5
20. y 5 Ï x 2 7 1 8
y
3
LESSON 11.1
}
19. y 5 2Ï x 1 6 1 2
1
y
y
10
6
6
6
2
26
22
22
2
2
6
x
2
2
6
10
14 x
22
22
26
2
Time (seconds)
}
Ïh .
t5}
7
a. Graph the function and identify its domain and range.
b. You are on a bridge that passes over a river. It takes about
1.5 seconds for a stone dropped from the bridge to reach
the river. About how high is the bridge?
23. Steel Pipe The radius of gyration of a steel pipe is a number
that describes a pipe’s resistance to buckling. The greater
value of r, the more resistance to buckling. The radius of
gyration r (in inches) of a steel pipe is given by the function
1 }
r 5 }4 Ï D2 1 d 2 where D is the outside diameter of the pipe
(in inches) and d is the inside diameter of the pipe (in inches).
One standard outside pipe diameter is 4 inches. Write a
function for r and d using D 5 4.
a. Graph the function and identify its domain and range.
b. If you want a pipe with a 4-inch outside diameter and a radius
of gyration of 1.3 inches, what must its inside diameter be?
Round your answer to the nearest tenth.
Radius of gyration
(inches)
Copyright © Holt McDougal. All rights reserved.
height h (in meters) to reach the ground is given by the function
Ï10
10
x
26
22. Bridge The time t (in seconds) it takes an object dropped from a
}
6
r
2.5
2.0
1.5
1.0
0.5
0
t
2.5
2.0
1.5
1.0
0.5
0
0 5 10 15 20 25 h
Height (meters)
0 1 2 3 4 5 6 7 d
Inside diameter (inches)
Algebra 1
Chapter 11 Resource Book
263
Name ———————————————————————
LESSON
11.1
Date ————————————
Review for Mastery
For use with pages 734–741
GOAL
Graph square root functions.
Vocabulary
A radical expression is an expression that contains a radical, such as a
square root, cube root, or other root.
A radical function involves a radical expression with the independent
variable in the radicand.
If the radical is a square root, then the function is called a square root
function.
LESSON 11.1
The most basic square root function in the family of all square root
}
functions, called the parent square root function, is y 5 Ï x .
EXAMPLE 1
}
Graph a function in the form y 5 aÏ x
}
Graph the function y 5 5Ï x and identify its domain and range. Compare
}
the graph with the graph of y = Ï x .
Solution
Make a table. Because the square root of
a negative number is undefined, x must be
non-negative. So the domain is x ≥ 0.
y
14
y55 x
10
x
0
1
2
3
6
y
0
5
7.1
8.7
2
y5 x
STEP 2
Plot the points.
STEP 3
Draw a smooth curve through the points. From either the table or the graph,
you can see the range of the function is y ≥ 0.
STEP 4
Compare the graph with the graph of y 5 Ï x . The graph of y 5 5Ï x is a
2
6
}
10
x
}
}
vertical stretch (by a factor of 5) of the graph of y 5 Ï x .
Exercises for Example 1
Graph the function and identify its domain and range. Compare the
}
graph with the graph of y 5 Ï x .
}
1. y 5 4Ï x
264
Algebra 1
Chapter 11 Resource Book
}
2. y 5 26Ï x
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
LESSON
11.1
Review for Mastery
For use with pages 734–741
EXAMPLE 2
Date ————————————
continued
}
Graph a function in the form y 5 Ï x 1 k
}
Graph the function y 5 Ï x 22 and identify its domain and range.
}
Compare the graph with the graph of y 5 Ï x .
Solution
To graph the function, make a table, plot the
points, and draw a smooth curve through the
points. The domain is x ≥ 0.
y
3
x
0
1
2
3
4
1
y
22
21
20.6
20.3
0
21
y5 x
5
x
y5 x22
LESSON 11.1
}
The range is y ≥ 22. The graph of y 5 Ï x 2 2 is
a vertical translation (of 2 units down) of the graph
}
of y 5 Ï x .
EXAMPLE 3
}
Graph a function in the form y 5 aÏx 2 h 1 k
}
Graph the function y 5 3Ï x 1 2 2 4.
Copyright © Holt McDougal. All rights reserved.
Solution
}
STEP 1
Sketch the graph of y 5 3Ï x .
STEP 2
Shift the graph h units horizontally
and k units vertically. Notice that
}
}
y 5 3Ï x 1 2 2 4 5 3Ï x 2 (22) 1 (24).
So, h 5 22 and k 5 24. Shift the graph
left 2 units and down 4 units.
y
y53 x
10
6
26
(0, 0)
y53 x1224
22
2
6
10
x
(22, 24)
Exercises for Examples 2 and 3
Graph the function and identify its domain and range. Compare the
}
graph with the graph of y 5 Ï x .
}
3. y 5 Ï x 1 1
}
4. y 5 Ï x 2 3
5. Identify the domain and range of the function in Example 3.
Algebra 1
Chapter 11 Resource Book
265
Name ———————————————————————
LESSON
11.1
Date ————————————
Challenge Practice
For use with pages 734–741
In Exercises 1–5, graph the function and identify its domain and range.
}
Compare the graph with the graph of y 5 Ï x .
}
1. y 5 Ï 6 2 x
}
2. y 5 2Ï 2x
}
3. y 5 2Ï 1 2 x
}
4. y 5
Ï}2 x 1 2
1
}
LESSON 11.1
5. y 5 Ï 2 2 x 1 3
In Exercises 6–10, write a rule for a radical function that has the
given properties.
6. The domain is all real numbers greater than or equal to 2. The range is all real
numbers greater than or equal to 1.
7. The domain is all real numbers less than or equal to 4. The range is all real numbers
greater than or equal to 0.
8. The domain is all real numbers greater than or equal to 0. The range is all real
numbers less than or equal to 1.
9. The domain is all real numbers less than or equal to 5. The range is all real numbers
less than or equal to 3.
10. The domain is all real numbers greater than or equal to 21. The range is all real
Copyright © Holt McDougal. All rights reserved.
numbers greater than or equal to 0.
266
Algebra 1
Chapter 11 Resource Book
Name ———————————————————————
LESSON
11.2
Date ————————————
Activity Support Master
For use with page 742
Values of
a and b
Value of
}
}
Ïa p Ïb
Value of
}
Ïab
Value of
}
}
Ïa p Ïb
Value of
}
Ïab
Value
of
}
Value of
Value
of
}
Value of
a 5 4, b 5 9
a 5 9, b 5 16
a 5 25, b 5 4
a 5 16, b 5 36
Values of
a and b
a 5 2, b 5 3
a 5 10, b 5 5
a 5 7, b 5 11
a 5 13, b 5 6
Values of
a and b
Ïa
}
}
Ïb
Î}ba
}
a 5 4, b 5 16
LESSON 11.2
Copyright © Holt McDougal. All rights reserved.
a 5 9, b 5 25
a 5 36, b 5 4
a 5 4, b 5 49
Values of
a and b
Ïa
Ïb
}
}
Î}ba
}
a 5 1, b 5 2
a 5 3, b 5 8
a 5 12, b 5 7
a 5 6, b 5 11
Algebra 1
Chapter 11 Resource Book
267
Name ———————————————————————
LESSON
11.2
Date ————————————
Practice A
For use with pages 742–750
Match the radical with the simplified expression.
}
}
1. Ï 150
}
2. Ï 90
}
3. Ï 60
}
A. 3Ï 10
}
B. 2Ï 15
C. 5Ï 6
Simplify the expression.
}
5. Ï 28
}
}
8. Ï 27a
4. Ï 99
}
7. Ï 50
10.
}
6. Ï 54
}
Ï100n 3
11.
9.
}
}
Ï16x 2
}
}
12. Ï 3 p Ï 15
Ï125p 3
Name the value of 1 that you would multiply the radical expression by to
rationalize the denominator.
3
14. }
}
Ï10
1
13. }
}
Ï23
1
15. }
}
Ï 5x
Simplify the expression by rationalizing the denominator.
1
16. }
}
Ï5
7
18. }
}
Ï3
1
17. }
}
Ï17
Simplify the expression.
}
20. 10Ï 2 2 3Ï 2
}
}
23. 5Ï 8 2 4Ï 8
}
26. Ï 3 (Ï 3 2 2)
LESSON 11.2
22. 4Ï 18 1 Ï 18
}
25. Ï 2 (1 1 Ï 2 )
}
}
}
}
}
}
}
}
21. Ï 7 2 4Ï 7
}
}
24. Ï 12 1 3Ï 3
}
}
27. Ï 3 (1 1 Ï 12 )
}
28. Electricity The voltage V (in volts) required for a circuit is given by V 5 Ï PR
where P is the power (in watts) and R is the resistance (in ohms). Find the volts
needed to light a 60-watt light bulb with a resistance of 110 ohms. Round your
answer to the nearest tenth.
29. Drum Heads The radius r (in inches) of a circle with an area A (in square inches)
Î
}
A
is given by the function r 5 }
.
π
a. The drum head on a conga drum has an area of 16π square inches. Find the
diameter of the drum head.
b. The drum head on a bongo has an area of 9π square inches. Find the diameter
of the drum head.
268
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
}
19. 3Ï 5 1 4Ï 5
Name ———————————————————————
Date ————————————
Practice B
LESSON
11.2
For use with pages 742–750
Simplify the expression.
}
}
1. Ï 200
}
}
4. Ï 400d
}
5.
}
}
Ï9y 2
}
7. Ï 3 p Ï 21
6.
}
8. Ï 20 p Ï 15
Î1681
}
10.
}
3. Ï 112
2. Ï 45
}
11.
}
9. Ï 10x p Ï 2x
Î
}
Î}495
}
}
Ï25n3
12.
x2
144
}
Simplify the expression by rationalizing the denominator.
Î503
}
4
13. }
}
Ï5
14.
2
16. }
}
Ïp
1
17. }
}
Ï3y
Î}759
}
}
15.
9
18. }
}
Ï2x
Simplify the expression.
}
}
}
19. 10Ï 7 1 3Ï 7
}
}
20. 4Ï 5 2 7Ï 5
}
}
}
22. Ï 5 (8Ï 10 1 1)
23.
(2Ï3 1 5)2
}
21. Ï 7 (4 2 Ï 7 )
}
24.
}
(6 1 Ï3 )(6 2 Ï3 )
25. Water Flow You can measure the speed of water by using an
h
L-shaped tube. The speed V of the water (in miles per hour) is
Î5
}
Copyright © Holt McDougal. All rights reserved.
V
column of water above the surface (in inches).
a. If you use the tube in a river and find that h is 6 inches, what is the speed of the
water? Round your answer to the nearest hundredth.
b. If you use the tube in a river and find that h is 8.5 inches, what is the speed of
the water? Round your answer to the nearest hundredth.
LESSON 11.2
given by the function V 5 }2 h where h is the height of the
26. Walking Speed The maximum walking speed S (in feet per second) of an animal
}
is given by the function S 5 Ï gL where g is 32 feet per second squared and L is
the length of the animal’s leg (in feet).
a. How fast can an animal whose legs are 9 inches long walk? Round your answer
to the nearest hundredth.
b. How fast can an animal whose legs are 3 feet long walk? Round your answer to
the nearest hundredth.
Algebra 1
Chapter 11 Resource Book
269
Name ———————————————————————
Date ————————————
Practice C
LESSON
11.2
For use with pages 742–750
Simplify the expression.
}
}
1.
Ï45s 3
4.
Ï124m4n10
2.
}
}
7.
}
Ï196r 4
3.
Ï450c5
6.
Ïa3b p Ïab
9.
Î
}
12.
Î
}3
}
}
5. 11Ï x7y 8
Ï27xy p Ï5y
Î
}
}
}
3
8.
}
121
}2
16m
5d 2
125
Simplify the expression by rationalizing the denominator.
Î
10.
Î
}
}
5
}
8
11.
}
7m5
}
11
125
4x
Simplify the expression.
}
}
}
13. Ï 15 1 5Ï 3 2 2Ï 27
}
}
}
14. Ï 7 (3 2 2Ï 7 )
}
}
15. Ï 2 (3Ï 14 2 Ï 7 )
Î
}
}
16.
2
(3Ï12 1 5)
2
5
19. }
}
} 1 }
Ï 14
Ï7
}
17.
}
}
(8Ï3 1 Ï2 )(1 2 Ï3 )
}
2
4Ï 10 2 }
20. }
}
}
Ï3
Ï30
18.
250m3
2n
}
5
4
21. }
}
} 1 }
2Ïx
Ïx
Î}PR
LESSON 11.2
where I is the current (in amps), P is the power (in watts), and R is the resistance
(in ohms).
a. A light bulb with a 283-ohm resistor is using 0.42 amp of current. What is the
wattage of the light bulb? Round your answer to the nearest whole watt.
b. A light bulb with a 145-ohm resistor is using 0.83 amp of current. What is the
wattage of the light bulb? Round your answer to the nearest whole watt.
23. Medicine A doctor may need to know a person’s body surface area to prescribe the
correct amount of medicine. A person’s body surface area A (in square meters) is
given by the function
Î hw
}
A5 }
3131
where h is the height (in inches) and w is the weight (in pounds).
a. Find the body surface area of a person who is 5 feet 5 inches tall and weighs
110 pounds. Round your answer to the nearest tenth of a square meter.
b. Find the body surface area of a person who is 5 feet 10 inches tall and weighs
120 pounds. Round your answer to the nearest tenth of a square meter.
270
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
}
22. Electricity Current, power, and resistance are related by the formula I 5
Name ———————————————————————
LESSON
11.2
Date ————————————
Review for Mastery
For use with pages 742–750
GOAL
Simplify radical expressions.
Vocabulary
A radical expression is in simplest form if the following conditions
are true:
• No perfect square factors other than 1 are in the radicand.
• No fractions are in the radicand.
• No radicals appear in the denominator of a fraction.
The process of eliminating a radical from an expression’s denominator
is called rationalizing the denominator.
EXAMPLE 1
Use the product property of radicals
Simplify the expression.
Solution
}
}
a. Ï 28 5 Ï 4 p 7
}
Factor using perfect square factor.
}
5 Ï4 p Ï7
Product property of radicals
}
5 2Ï 7
}
Copyright © Holt McDougal. All rights reserved.
}}
Ï50y 3 5 Ï25 p 2 p y2 p y
}
}
}
}
5 Ï 25 p Ï 2 p Ï y 2 p Ï y
Factor using perfect square factors.
Product property of radicals
}
5 5y Ï 2y
EXAMPLE 2
Simplify.
LESSON 11.2
b.
Simplify.
Multiply radicals
Simplify the expression.
Solution
}
}
}
a. Ï 2 p Ï 18 5 Ï 2 p 18
Product property of radicals
}
}
5 Ï36
Multiply.
56
Simplify.
}
}
b. 5Ï 2xy p Ï 32y 5 5Ï 2xy p 32y
Product property of radicals
}
5 5Ï64xy 2
}
Multiply.
}
}
55Ï 64 p Ï x p Ï y 2
}
5 40yÏ x
Product property of radicals
Simplify.
Algebra 1
Chapter 11 Resource Book
271
Name ———————————————————————
LESSON
11.2
Review for Mastery
For use with pages 742–750
EXAMPLE 3
Date ————————————
continued
Use the quotient property of radicals
a.
Î1725 5 ÏÏ1725
}
}
}
}
}
Quotient property of radicals
}
Ï17
Î
}
b.
Simplify.
5}
5
}
Ï4
4
}2 5 }
}
49y
Ï49y 2
Quotient property of radicals
2
Simplify.
5}
7y
Exercises for Examples 1, 2, and 3
Simplify the expression.
}
1. Ï 72
2.
}
}
}
5. Ï 5 p Ï 10
EXAMPLE 4
}
Ï3x 2
3.
}
6. Ï 3x p Ï 15xy
7.
}
}
4. 3Ï 12x 2
Ï45y 5
Ï
}
5
}
81
8.
Ï
}
2x 2
9y
}2
Rationalize the denominator
3
Ï6x
3
Ï6x
}
}
Ï6x
Ï6x
Ï6x
Ï6x
}
} p }
}
} 5 }
Multiply by }
}.
Product property of radicals
5}
}
Ï36x 2
}
3Ï6x
}
Ï6x
Simplify.
5}
5}
6x
2x
EXAMPLE 5
Add and subtract radicals
}
}
}
}
3Ï3 1 6Ï 27 5 3Ï 3 1 6Ï 9 p 3
}
}
Factor using perfect square factor.
}
5 3Ï 3 1 6 p Ï 9 p Ï 3
}
}
Product property of radicals
}
5 3Ï 3 1 18Ï 3 5 21Ï 3
Simplify.
Exercises for Examples 4 and 5
Simplify the expression.
3
9. }
}
Ï2x
272
Algebra 1
Chapter 11 Resource Book
10.
}
}
}
6Ï 7 1 8Ï 10 2 3Ï 7
}
}
11. 3Ï 5 1 2Ï 500
Copyright © Holt McDougal. All rights reserved.
LESSON 11.2
}
3Ï 6x
Name ———————————————————————
Date ————————————
Challenge Practice
LESSON
11.2
For use with pages 742–750
In Exercises 1–5, simplify the expression.
1
1
1. }
}
} 2 }
Ï8
Ï2
2.
}
}
}
}
(3Ï6 2 Ï18 )(2Ï6 1 2Ï18 )
}
}
}
3. Ï 5 (2Ï 10 2 3Ï 15 )
}
}
}
}
4.
(4Ï2x 2Ïx ) (Ï3x 1 3Ïx ); x > 0
5.
Ïy (Ï2y 1 5Ï4y ); y > 0
}
}
}
In Exercises 6–8, use the following information.
A student studying the falling velocity of a skydiver jumping out of an airplane at a height
Îd 2 h
}
of d feet above the ground decides to model the velocity by the equation v 5 }
c
where v is the velocity in feet per second, c is a constant measuring the coefficient of drag
caused by the air resistance of the skydiver, and h is the height of the skydiver above the
ground in feet.
6. Suppose a skydiver jumps from a height of 10,000 feet wearing a normal jumpsuit
1
with a coefficient of drag c 5 }2. What is the velocity of the skydiver, in miles per
hour, when the skydiver is 1000 feet above the ground? Round your answer to the
nearest tenth.
7. Suppose a skydiver jumps from a height of 10,000 feet wearing a low drag
Copyright © Holt McDougal. All rights reserved.
when the skydiver is 1000 feet above the ground? Round your answer to the
nearest tenth.
8. Suppose two skydivers, Ann and Bob, jump simultaneously from two different
LESSON 11.2
1
jumpsuit with a coefficient of drag c 5 }4 . What is the velocity of the skydiver
planes. Ann jumps from a height of 12,000 feet wearing a jumpsuit with a
1
coefficient of drag c 5 }3 . Bob is wearing a jumpsuit with a coefficient of drag
3
c 5 }4 . Ann and Bob both plan to open their parachutes at a height of 2000 feet.
From what height should Bob jump if he wants his velocity to be the same as Ann’s
velocity when they open their parachutes? Round your answer to the nearest foot.
Algebra 1
Chapter 11 Resource Book
273
Name ———————————————————————
FOCUS ON
11.2
Date ————————————
Practice
For use with pages 751–752
Simplify the expression. Assume variables are nonzero.
}
3 }
3 }
3z
1. Î 27 ? Î 27
2. 3 }
281
Î
Î
3.
3 }
Î3 }
7 ? Î49x
}
4.
3 }
Î3 }
2 ? Î232
5.
4
7. }
3}
Ï2
10.
3}
3}
3}
13. Ï 2z 2 Ï z
}
16.
5y
40
}
14.
}
3
3
1Ï
32 1 4 21 Ï2 2 1 2
}
3}
3}
Ï3 1 4 1 Ï9x 2
3}
3}
17. Ï 25 1 4 2 Ï 225 2
1
3}
12.
3}
3}
3}
3}
Ï254 1 3Ï2
15. Ï 22 1 2 1 Ï 4 2
}
18.
}
3
3
1Ï
8 1 2 21 Ï 28 2 2 2
3}
2 Ï281 5 3.
Copyright © Holt McDougal. All rights reserved.
FOCUS ON 11.2
3}
9. 7Ï x 2 5Ï x
3
3
11. 3Ï
p 1Ï
27p
x
2
}
22. Challenge Solve the equation }
3} 1 3}
Ï3 Ï23
274
6
6. }
3}
Ï72
}
12
8. }
3}
Ï29
Ï128 2 4Ï2
3}
3
Algebra 1
Chapter 11 Resource Book
Name ———————————————————————
FOCUS ON
11.2
Date ————————————
Review for Mastery
For use with pages 751–752
GOAL
Perform operations with cube roots.
Key Concepts
The Product Property of Cube Roots states that the cube root of a
product equals the product of the cube roots.
The Quotient Property of Cube Roots states that the cube root of
a quotient equals the quotient of the cube roots of the numerator and
denominator.
When rationalizing a denominator, multiply by a form of 1 that will
make the radicand in the denominator a perfect cube.
You can use the distributive property to simplify sums and differences
of cube roots when the expressions have the same radicand.
EXAMPLE 1
Use properties of radicals
3}
3}
3}
a. Ï 4 p Ï 54 5 Ï 4 p 54
Product property of cube roots
3}
Multiply.
56
Simplify.
3}
}
b.
5 Ï216
a
a
5}
Î}
28
Ï28
Ï
3
Quotient property of cube roots
3}
Simplify.
Copyright © Holt McDougal. All rights reserved.
Exercises for Example 1
Simplify the expression. Assume variables are nonzero.
1.
EXAMPLE 2
3}
Î648x
}
3}
Ï16 p Ï32
2.
3
3}
}
3
3. Ï 4 p Ï
2y
}
FOCUS ON 11.2
3}
Ïa
5}
22
Rationalize the denominator
4
4
3}
Ï2
Ï2
}
} }
3} 5 3} p 3}
Ï4
Ï4
3}
Ï2
Ï2
Multiply by }
3 }.
3}
4Ï 2
5}
3}
Ï8
3}
5 2Ï 2
Product property of cube roots
Simplify.
Algebra 1
Chapter 11 Resource Book
275
Name ———————————————————————
FOCUS ON
11.2
Review for Mastery
Date ————————————
continued
For use with pages 751–752
Exercises for Example 2
Simplify the expression. Assume variables are nonzero.
2
4. }
3}
Ï2
EXAMPLE 3
3
6. }
3}
Ï3
1
}
3}
5.
Ï5
Add and subtract cube roots
3}
3}
3}
a. 3Ï n 1 2Ï n 5 (3 1 2)Ï n
Distributive property
3}
5 5Ï n
b.
3}
3}
Simplify.
3}
3}
Ï81 2 2Ï3 5 Ï3 p 27 2 2Ï3
3}
Factor using perfect cube factor.
3}
5 3Ï 3 2 2Ï 3
Product property of cube roots
3}
5 (3 2 2)Ï3
Distributive property
3}
5 Ï3
Simplify.
Exercises for Example 3
Simplify the expression. Assume variables are nonzero.
}
EXAMPLE 4
8.
3}
3}
Ï135 1 3Ï5
Multiply expressions involving cube roots
a.
3}
3}
3}
3}
3}
3}
3}
Ï6 1 4 1 Ï36x 2 5 4Ï6 1 Ï6 p Ï36x
Distributive property
Product property of
cube roots
5 4Ï6 1 Ï 216x
3}
3}
5 4Ï6 1 6Ï x
}
b.
Simplify.
}
3
3
1Ï
9 2 3 21 Ï3 1 1 2
3}
3}
3}
3}
5 Ï 9 1 Ï3 2 1 Ï9 (1) 1 (23)Ï3 1 (23)(1)
3}
3}
Multiply.
Product property of
cube roots
3}
5 Ï 27 1 Ï 9 2 3Ï3 2 3
3}
3}
5 3 1 Ï 9 2 3Ï 3 2 3
3}
Simplify.
3}
5 Ï 9 2 3Ï 3
Combine like terms.
Exercises for Example 4
Simplify the expression. Assume variables are nonzero.
3}
3}
9. Ï 4 1 2 2 Ï 16x 2
276
Algebra 1
Chapter 11 Resource Book
}
10.
}
3
3
1Ï
25 1 5 2 1 Ï5 2 1 2
Copyright © Holt McDougal. All rights reserved.
FOCUS ON 11.2
}
3
3
7. 4Ï
p 2 2Ï
p
Name ———————————————————————
LESSON
11.3
Date ————————————
Practice A
For use with pages 7532758
Determine whether the given value is a solution of the equation.
}
}
1. Ï 2x 1 5 5 3; 2
2. Ï 3x 2 1 5 4; 25
}
}
3. Ï 7x 1 3 5 10; 1
4. Ï 2x 1 10 5 4; 23
}
}
5. Ï 1 2 4x 5 5; 26
6. Ï 6 1 3x 5 12; 22
Isolate the radical expression on one side of the equation. Do not
solve the equation.
}
}
7. 7Ï x 2 21 5 0
8. 22Ï x 1 8 5 0
}
}
9. 3Ï x 1 5 5 14
10. Ï x 1 5 2 1 5 8
}
}
11. Ï x 2 4 2 6 5 22
12. Ï 2x 1 3 2 10 5 3
Solve the equation. Check for extraneous solutions.
}
}
13. Ï x 2 2 5 13
}
14. Ï x 1 6 5 14
}
15. 8Ï x 2 24 5 0
}
16. 5Ï x 2 15 5 0
}
17. Ï 4x 1 3 5 15
}
18. Ï 2x 2 7 5 5
}
19. Ï 2x 2 1 5 7
}
20. Ï 3x 1 7 5 4
21. 2Ï x 1 5 5 12
Simplify each side of the equation.
}
}
22.
(Ï7x 1 3 )2 5 (Ï7x 2 1 )2
24.
(Ï9 2 2x )2 5 (5x)2
}
23.
}
}
}
25. (2x)2 5 (Ï 3x 1 1 )
2
}
2
27.
(Ï4x 2 3 )2 5 (x 2 2)2
Solve the equation. Check for extraneous solutions.
}
}
28. Ï 2x 1 5 5 Ï 3x 1 4
}
}
29. Ï 9x 2 3 5 Ï 7x 1 9
}
30. x 5 Ï 6 2 x
31. Free-Falling Object The velocity v of a free-falling object (in feet per second),
the height h in which it falls (in feet), and the acceleration
due to gravity, 32 feet
}
per second squared, are related by the function v 5 Ï 64h .
a. Find the height from which a tennis ball was dropped if it hits the ground with
a velocity of 32 feet per second.
b. How much higher than the ball in part (a) was a tennis ball dropped from if it
hits the ground with a velocity of 40 feet per second?
32. Children’s Museum A new children’s museum opens. For the first 12 weeks,
the number of people N (in hundreds of people) that visit the museum can be
}}
modeled by the function N 5 Ï 1000 1 300t where t is the number of weeks
since the opening week.
a. After how many weeks did 4000 (or 40 hundred) people visit the museum?
b. After how many weeks did 5000 (or 50 hundred) people visit the museum?
Algebra 1
Chapter 11 Resource Book
LESSON 11.3
Copyright © Holt McDougal. All rights reserved.
26. (x 1 1)2 5 (Ï 1 2 3x )
}
(Ï5x 2 8 )2 5 (Ï1 2 6x )2
277
Name ———————————————————————
LESSON
11.3
Date ————————————
Practice B
For use with pages 7532758
Determine whether the given value is a solution of the equation.
}
}
1. 4Ï 2x 2 3 5 12; 2
}
2. 2Ï 9x 2 1 5 20; 7
}
}
3. Ï 4x 1 8 5 Ï 6 1 2x ; 21
}
4. Ï 7x 2 2 5 Ï 8 2 3x ; 21
}
}
5. x 5 Ï 4x 2 3 ; 3
6. Ï 4x 2 3 5 x 2 2; 7
Describe the steps you would use to solve the equation. Do not
solve the equation.
}
}
8. 6Ï 4 2 x 2 3 5 1
7. Ï 7x 1 3 2 5 5 2
}
}
}
}
}
10. 10Ï 6 2 x 5 2Ï x 1 4
9. Ï 12x 2 7 5 Ï 9x 1 3
}
}
11. Ï 5x 2 3 2 Ï 10 2 4x 5 0
12. Ï 9x 1 1 2 2 5 x
Solve the equation. Check for extraneous solutions.
}
16. Ï x 2 6 2 2 5 4
}
19. Ï 5x 1 4 2 12 5 26
}
}
22. Ï x 5 Ï 5x 2 1
}
}
25. Ï 7x 2 5 5 Ï 3x 1 19
}
28. Ï 5x 2 6 5 x
}
14. Ï 5x 2 4 5 16
}
17. Ï x 1 9 2 5 5 2
}
20. 3Ï x 1 5 2 3 5 6
}
}
}
}
23. Ï 7x 2 6 5 Ï x
26. Î x 2 15 2 Î x 2 7 5 0
}
29. x 5 Ï 2x 1 24
}
15. Ï x 1 3 1 8 5 15
}
18. Ï 8 2 3x 1 5 5 6
}
21. 4Ï 2x 1 1 2 7 5 1
}
}
24. Ï 6x 2 8 5 Ï 4x 2 10
}
}
27. Ï 10x 2 3 2 Ï 8x 2 11 5 0
}
30. Ï 2x 2 15 5 x
31. Market Research A marketing department determines that the price of a
magazine subscription
and the demand to subscribe are related by the function
}}
P 5 40 2 Ï 0.0004x 1 1 where P is the price per subscription and x is the number
of subscriptions sold.
a. If the subscription price is set at $25, how many subscriptions would be sold?
Round your answer to the nearest whole subscription.
b. If the subscription price is set at $30, how many more subscriptions are sold
in part (a) than when the price is $30. Round your answer to the nearest whole
subscription.
LESSON 11.3
32. Awning The area A of a portion of a circle bounded by two radii r and angle t
278
of a sector of a circle are related by the function
5p
3
}
2A
r 5 Î}
t .
The length of a side (radius) of the top view of the awning shown at the right
5π
is 6 feet and the angle that is formed by the awning is }
. Find the area of the
3
awning. Round your answer to the nearest hundredth.
Algebra 1
Chapter 11 Resource Book
6 ft
Copyright © Holt McDougal. All rights reserved.
}
13. 8Ï x 2 32 5 0
Name ———————————————————————
LESSON
11.3
Date ————————————
Practice C
For use with pages 7532758
Describe how you would solve the equation. Do not solve the equation.
}
1. 1 1 Ï x 1 6 5 13
}
4. 6Ï 5x 1 3 2 5 5 2
}
}
7. 6Ï 5 2 2x 5 3Ï 5x 2 2
}
2. 15 2 Ï 2x 1 2 5 13
Î}32 x 2 1
}
}
5. Ï 10 2 6x 5
}
8. x 1 1 5 Ï 3 2 2x
}
3. 4 2 2Ï 1 2 4x 5 26
}
}
6. Ï 3 2 2x 2 Ï 2 1 4x 5 0
}
9. x 1 Ï 1 2 3x 5 25
Solve the equation. Check for extraneous solutions.
}
}
10. 3Ï x 1 9 5 4
11. 7Ï 3x 2 4 1 7 5 35
}
13. 3Ï 5 1 x 2 8 5 4
}
}
19. 2x 5 Ï 11x 1 3
}
22. Ï x 1 3 5 Ï x 1 12
Î}14 x 2 5 2 Îx 2 9 5 0
}
}
16. Ï 4x 2 3 2 Ï 6x 2 11 5 0
}
}
14. 10 1 4Ï 3 2 2x 5 14
17.
}
}
20. Ï 3x 1 6 5 x 2 4
}
}
23. 4 2 Ï x 2 3 5 Ï x 1 5
}
12. 14 2 5Ï 8 2 3x 5 19
}
15. 2Ï 5 2 2x 2 13 5 217
}
18. Ï 8 2 6x 5 3x
}
21. x 1 3 5 Ï 2x 1 21
}
}
24. Ï 4x 1 3 1 Ï 4x 5 3
25. Write a radical equation that has 22 and 3 as solutions.
temperature.
The speed v (in meters per second) is given by the function
}
v 5 20Ï t 1 273 where t is the temperature (in degrees Celsius).
a. A friend is throwing a tennis ball against a wall 200 meters from you. You hear
the sound of the ball hitting the wall 0.6 second after seeing the ball hit the wall.
What is the temperature? Round your answer to the nearest tenth.
b. The temperature 2273°C is called absolute zero. What is the speed of sound at
this temperature?
27. Pendulum The period T (in seconds) of a pendulum is the time it takes for the
pendulum to swing back and forth. The period is related to the length L (in inches)
Î
}
L
of the pendulum by the model T 5 2π }
.
384
a. Find the length of a pendulum with a period of 2 seconds. Round your
answer to the nearest tenth.
b. What is the length of a pendulum whose period is double the period of the
pendulum in part (a)? Round your answer to the nearest tenth.
L
LESSON 11.3
Copyright © Holt McDougal. All rights reserved.
26. Speed of Sound The speed of sound near Earth’s surface depends on the
Algebra 1
Chapter 11 Resource Book
279
Name ———————————————————————
LESSON
11.3
Date ————————————
Review for Mastery
For use with pages 7532758
GOAL
Solve radical equations.
Vocabulary
An equation that contains a radical expression with a variable in the
radicand is a radical equation.
Squaring both sides of the equation a 5 b can result in a solution of
a2 5 b2 that is not a solution of the original equation. Such a solution
is called an extraneous solution.
EXAMPLE 1
Solve a radical equation
}
Solve 16Ï x 2 4 5 0.
Solution
}
16Ï x 2 4 5 0
Write original equation.
}
16Ï x 5 4
4
Ïx 5 }
16
}
Divide each side by 16.
1
Ïx 5 }
4
Simplify.
2
142
}
1
1 Ï x 22 5 }
1
Square each side.
Simplify.
x5}
16
1
The solution is }
.
16
CHECK
Check your solution by substituting it in the original equation.
}
16Ï x 2 4 5 0
Î
Write original equation.
}
1
2400
16 }
16
16 p 1 }4 2 2 4 0 0
LESSON 11.3
1
280
050✓
1
Substitute }
for x.
16
Simplify.
Solution checks.
Exercise for Example 1
}
1. Solve 5Ï x 2 15 5 0.
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
}
Add 4 to each side.
Name ———————————————————————
LESSON
11.3
Review for Mastery
For use with pages 7532758
EXAMPLE 2
Date ————————————
continued
Solve a radical equation
}
Solve 3Ï x 1 2 1 17 5 32.
Solution
}
3Ïx 1 2 1 17 5 32
Write original equation.
}
3Ïx 1 2 5 15
Subtract 17 from each side.
}
Ïx 1 2 5 5
}
1 Ïx 1 2
22
Divide each side by 3.
5 52
Square each side.
x 1 2 5 25
Simplify.
x 5 23
Subtract 2 from each side.
The solution is 23.
CHECK To check the solution using a graphing
calculator, first
rewrite the equation so that on one
}
2 2 15 5 0. Then graph the related
side is 0: 3Ï x 1}
equation y 5 3Ï x 1 2 2 15. You can see that the
graph crosses the x-axis at x 5 23.
EXAMPLE 3
X=23
Y=0
Solve an equation with radicals on both sides
}
}
Solution
}
}
Ïx 1 3 5 Ï3x 2 5
}
}
1 Ï x 1 3 22 5 1 Ï 3x 2 5 22
x 1 3 5 3x 2 5
Write original equation.
Square each side.
Simplify.
22x 1 3 5 25
Subtract 3x from each side.
Subtract 3 from each side.
22x 5 28
x54
Divide each side by 22.
The solution is 4. Check the solution.
LESSON 11.3
Copyright © Holt McDougal. All rights reserved.
Solve Ï x 1 3 5 Ï 3x 2 5 .
Exercises for Examples 2 and 3
Solve the equation.
}
2. 5Ï x 2 3 2 12 5 18
}
}
3. Ï x 1 2 5 Ï 4x 2 7
}
}
4. Ï 5x 2 12 2 Ï 2x 1 9 5 0
Algebra 1
Chapter 11 Resource Book
281
Name ———————————————————————
LESSONS
11.1–11.3
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 7342758
1999–2005, the annual revenue y
(in millions of dollars) of a company
can be
}
modeled by y 5 170 1 38Ï x where x is the
number of years since 1999.
a. Graph the function.
b. In what year was the revenue about
$255 million?
2. Multi-Step Problem The final velocity
v (in meters per second) of an object after
traveling a distance of 200 meters with a
constant acceleration of 0.5 meter per
}
second squared is given by v 5 Ï v02 1 200
where v0 is the initial velocity of the object.
a. Graph the function.
b. What is the final velocity of an object
after 200 meters that has an initial
velocity of 20 meters per second?
c. What is the initial velocity of an
object that travels 200 meters and
has a final velocity of 35 meters per
second?
3. Open-Ended The velocity v (in meters per
LESSON 11.3
second) of a car moving in a circular path
that has
radius r (in meters) is given by
}
v 5 Ï ar where a is the centripetal
acceleration (in meters per second squared)
of the car. A car is traveling at a constant
velocity of 15 meters per second in a
circular path of radius r where r ≥ 30.
Choose two different values of r to show
how the centripetal acceleration a of the car
changes as the radius increases.
4. Gridded Response Many birds drop
clams or other shellfish in order to break
the shells and get the food inside. The
time t (in seconds) it takes for a clam
to fall}a distance d (in feet) is given by
Ïd
t5}
. A bird drops a clam and it takes
4
1.75 seconds to hit the ground. What is the
height of the bird, in feet?
282
Algebra 1
Chapter 11 Resource Book
5. Short Response The velocity v
(in meters per second) of an object
moving in a straight path can be modeled
Î 2E
}
by the equation v 5 }
m where E is the
kinetic energy (in joules) of the object and
m is the mass (in kilograms) of the object.
a. A 50-kilogram boy is on a moped
that is moving at 5 meters per second.
What is the kinetic energy of the boy?
b. What happens to the kinetic energy
of an object as its mass stays constant
and its velocity increases? Explain.
6. Open-Ended Write a problem involving
distance that can be solved by simplifying
a radical expression. Find a solution of the
expression. Explain what the solution means
in the context of the problem.
7. Extended Response In chemistry,
Graham’s Law of Effusion shows the
relationship between the molecular mass
of a gas and the rate at which it will effuse.
Effusion is the process of gas molecules
escaping through tiny holes in a container.
To determine how many times greater the
rate of a gas is to the rate of oxygen, use
Î32
}
the equation r 5 }
where r is how many
M
times greater the rate of effusion is for a gas
compared to the rate of effusion for oxygen
and M is the molecular mass (in grams) of
the gas.
a. Helium has a molecular mass of
2 grams. How many times greater than
the rate of effusion for oxygen is the
rate of effusion for helium?
b. Nitrogen has a molecular mass of
28 grams. How many times greater
than the rate of effusion for oxygen is
the rate of effusion for nitrogen?
c. What happens to rate of effusion when
using a gas that has a molecular mass
greater than 28? Explain.
Copyright © Holt McDougal. All rights reserved.
1. Multi-Step Problem For the period
Name ———————————————————————
LESSON
11.3
Date ————————————
Challenge Practice
For use with pages 7532758
In Exercises 1–5, write a radical equation that has the given solutions.
1. The solutions are 1 and 2.
2. The solutions are 3, 6, and 22.
3. The solutions are 21, 0, and 1.
4. The solutions are 0, 1, and 1.
1 3
1
5. The solutions are 2}, }, and }.
2 2
2
In Exercises 6–15, write a radical equation that has the given solution(s)
and the given extraneous solution(s).
6. 1 is a solution; 23 is an extraneous solution.
7. 23 is a solution; 1 is an extraneous solution.
8. 24 is a solution; 5 is an extraneous solution.
9. 5 is a solution; 24 is an extraneous solution.
10. 0 and 2 are solutions; 25 is an extraneous solution.
11. 0 and 25 are solutions; 2 is an extraneous solution.
12. 1 is a solution; 23 and 25 are extraneous solutions.
13. 23 and 25 are solutions; 1 is an extraneous solution.
15. 2 and 6 are solutions; 27 is an extraneous solution.
LESSON 11.3
Copyright © Holt McDougal. All rights reserved.
14. 27 is a solution; 2 and 6 are extraneous solutions.
Algebra 1
Chapter 11 Resource Book
283
Name ———————————————————————
LESSON
11.4
Date ————————————
Practice A
For use with pages 760–766
LESSON 11.4
Name the legs and hypotenuse of the right triangle.
1.
2.
3.
m
z
x
n
r
t
p
y
c
Let a and b represent the lengths of the legs of a right triangle, and let c
represent the length of the hypotenuse. Find the unknown length.
4.
5.
c
a52
b53
a51
6.
a55
c
b54
7. a 5 6, b 5 4
10. a 5 9, c 5 12
b53
c
8. a 5 3, b 5 7
9. a 5 5, b 5 5
11. a 5 8, b 5 6
12. b 5 2, c 5 10
14.
15.
Find the unknown lengths.
x
x
2x
3 2
x
2 5
4 2
x
x
Tell whether the triangle with the given side lengths is a right triangle.
16. 3, 3, 9
17. 12, 16, 20
18. 6, 9, 12
19. Window A window in a house is in the shape of a square.
The side length of the window is 20 inches. What is the
length of the diagonal from one corner of the window to
the opposite corner? Round your answer to the nearest tenth.
20 in.
20 in.
20. Table Top Soccer The top of a soccer table is in the shape
of a rectangle. If the tabletop measures 60 inches by
42 inches, what is the length of the diagonal from one
corner of the table to the opposite corner? Round your
answer to the nearest tenth.
284
Algebra 1
Chapter 11 Resource Book
42 in.
60 in.
Copyright © Holt McDougal. All rights reserved.
13.
Name ———————————————————————
Date ————————————
Practice B
LESSON
11.4
For use with pages 760–766
1. a 5 1, b 5 5
2. b 5 4, c 5 9
3. a 5 6, b 5 6
4. b 5 7, c 5 12
5. a 5 2, b 5 8
6. a 5 6, b 5 30
7. a 5 4, b 5 15
8. b 5 7, c 5 11
9. a 5 10, b 5 20
10. a 5 30, b 5 40
11. a 5 15, c 5 25
12. a 5 11, b 5 22
14.
15.
LESSON 11.4
Let a and b represent the lengths of the legs of a right triangle, and let c
represent the length of the hypotenuse. Find the unknown length.
Find the unknown lengths.
x
13.
x26
2 17
3x 2 2
4x
3x 1 2
x16
4x 1 3
2x
16. A right triangle has one leg that is 3 inches longer than the other leg. The hypotenuse
}
is Ï65 inches. Find the lengths of the legs.
Tell whether the triangle with the given side lengths is a right triangle.
17. 4, 5, 6
18. 15, 20, 25
19. 9, 15, 20
Copyright © Holt McDougal. All rights reserved.
20. Shuffleboard The playing bed of a shuffleboard table is in the shape of a
20 in.
rectangle. If the playing bed measures 154 inches by 20 inches, what is the
length of the diagonal from one corner of the playing bed to the opposite
corner? Round your answer to the nearest inch.
21. Indirect Measurement You are trying to determine the distance
across a pond. You put posts into the ground at A, B, and C so that
}
angle B is a right angle. You measure and find that the length of AB
}
is 18 feet and the length of CB is 28 feet. How wide is the pond from
A to C? Round your answer to the nearest foot.
154 in.
A
C
18 ft
28 ft
B
22. Badminton You are setting up a badminton net. To keep each pole
standing straight, you use two ropes and two stakes as shown. How
long is each piece of rope? Round your answer to the nearest tenth.
8 ft
4.5 ft
4.5 ft
Algebra 1
Chapter 11 Resource Book
285
Name ———————————————————————
LESSON
LESSON 11.4
11.4
Date ————————————
Practice C
For use with pages 760–766
Let a and b represent the lengths of the legs of a right triangle, and let c
represent the length of the hypotenuse. Find the unknown length.
1. a 5 9, b 5 12
2. b 5 25, c 5 30
3. a 5 4, b 5 1.5
4. b 5 2.5, c 5 7
5. a 5 4, b 5 1.8
6. a 5 2.6, b 5 3.5
7. a 5 14, b 5 8.8
8. b 5 1.4, c 5 2.5
9. a 5 0.2, b 5 0.6
10. a 5 10.5, b 5 6.4
11. a 5 14.1, c 5 20.5
12. a 5 0.3, b 5 0.7
Find the unknown lengths.
13. A right triangle has one leg that is 4 inches shorter than the other leg.
}
The hypotenuse is Ï106 inches. Find the lengths of the legs.
14. A right triangle has one leg that is 2 times as long as the other leg.
}
The hypotenuse is Ï80 inches. Find the lengths of the legs.
3
15. A right triangle has one leg that is } of the length of the other leg.
5
}
The hypotenuse is 2Ï 34 inches. Find the lengths of the legs.
Tell whether the triangle with the given side lengths is a right triangle.
16. 4.5, 6, 7.5
17. 15, 60, 61
18. 12, 71, 72
The correct length of the guy wire that will help tether the tower
should be 39 feet long. If the correct length wire is used, how far
away from the tower should the guy wire be attached to the ground?
Round your answer to the nearest foot.
20. Shortest Route You are traveling from Valmont to Milesburg.
You can avoid the city traffic by taking the L-shaped route shown.
If you could travel straight through the city, how many miles
could you save? Round your answer to the nearest mile.
30 ft
Valmont
39 ft
13.5 mi
25.75 mi
Milesburg
21. Flag Each wilderness troop at a camping outing has created its own flag. Your
troop’s flag is triangular with side lengths of 15 inches, 18 inches, and 23 inches.
Is the flag a right triangle? Explain.
286
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
19. Guy Wire A tower that is being constructed will be 30 feet tall.
Name ———————————————————————
LESSON
11.4
Date ————————————
Review for Mastery
For use with pages 760–766
Use the Pythagorean theorem and its converse.
LESSON 11.4
GOAL
Vocabulary
The hypotenuse of a right triangle is the side opposite the right angle.
It is the longest side of a right triangle.
The legs are the two sides that form the right angle.
The Pythagorean theorem states the relationship among the lengths
of the sides of a right triangle.
The Pythagorean Theorem
Words If a triangle is a right triangle, then the sum of the squares
of the lengths of the legs equals the square of the length of the
hypotenuse.
Algebra a 2 1 b 2 5 c 2
Converse of the Pythagorean Theorem
If a triangle has side lengths a, b, and c such that a 2 1 b 2 5 c 2, then
the triangle is a right triangle.
EXAMPLE 1
Use the Pythagorean theorem
Copyright © Holt McDougal. All rights reserved.
Find the unknown length of the triangle shown.
a2 1 b2 5 c2
2
2
c 5 12
b58
Solution
Pythagorean theorem
2
a 1 8 5 12
Substitute 8 for b and 12 for c.
a2 1 64 5 144
Simplify.
a2 5 80
}
a
Subtract 64 from each side.
}
a 5 Ï 80 5 4Ï 5
Take positive square root of each side.
}
The side length a is 4Ï 5 .
Exercise for Example 1
1. The lengths of the legs of a right triangle are a 5 9 and b 5 12. Find c.
Algebra 1
Chapter 11 Resource Book
287
Name ———————————————————————
LESSON
LESSON 11.4
11.4
Review for Mastery
continued
For use with pages 760–766
EXAMPLE 2
Date ————————————
Use the Pythagorean theorem
A right triangle has one leg that is 3 inches shorter than the other leg.
}
The hypotenuse is Ï 29 inches. Find the unknown lengths.
Solution
Sketch a right triangle and label the sides with their lengths.
Let x be the length of the longer leg.
a2 1 b2 5 c 2
2
22
Substitute.
x 2 1 x 2 2 6x 1 9 5 29
Simplify.
2x 2 2 6x 2 20 5 0
Write in standard form.
2(x 2 5)(x 1 2) 5 0
x2550
or
x55
or
x
Pythagorean theorem
}
x 1 (x 2 3) 5 1 Ï 29
2
29
x23
Factor.
x1250
Zero-product property
Solve for x.
x 5 22
Because the length is non-negative, the solution x 5 22 does not make sense. The legs
have lengths of 5 inches and 5 2 3 5 2 inches.
EXAMPLE 3
Determine right triangles
Tell whether the triangle with the given side lengths is a right triangle.
b. 8, 13, 14
Solution
a.
142 1 482 0 502
196 1 2304 0 2500
2500 5 2500 ✓
The triangle is a right triangle.
b.
82 1 132 0 142
64 1 169 0 196
233 5 196 ✗
The triangle is not a right triangle.
Exercises for Examples 2 and 3
2. A right triangle has one leg that is 6 inches shorter than the other leg. The
}
hypotenuse is 5Ï 2 inches. Find the unknown lengths.
Tell whether the triangle with the given side lengths is a right triangle.
3. 4, 7, 9
288
Algebra 1
Chapter 11 Resource Book
4.
10, 12, 26
5. 33, 180, 183
Copyright © Holt McDougal. All rights reserved.
a. 14, 48, 50
Name ———————————————————————
LESSON
11.4
Date ————————————
Challenge Practice
For use with pages 760–766
1. (x, x 1 1, x 1 2)
2.
x22
, x, x 1 1 2
1}
2
3.
1 }2 1 1, x 2 2, x 2
LESSON 11.4
In Exercises 1–5, find the values of x so that the given set of values forms
a Pythagorean triple.
x
4. (x, x 1 3, x 1 6)
5.
x12
1 10, 2x 2
1 x 1 2, }
2
6. The circumference of a circle with radius 1 can be roughly approximated using the
Pythagorean theorem in the following way. Within the circle of radius 1, draw a
square whose corners just touch the circle.
y
0.25
Copyright © Holt McDougal. All rights reserved.
20.25
0.25
x
Four right triangles with legs of length 1 are formed within the diagram. Use the
Pythagorean theorem to find the hypotenuse of the triangles, then approximate the
circumference of the circle by the sum of the hypotenuses. Round your answer to
the nearest tenth.
Algebra 1
Chapter 11 Resource Book
289
Name ———————————————————————
LESSON
11.5
Date ————————————
Practice A
For use with pages 767–774
Match the pair of points with the expression that gives the distance
between the points.
1. (26, 3), (24, 2)
2. (6, 23), (24, 2)
}}
A.
Ï(24 2 6)2 1 (2 1 3)2
3. (6, 23), (4, 22)
}}
B.
Ï(4 ⫺ 6)2 1 (22 ⫹ 3)2
}}
C.
Ï(24 1 6)2 1 (2 2 3)2
Use the coordinate plane to estimate the distance between the two points.
Then use the distance formula to find the distance between the points.
4.
3
5.
y (3, 3)
21
21
LESSON 11.5
1
23
21
y
1
(4, 1)
1
3 x
6.
1
3
5 x
1
23
23
(23, 22)
y
3
(23, 2)
25
(3, 24)
3 x
1
23
(1, 23)
Find the distance between the two points.
7. (2, 4), (5, 6)
8. (7, 3), (1, 5)
9. (8, 2), (4, 1)
10. (0, b), (5, 12); d 5 13
11. (1, b), (4, 5); d 5 5
12. (2, 3), (b, 9); d 5 10
13. (1, 4), (10, b); d 5 15
14. (5, 2), (21, b); d 5 6
15. (b, 6), (3, 22); d 5 8
Find the midpoint of the line segment with the given endpoints.
16. (5, 3), (7, 11)
17. (23, 10), (9, 2)
18. (22, 24), (8, 4)
19. Bus Stop A student is taking the bus home. The student can get
off at one of two stops, as shown on the map. The distance between
consecutive grid lines represents 0.5 mile.
y
Stop 2
2.5
a. Find the distance between stop 1 and home. Round your answer
to the nearest hundredth.
b. Find the distance between stop 2 and home. Round your answer
to the nearest hundredth.
c. Which distance is shorter? By how much?
1.5
Home
0.5
Stop 1
0.5
20. Sales Use the midpoint formula to estimate the sales of a company in 2000,
given the sales in 1995 and 2005. Assume that the sales followed a linear pattern.
Year
Sales (dollars)
290
Algebra 1
Chapter 11 Resource Book
1995
2005
740,000
980,000
1.5
2.5
x
Copyright © Holt McDougal. All rights reserved.
The distance d between two points is given. Find the value of b.
Name ———————————————————————
LESSON
11.5
Date ————————————
Practice B
For use with pages 767–774
Find the distance between the two points.
1. (8, 3), (10, 4)
2. (2, 7), (5, 6)
3. (9, 6), (4, 1)
4. (0, 4), (8, 22)
5. (25, 3), (1, 2)
6. (1, 26), (22, 4)
7. (8, 27), (4, 23)
8. (210, 22), (6, 5)
9. (21, 28), (25, 22)
The distance d between two points is given. Find the value of b.
10. (b, 4), (2, 21); d 5 5
}
13. (4, 1), (5, b); d 5 Ï 17
11. (23, 2), (7, b); d 5 10
}
14. (b, 2), (3, 21); d 5 Ï 58
12. (3, 2), (b, 29); d 5 11
}
15. (24, b), (5, 22); d 5 Ï 106
Find the midpoint of the line segment with the given endpoints.
17. (27, 2), (210, 14)
18. (29, 25), (7, 214)
19. (8, 28), (3, 5)
20. (20, 5), (30, 25)
21. (211, 7), (8, 23)
LESSON 11.5
16. (2, 5), (4, 12)
Use the distance formula and the converse of the Pythagorean theorem to
determine whether the points are vertices of a right triangle.
22. (1, 1), (4, 4), (1, 4)
23. (6, 0), (6, 4), (2, 4)
24. (22, 1), (3, 5), (6, 22)
25. (6, 4), (21, 22), (24, 3)
26. (5, 3), (4, 22), (10, 2)
27. (2, 24), (2, 23), (6, 1)
Copyright © Holt McDougal. All rights reserved.
28. Walking Trail A walking trail follows the path shown on the
map. The distance between consecutive grid lines is 1 mile.
Find the total distance of the trail from start to finish. Round
your answer to the nearest mile.
y
5
Stop 2
Stop 3
Finish
Stop 1
21
Start
21
29. Amusement Park An amusement park designer wants to place
a Ferris wheel midway between the two largest coasters. The
distance between consecutive grid lines is 500 feet.
a. Determine the coordinates of where the Ferris wheel should be.
b. How far will the Ferris wheel be from each of the coasters?
Round your answer to the nearest foot.
3
5
x
y
3500
Big coaster 2
2500
1500
500
Big coaster 1
500
1500
2500
3500
x
30. Reading You have 30 days left to read the books on your summer reading list. As
of today, you have read 5 books. By the end of the 30 days, you have to have read
12 books. Assume that the books are all approximately the same length and you read
at a relatively constant pace. After 15 days, how many books should you have read?
Algebra 1
Chapter 11 Resource Book
291
Name ———————————————————————
LESSON
11.5
Date ————————————
Practice C
For use with pages 767–774
Find the distance between the two points.
1. (1, 25), (6, 7)
2. (23, 23), (8, 22)
3. (14, 25), (23, 8)
4. (211, 24), (9, 22)
5. (4, 215), (22, 10)
6. (1.5, 6), (1.5, 22)
7. (4.1, 6), (5.1, 17)
8.
1 }12, 8 2, 1 }32, 5 2
9.
1 2}13, }23 2, 1 }53, }13 2
The distance d between two points is given. Find the value of b.
}
10. (7, b), (21, 3); d 5 2Ï 17
}
13. (9, 25), (b, 6); d 5 Ï 290
}
}
12. (b, 1), (22, 8); d 5 5Ï 2
11. (4, 22), (b, 9); d 5 5Ï 5
}
}
14. (28, b), (1, 23); d 5 3Ï 10 15. (10, 210), (b, 22); d 5 2Ï 65
16. (214, 3), (10, 24)
17. (211, 26), (16, 22)
18. (105, 2214), (97, 45)
19. (3.5, 8), (4, 10.5)
20. (7.25, 21.5), (2.25, 22)
21. (28.4, 3.5), (22.6, 4.5)
Use the distance formula and the converse of the Pythagorean theorem to
determine whether the points are vertices of a right triangle.
22. (1, 24), (5, 6), (22, 3)
23. (22, 4), (5, 3), (0, 21)
24. (2, 1), (6, 23), (25, 1)
25. (22, 23), (4, 3), (3, 28)
26. (4, 22), (2, 3), (23, 1)
27. (7, 21), (26, 3), (29, 27)
28. Treasure Hunt You set up a treasure hunt with the items placed
according to the map shown. The distance between consecutive
grid lines is 200 feet.
a. Which two objects are closest together? What is the distance
between these two objects? Round your answer to the
nearest foot.
b. Which two objects are farthest apart? What is the distance
between these two objects? Round your answer to the
nearest foot.
y
1000
Backpack
200
Pen
200
29. Biking You are biking a straight-line distance between the two
towns shown on the map. The distance between consecutive grid
lines is 1 mile.
a. How far is your bike ride one way? Round your answer to
the nearest mile.
b. You stop halfway between the two towns to eat a snack.
What are the coordinates of your location?
c. On the way back, you stop one-quarter of the way from your
destination to visit a friend. How far are you from your
destination? Round your answer to the nearest mile. What
are the coordinates of your location? Explain how you got
your answers.
Basket
Book
y
Algebra 1
Chapter 11 Resource Book
1000
x
Larkin
11
9
7
5
3
Tipton
1
1
292
600
3
5
7 x
Copyright © Holt McDougal. All rights reserved.
LESSON 11.5
Find the midpoint of the line segment with the given endpoints.
Name ———————————————————————
LESSON
11.5
Date ————————————
Review for Mastery
For use with pages 767–774
GOAL
Use the distance and midpoint formulas.
Vocabulary
The Distance Formula
The distance d between any two points (x1, y1) and (x2, y2) is
}}
d 5 Ï (x2 2 x1)2 1 (y2 2 y1)2 .
The midpoint of a line segment is the point on the segment that is
equidistant from the endpoints.
The Midpoint Formula
The midpoint M of the line segment with endpoints A(x1, y1) and
x 1x y 1y
2
LESSON 11.5
1
1
2 1
2
,}
.
B(x2, y2) is }
2
2
EXAMPLE 1
Find the distance between two points
Find the distance between (3, 22) and (22, 4).
Solution
Let (x1, y1) 5 (3, 22) and (x2, y2) 5 (22, 4).
}}
d 5 Ï (x2 2 x1)2 1 ( y2 2 y1)2
Distance formula
Copyright © Holt McDougal. All rights reserved.
}}}
5 Ï (22 2 3)2 1 [4 2 (22)]2
}}
Substitute.
}
5 Ï (25)2 1 (6)2 5 Ï 61
Simplify.
}
The distance between the points is Ï 61 units.
Exercises for Example 1
Find the distance between the points.
1. (5, 2), (3, 8)
2. (22, 0), (24, 5)
3. (7, 21), (25, 3)
Algebra 1
Chapter 11 Resource Book
293
Name ———————————————————————
LESSON
11.5
Review for Mastery
For use with pages 767–774
EXAMPLE 2
Date ————————————
continued
Find a missing coordinate
}
The distance between (4, 1) and (a, 23) is Ï 52 units. Find
the value of a.
Solution
}
Use the distance formula with d 5 Ï52 . Let (x1, y1) 5 (4, 1) and (x2, y2 ) 5 (a, 23).
Then solve for a.
}}
d 5 Ï (x2 2 x1)2 1 ( y2 2 y1)2
}
}}
}
}}
}
}}
Ï52 5 Ï(a 2 4)2 1 (23 2 1)2
LESSON 11.5
Ï52 5 Ïa2 2 8a 1 16 1 16
Ï52 5 Ïa2 2 8a 1 32
Distance formula
Substitute.
Multiply.
Simplify.
52 5 a2 2 8a 1 32
Square each side.
0 5 a2 2 8a 2 20
Write in standard form.
0 5 (a 2 10)(a 1 2)
Factor.
a 2 10 5 0
a 5 10
or
a1250
or
a 5 22
Zero-product property
Solve for a.
The value of a is 10 or 22.
Exercise for Example 2
EXAMPLE 3
Find a midpoint of a line segment
Find the midpoint of the line segment with endpoints (7, 21) and (5, 7).
Solution
Let (x1, y1) 5 (7, 21) and (x2, y2) 5 (5, 7).
x1 1 x2 y1 1 y2
7 1 5 21 1 7
,}
5 }, }
1}
2
2 2 1 2
2 2
5 (6, 3)
Substitute.
Simplify.
The midpoint of the line segment is (6, 3).
Exercises for Example 3
Find the midpoint of the line segment with the given endpoints.
5. (14, 3), (6, 9)
294
Algebra 1
Chapter 11 Resource Book
6. (211, 23), (2, 25)
Copyright © Holt McDougal. All rights reserved.
4. The distance between (5, 7) and (23, b) is 17 units. Find the value of b.
Name ———————————————————————
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
LESSONS
11.4–11.5
For use with pages 760–774
1. Multi-Step Problem Use the triangle
below.
frame for a rectangular garden. He wants the
frame to have a diagonal that is 25 feet long
and connects opposite corners of the frame.
What is one possibility for the length and
width of the frame?
y
3
B
23
21
A
4. Open-Ended Andrew wants to build a
1
3 x
5. Multi-Step Problem You and a friend go
hiking. You hike 2 miles north and 3 miles
east. Starting from the same point, your
friend hikes 2 miles west and 1 mile south.
C
23
a. Find the length of each side of the
a. How far apart are you and your
friend? (Hint: Draw a diagram on a
grid.)
b. You and your friend want to meet for
lunch. Where should you meet so that
both of you hike the same minimum
distance? How far do you have to
hike?
LESSON 11.5
triangle.
b. Find the midpoint of each side of the
triangle.
c. Join the midpoints to form a new
triangle. Find the length of each of its
sides.
d. Compare the perimeters of the two
triangles.
6. Short Response You have just planted a
2. Multi-Step Problem A rescue helicopter
Copyright © Holt McDougal. All rights reserved.
and an ambulance are both traveling from
the scene of an accident to the hospital. The
distance between consecutive grid lines
represents 1 mile.
y
7
5
C(7, 5)
B(3, 5)
Hospital
3
new tree. To support the tree in bad weather,
you attach guy wires from the trunk of the
tree to stakes in the ground. You cut 25 feet
of wire into four equal lengths to make the
guy wires. You attach the four guy wires so
they are evenly spaced around the tree. You
put the stakes in the ground four feet from
the base of the trunk. Approximately how
far up the trunk should you attach the guy
wires? Explain.
7. Extended Response Molly and Julie
1
21
21
A(3, 0)
1
Accident scene
3
5
7
9 x
a. Find the distance that the ambulance
traveled (solid route).
b. How much farther did the ambulance
travel than the helicopter (dashed
route)?
3. Gridded Response A lacrosse field is a
rectangle 60 yards by 110 yards. What is the
length of the diagonal from one corner of
the field to the opposite corner? Round your
answer to the nearest yard.
leave from the same point at the same time.
Julie bicycles east at a rate that is 3 miles per
hour faster than Molly, who bicycles north.
After one hour they are 15 miles apart.
a. Let r represent Molly’s rate in miles
per hour. Write an expression for the
distance each girl has traveled in one
hour.
b. Use the Pythagorean theorem to find
how fast each person is traveling.
c. They continue to bike at the same rate
for another hour. How far apart are
they after two hours? Explain how you
found your answer.
Algebra 1
Chapter 11 Resource Book
295
Name ———————————————————————
LESSON
11.5
Date ————————————
Challenge Practice
For use with pages 767–774
In Exercises 1–10, find the values of a and b to fit the given conditions.
1
2
1
1. (2, a) is the midpoint of 3, 2} and (1, 3a).
a
2. (a, 3) is the midpoint of (1, 5) and (4, b).
1
2
1
3. (23, 2a) is the midpoint of 1, } and (b, 24a).
a
4. (a, b) is the midpoint of (21, 1) and (2a2, b2).
5.
1 }1a, }1b 2 is the midpoint of (3a, 2b) and (2a, 2b).
}
LESSON 11.5
6. The distance between (4a, a) and (3, 7) is Ï 37 units.
}
7. The distance between (25a, 2) and (21, 26a) is Ï 221 units.
1
2
3
1
8. The distance between }, 3 and 22, } is
2a
4a
1
2
}
units.
Ï}
400
5013
}
9. The distance between (6, 22) and (3, a) is Ï 13 units.
10. The distance between (a, b) and (3a, 5b) is 2a units.
Park rangers in Yellowstone National Park receive word that there is a lost hiker
somewhere in the Lamar valley. Two rangers are sent out on foot to search the trails
nearest their ranger stations. One ranger heads directly south hiking at a speed of
4 miles per hour. The other ranger heads directly east hiking at a rate of 3 miles per
hour. At these speeds the rangers should meet each other after 5 hours of hiking. Both
rangers leave their stations at the same time, and plan to hike until their paths intersect.
After hiking for three hours the first ranger finds the lost hiker and stops hiking. The
ranger decides to stay with the lost hiker and wait until the second ranger is within
radio communication distance, which is 9 miles.
11. How far apart are the two ranger stations?
12. How far apart are the two rangers when the hiker is found?
13. How long must the first ranger wait after finding the hiker until the second ranger is
within radio communication distance? Round your answer to the nearest minute.
296
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
In Exercises 11–13, use the following information.
Name ———————————————————————
CHAPTER
11
Date ————————————
Chapter Review Game
For use after Chapter 11
Word Search
Use the clues at the bottom of the page to find and circle the vocabulary
words from Chapter 11 in the puzzle. Words can be found forward,
backward, upward, downward, and diagonal.
F R A T I
L L A S E
X E X D I
P A G T I
O U F S M
N Y M D I
E R I W D
M T D A P
E C P E O
R K O X I
O Z T D N
E S Z R T
H P E S U
T R I P L
J W Q P S
O
F
S
F
C
D
R
I
S
T
A
C
N
E
U
N
S
T
A
M
A
P
L
E
F
S
T
E
U
O
A
H
A
H
J
H
L
Q
U
Q
N
C
T
F
E
L
Q
N
F
C
G
K
B
U
R
P
Y
O
I
N
I
N
C
I
M
U
Y
A
K
Q
O
A
P
Z
A
Z
K
E
N
A
E
R
O
G
A
H
T
Y
P
R
I N G
P M S
F C I
R O M
U N P
E J L
K U E
M G S
V A T
R T F
S E O
A S R
H I M
L X N
T X E
1. Eliminating a radical from the denominator
2. Side opposite the right angle of a triangle
3. The point on a line segment that is
4. A group of integers a, b, and c that represent
equidistant from the endpoints
the side lengths of a right triangle is called
a Pythagorean _______.
}
5. y 5 Ï x is a _______ root function.
6. A function involving a radical expression
with the independent variable in the radicand
is called a _______ function.
}
}
7. The expressions 3 1 Ï 5 and 3 2 Ï 5
8. d 5
}}
Ï (x2 2 x1)2 1 ( y2 2 y1)2 represents
the _______ formula.
9. Two sides of a triangle that form a
10. a2 1 b2 5 c2 represents the _______ theorem.
right angle
11. A solution that is not a solution of an
12. A statement that can be proved true
original equation is called _______.
13. No perfect square factors in the radicand,
no fractions in the radicand, no radicals
appear in the denominator of a fraction
Algebra 1
Chapter 11 Resource Book
CHAPTER REVIEW GAME
Copyright © Holt McDougal. All rights reserved.
of an expression is called _______
the denominator.
297
Name ———————————————————————
CHAPTER
12
Date ————————————
Family Letter
For use with Chapter 12
Lesson Title
Lesson Goals
Key Applications
12.1: Model Inverse Variation
Write and graph inverse variation
equations.
• Theater
• Bicycles
• Sports
12.2: Graph Rational Functions
Graph rational functions.
• Trip Expenses
• Team Sports
• Charity Events
12.3: Divide Polynomials
Divide polynomials.
• Printing Costs
• Movie Rentals
• Membership Fees
Copyright © Holt McDougal. All rights reserved.
Focus on Operations
Use synthetic division to divide
polynomials.
12.4: Simplify Rational
Expressions
Simplify rational expressions.
• Cell Phone Costs
• Television
• Car Radios
12.5: Multiply and Divide
Rational Expressions
Multiply and divide rational
expressions.
Focus on Operations
Simplify complex fractions.
• Advertising
• Vehicles
• Consumer Spending
12.6: Add and Subtract
Rational Expressions
Add and subtract rational
expressions.
• Boat Travel
• Canoeing
• Driving
12.7: Solve Rational Equations
Solve rational equations.
• Paint Mixing
• Ice Hockey
• Running Times
CHAPTER SUPPORT
Chapter Overview One way you can help your student succeed in Chapter 12
is by discussing the lesson goals in the chart below. When a lesson is completed, ask your
student the following questions. “What were the goals of the lesson? What new words and
formulas did you learn? How can you apply the ideas of the lesson to your life?”
Key Ideas for Chapter 12
In Chapter 12, you will apply the key ideas listed in the Chapter Opener (see page 789)
and reviewed in the Chapter Summary (see page 859).
1. Graphing rational functions
2. Performing operations on rational expressions
3. Solving rational equations
Algebra 1
Chapter 12 Resource Book
299
Name ———————————————————————
CHAPTER
Family Letter
continued
For use with Chapter12
Key Ideas Your student can demonstrate understanding of key concepts by working
through the following exercises with you.
Lesson
Exercise
12.1
A public pool plans to hire lifeguards for the summer season. The work time t (in
hours per person) varies inversely with the number g of lifeguards hired. They
estimate that they will need 20 lifeguards working 170 hours each to meet their
needs. Find the total work time per lifeguard if the pool hires 25 lifeguards.
12.2
Graph y 5 }
2 2.
x13
12.3
Divide 6x 2 2 x 2 12 by 3x 1 4.
Focus on
Operations
4
Divide 2x3 2 x2 1 x 1 4 by x 1 1 using synthetic division.
3x 1 4x
12.4
Simplify the rational expression, }
, if possible. State the excluded values.
x24
12.5
Find the quotient }
4}
.
2
2
Focus on
Operations
3x 2 2 9x
x 1 3x 2 18
x17
x 1 5x 2 6
4x
3
}3 .
}
Simplify the complex fraction
4x 1 5
x 29
28x
2x 2 3
x 29
12.6
Find the sum of }
1}
.
2
2
12.7
Solve the equation }
225}
.
x18
x18
5x
5
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
12
Date ————————————
Home Involvement Activity
Directions Compare the cost of a season pass and additional expenses, such as
parking and food, for a local amusement park or other summer attraction, to a per
usage cost of the same attraction. Write an equation that gives the average cost C per
use as a function of the number of times n you use the attraction. Graph the equation.
How many times must you go for the season pass to save you money?
26
22
22
x
2
12.1: 136 h 12.2:
3x(x 2 1)
7x
; 4 12.5: }
12.4: }
x17
x24
6x 1 2
x 29
12.7: x 5 7
12.6: }
2
26
y
1
6x
Focus on Operations:2}2
12.3: 2x 2 3 Focus on Operations: 2x2 2 3x 1 4
Answers
300
Algebra 1
Chapter 12 Resource Book
Nombre ——————————————————————
CAPÍTULO
12
Fecha ———————————
Carta para la familia
Usar con el Capítulo 12
Título de la lección
Objetivos de la lección
Aplicaciones clave
12.1: Modelar variación inversa
Escribir y graficar ecuaciones de
variación inversa
• Teatro
• Bicicletas
• Deportes
12.2: Graficar funciones
racionales
Graficar funciones racionales
• Gastos de viaje
• Deportes en equipo
• Eventos de caridad
12.3: Dividir polinomios
Dividir polinomios
• Gastos de impresa
• Alquiler de películas
• Gastos de membresía
Copyright © Holt McDougal. All rights reserved.
Enfoque en las
operaciones
Usar división sintética para
dividir polinomios
12.4: Simplificar expresiones
racionales
Simplificar expresiones
racionales
• Teléfonos celulares
• Televisión
• Radios de carro
12.5: Multiplicar y dividir
expresiones racionales
Multiplicar y dividir expresiones
racionales
• Publicidad
• Vehículos
• Consumidores
Enfoque en las
operaciones
Simplificar fracciones complejas
12.6: Sumar y restar
expresiones racionales
Sumar y restar expresiones
racionales
• Viaje en bote
• Ir en canoa
• Manejar
12.7: Sumar y restar
expresiones racionales
Resolver ecuaciones racionales
• Mezclar pintura
• Hockey sobre hielo
• Tiempos de carrera
CHAPTER SUPPORT
Vistazo al capítulo Una manera en que puede ayudar a su hijo a tener éxito en
el Capítulo 12 es hablar sobre los objetivos de la lección en la tabla a continuación.
Cuando se termina una lección, pregúntele a su hijo lo siguiente: “¿Cuáles fueron los
objetivos de la lección? ¿Qué palabras y fórmulas nuevas aprendiste? ¿Cómo puedes
aplicar a tu vida las ideas de la lección?”
Ideas clave para el Capítulo 12
En el Capítulo 12, aplicarás las ideas clave enumeradas en la Presentación del capítulo
(ver la página 789) y revisadas en el Resumen del capítulo (ver la página 859).
1. Graficar funciones racionales
2. Hacer operaciones en expresiones racionales
3. Resolver ecuaciones racionales
Algebra 1
Chapter 12 Resource Book
301
Nombre ——————————————————————
CAPÍTULO
Carta para la familia
continúa
Usar con el Capítulo 12
Ideas clave Su hijo puede demostrar la comprensión de las ideas clave al hacer los
siguientes ejercicios con usted.
Lección
Ejercicio
12.1
Una piscina comunitaria piensa emplear unos salvavidas para el verano. El tiempo
de trabajo t (en horas por persona) varía inversamente con el número g de
salvavidas que se emplean. Se calcula que se necesitarán 20 salvavidas trabajando
170 horas para satisfacer las necesidades. Halla el total del tiempo trabajado por
salvavidas si se emplean 25 salvavidas.
12.2
Grafica y 5 }
2 2.
x13
12.3
Divide 6x 2 2 x 2 12 por 3x 1 4.
Enfoque en las
operaciones
4
Divide 2x3 2 x2 1 x 1 4 por x 1 1 usando división sintética.
3x 1 4x
12.4
Simplifica la expresión }
, si es posible. Nombra los valores excluidos.
x24
12.5
Halla el cociente de }
4}
.
2
2
Enfoque en las
operaciones
3x 2 2 9x
x 1 3x 2 18
x17
x 1 5x 2 6
4x
3
}3 .
}
Simplifica la fracción compleja
4x 1 5
x 29
28x
2x 2 3
x 29
12.6
Halla la suma de }
1}
.
2
2
12.7
Resuelve la ecuación }
225}
.
x18
x18
5x
Copyright © Holt McDougal. All rights reserved.
CHAPTER SUPPORT
12
Fecha ———————————
5
Actividad para la familia
Instrucciones Compara el costo de una entrada de temporada y gastos adicionales,
tales como estacionamiento y comida, para un parque de atracciones local u otra
atracción, a un costo de uso por la misma atracción. Escribe una ecuación que indique el
costo promedio C por uso como una función del número de veces n que usas la
atracción. Grafica la ecuación. ¿Cuántas veces tendrías que entrar para que una entrada
de temporada te ahorre dinero?
26
22
22
x
2
12.1: 136 h 12.2:
3x(x 2 1)
7x
; 4 12.5: }
12.4: }
x17
x24
6x 1 2
x 29
12.7: x 5 7
12.6: }
2
26
y
1
6x
Enfoque en las operaciones: 2}2
12.3: 2x 2 3 Enfoque en las operaciones: 2x2 2 3x 1 4
Respuestas
302
Algebra 1
Chapter 12 Resource Book
Name ———————————————————————
LESSON
12.1
Date ————————————
Activity Support Master
For use with page 790
Copyright © Holt McDougal. All rights reserved.
LESSON 12.1
Algebra 1
Chapter 12 Resource Book
303
Name ———————————————————————
LESSON
12.1
Date ————————————
Practice A
For use with pages 790–798
Tell whether the equation represents direct variation, inverse variation,
or neither.
1. y 5 27x
2. xy 5 21
3. y 5 x 1 2
23
4. x 5 }
y
5. xy 5 8
y
6. } 5 9
x
7. x 5 11y
8. 2x 1 y 5 8
9. y 5 13x
Match the inverse variation equation with its graph.
10. xy 5 10
A.
B.
y
6
LESSON 12.1
12. xy 5 5
11. xy 5 210
C.
y
3
2
22
22
2
6 x
29
y
9
3
x
23
23
3
23
3
9 x
3
9 x
2
6 x
29
Graph the inverse variation equation. Then find the domain and range of
the function.
8
14. y 5 }
x
11
15. y 5 }
x
y
23
y
6
9
1
2
3
21
21
1
3 x
26
22
22
23
6 x
29
23
23
29
7
18. y 5 }
x
29
17. y 5 }
x
y
y
y
9
9
6
3
3
2
23
23
3
29
304
2
26
210
16. y 5 }
x
29
y
3
Algebra 1
Chapter 12 Resource Book
9 x
29
23
23
29
3
9 x
26
22
22
26
Copyright © Holt McDougal. All rights reserved.
22
13. y 5 }
x
Name ———————————————————————
Practice A
LESSON
12.1
For use with pages 790–798
Date ————————————
continued
Match the inverse variation equation with its description.
19. y varies inversely with x and the constant of variation is 4.
24
A. y 5 }
x
1
20. y varies inversely with x and the constant of variation is }.
4
1
B. y 5 }
4x
21. y varies inversely with x and the constant of variation is 24.
4
C. y 5 }
x
Given that y varies inversely with x, use the specified values to write an inverse
variation equation that relates x and y. Then find the value of y when x 5 2.
22. x 5 1, y 5 3
23. x 5 4, y 5 2
24. x 5 3, y 5 6
25. x 5 22, y 5 8
26. x 5 7, y 5 22
27. x 5 5, y 5 21
28.
x
0
1
2
3
4
y
0
3
6
9
12
29.
x
24
22
2
4
8
y
0.5
1
21
20.5
20.25
LESSON 12.1
Tell whether the table represents inverse variation. If so, write the inverse
variation equation.
In Exercises 30 and 31, tell whether the variables in the situation described
have direct variation, inverse variation, or neither.
30. Bike Ride You are riding your bike at an average speed of 14 miles per hour.
31. Earning Money You want to find out how many hours you need to work at your job to
earn $500. The number of hours h you have to work at pay rate p is given by ph 5 500.
32. Volunteer Work Every spring, a volunteer group plants flowers to beautify
different areas of a city. The planting time t (in hours per person) varies inversely
with the number p of people volunteering. The group estimates that 20 people
working for 200 hours can get all of the flower beds planted.
a. Write an inverse variation equation that relates t and p.
b. Find the total amount of time it will take if 32 people
volunteer to plant.
33. Walking You are walking to a bookstore that is 3 miles from
your home. Write and graph an equation that relates your
walking speed s (in miles per hour) and the time t (in hours)
that it takes for you to get to the bookstore. Is the equation
an inverse variation equation? Explain.
Walking speed (mi/hour)
Copyright © Holt McDougal. All rights reserved.
The number of miles you ride d during t hours is given by d 5 14t.
s
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 t
Time (hours)
Algebra 1
Chapter 12 Resource Book
305
Name ———————————————————————
LESSON
12.1
Date ————————————
Practice B
For use with pages 790–798
Tell whether the equation represents direct variation, inverse variation,
or neither.
1. y 5 211x
2. xy 5 25
3. y 5 x 2 4
28
4. x 5 }
y
5. xy 5 14
y
6. } 5 13
x
7. 2x 1 y 5 8
9
8. 3y 5 }
x
9. 4x 2 4y 5 0
Graph the inverse variation equation. Then find the domain and range of
the function.
10. xy 5 12
12. xy 5 7
11. xy 5 26
y
LESSON 12.1
y
9
3
y
6
2
23
23
3
9 x
26
2
x
22
22
22
22
2
2
6 x
4
12 x
2
6 x
26
15
14. y 5 }
x
y
6
2
26
14
15. y 5 }
x
y
12
y
12
4
4
x
22
22
24
2
4
12 x
26
29
16. y 5 }
x
5
18. y 5 }
x
212
17. y 5 }
x
y
3
29
23
23
3
x
3
29
306
y
6
y
Algebra 1
Chapter 12 Resource Book
29
23
23
29
x
3
2
22
22
Copyright © Holt McDougal. All rights reserved.
28
13. y 5 }
x
Name ———————————————————————
LESSON
12.1
Practice B
Date ————————————
continued
For use with pages 790–798
Given that y varies inversely with x, use the specified values to write an inverse
variation equation that relates x and y. Then find the value of y when x 5 2.
19. x 5 7, y 5 2
20. x 5 3, y 5 9
21. x 5 23, y 5 1
22. x 5 11, y 5 21
23. x 5 212, y 5 212
24. x 5 218, y 5 24
25. x 5 10, y 5 5
26. x 5 7, y 5 24
27. x 5 6, y 5 6
28. x 5 23, y 5 12
29. x 5 25, y 5 40
30. x 5 25, y 5211
Tell whether the table represents inverse variation. If so, write the inverse
variation equation.
31.
2
4
6
8
10
y
11
21
31
41
51
x
10
23
25
28
50
y
160
368
400
448
800
34.
x
25
24
1
2
10
y
24
25
20
10
2
x
210
29
26
25
24
y
21.8
22
23
23.6
24.5
LESSON 12.1
33.
32.
x
Internet or by phone. The orders must be entered into the computer inventory
system. The amount of time t needed to enter 1000 orders varies inversely with
the number p of people working. The company estimates that 10 people
can enter 1000 orders in 240 minutes.
a. Write an inverse variation equation that relates t and p.
b. Find the time needed to enter 1000 orders if 20 people are working.
c. Find the time needed to enter 1000 orders if 8 people are working.
36. Volume and Pressure The volume V of a gas at a constant temperature varies
inversely with the pressure P. When the volume is 125 cubic inches, the pressure
is 20 pounds per cubic inch.
a. Write the inverse variation equation that relates P and V.
b. Find the pressure of a gas with a volume of 250 cubic inches.
37. Running Every other day, weather permitting, you run 5 miles.
Write and graph an equation that relates your average running
speed s (in miles per hour) and the time t (in hours) that it takes
for you to complete the run. Is the equation an inverse variation
equation? Explain.
Average speed (mi/hour)
Copyright © Holt McDougal. All rights reserved.
35. Catalog Orders A clothing company allows customers to place orders on the
s
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 t
Time (hours)
Algebra 1
Chapter 12 Resource Book
307
Name ———————————————————————
Date ————————————
Practice C
LESSON
12.1
For use with pages 790–798
Tell whether the equation represents direct variation, inverse variation,
or neither.
1. y 5 20.5x
2. xy 5 20.25
3. 4y 5 x 2 8
27
4. x 5 }
y
5. xy 5 22
y
6. } 5 4.5
x
7. x 5 24
8. 8x 2 8y 5 0
9. 5xy 5 30
Graph the inverse variation equation. Then find the domain and range of
the function.
10. xy 5 0.75
12. 2xy 5 14
11. xy 5 23
LESSON 12.1
y
y
y
3
6
1
2
1
21
1
x
23
21
21
1
3 x
26
22
22
6 x
1
3 x
4
12 x
21
23
24
15. y 5 }
x
y
15
3
3
5
1
23
23
3
9 x
29
215
25
25
5
15 x
23
21
21
23
215
1.5
16. y 5 }
x
13
18. y 5 }
x
20.2
17. y 5 }
x
y
23
y
9
y
y
3
1.2
12
1
0.4
4
21
21
1
23
Algebra 1
Chapter 12 Resource Book
3 x
21.2 20.4
20.4
21.2
0.4
1.2 x
212
24
24
212
Copyright © Holt McDougal. All rights reserved.
y
29
26
20
14. y 5 }
x
211
13. y 5 }
x
308
2
Name ———————————————————————
Practice C
LESSON
12.1
For use with pages 790–798
Date ————————————
continued
Given that y varies inversely with x, use the specified values to write
an inverse variation equation that relates x and y. Then find the value
of y when x 5 2.
19. x 5 17, y 5 23
20. x 5 212, y 5 212
21. x 5 26, y 5 7
22. x 5 9, y 5 4
23. x 5 10, y 5 23
24. x 5 7, y 5 7
25. x 5 23, y 5 50
26. x 5 26, y 5 220
27. x 5 4, y 5 211
28. x 5 219, y 5 6
29. x 5 7, y 5 15
30. x 5 214, y 5 25
Tell whether the table represents inverse variation. If so, write the inverse
variation equation.
x
232
220
216
210
25
y
20.5
20.8
21
21.6
23.2
32.
x
2
4
20
25
40
y
25
22.5
20.5
20.4
20.25
LESSON 12.1
31.
33. Radio Waves The frequency f in hertz (vibrations per second) of a radio wave varies
inversely with the wavelength w (in meters per vibration). When the frequency is
2.336 3 105 hertz, the wavelength is 1.28 meters.
a. Write the inverse variation equation that relates f and w.
b. What is the frequency when the wavelength is 2.92 meters?
you can afford a season pass to a local ski area. One season pass costs $400.
a. Let a represent the amount of money that you plan to save each month.
Complete the table that gives the number m of months that you need to
save money for different values of a. Describe how the number of months
changes as the amount of money you save each month increases.
a
40
50
80
100
200
400
m
b. Use the values in the table to draw a graph of the situation.
Does the graph suggest a situation that represents direct
variation or inverse variation? Explain your choice.
c. Write the equation that relates a and m.
Number of months
Copyright © Holt McDougal. All rights reserved.
34. Saving Money You plan to save the same amount of money each month so that
m
9
8
7
6
5
4
3
2
1
0
0
100
200
300
400
a
Amount saved each month
(dollars)
Algebra 1
Chapter 12 Resource Book
309
Name ———————————————————————
LESSON
12.1
Date ————————————
Review for Mastery
For use with pages 790–798
GOAL
Write and graph inverse variation equations.
Vocabulary
a
The variables x and y show inverse variation if y 5 }x and a Þ 0.
The number a is the constant of variation, and y is said to vary
inversely with x.
a
LESSON 12.1
The graph of the inverse variation equation y 5 }x (a Þ 0) is a
hyperbola. The two symmetrical parts of a hyperbola are called the
branches of a hyperbola. The hyperbola also has two asymptotes,
which are lines that a hyperbola approaches but does not intersect.
EXAMPLE 1
Identify direct and inverse variation
Tell whether the equation represents direct variation, inverse variation,
or neither.
1
a. xy 5 }
5
y
c. } 5 x
3
b. y 5 3x 2 1
Solution
1
a. xy 5 }
5
Write original equation.
1
a
1
1
Because xy 5 }5 can be written in the form y 5 }x , xy = }5
represents inverse variation.
a
b. Because y 5 3x 2 1 cannot be written in the form y 5 } or
x
y 5 ax, y 5 3x 2 1 does not represent either direct variation or
inverse variation.
c.
}5x
y
3
Write original equation.
y 5 3x
Multiply each side by 3.
y
y
Because }3 5 x can be written in the form y 5 ax, }3 5 x represents
direct variation.
Exercises for Example 1
Tell whether the equation represents direct variation, inverse variation,
or neither.
1. 8x 5 y 2 3
310
Algebra 1
Chapter 12 Resource Book
2. 2x 5 8y
3. xy 5 3
y
x
4. } 5 }
2
3
Copyright © Holt McDougal. All rights reserved.
Divide each side by x.
y5}
5x
Name ———————————————————————
LESSON
12.1
Review for Mastery
continued
For use with pages 790–798
EXAMPLE 2
Date ————————————
Graph an inverse variation equation
6
Graph y 5 }
. Then find the domain and range of the function.
x
STEP 1
Make a table by choosing several integer values of x and finding the values of
y. Then plot the points. To see how the function behaves for values of x closer
to 0 and farther from 0, make a second table for such values and plot the points.
x 26 23 21
0
y
12
1 3 6
y 21 22 26 undefined 6 2 1
x 212
210 20.6 20.5 0.5 0.6 10
STEP 2
212
12
12
4
12 x
10 0.6 0.5
Connect the points in Quadrant I by drawing a smooth curve through them.
Repeat for the points in Quadrant III.
LESSON 12.1
y 20.5 20.6 210
4
Both the domain and the range of the function are all real numbers except 0.
EXAMPLE 3
Use an inverse variation equation
The variables x and y vary inversely, and y 5 22 when x 5 4.
a. Write an inverse variation equation that relates x and y.
Copyright © Holt McDougal. All rights reserved.
b. Find the value of y when x 5 210.
Solution
a
a. Because y varies inversely with x, the equation has the form y 5 } .
x
Use the fact that x 5 4 and y 5 22 to find the value of a.
a
y 5 }x
a
Write inverse variation equation.
a
22 5 }4
Substitute 4 for x and 22 for y in y 5 }x .
28 5 a
Multiply each side by 4.
28
An equation that relates x and y is y 5 }
.
x
28
4
b. When x 5 210, y 5 } 5 } .
5
210
Exercises for Examples 2 and 3
Graph the inverse variation equation. Then find the domain and range
of the function.
20
215
12
22
5. y 5 }
6. y 5 }
7. y 5 }
8. y 5 }
x
x
x
x
9. The variables x and y vary inversely. Write an inverse variation equation that
relates x to y when x 5 2 and y 5 8. Then find y when x 5 24.
Algebra 1
Chapter 12 Resource Book
311
Name ———————————————————————
LESSON
12.1
Date ————————————
Challenge Practice
For use with pages 790–798
In Exercises 1–10, use the following information.
The variables u and v vary inversely with a constant of variation a.
The variables x and y vary inversely with a constant of variation b.
The variables w and z vary inversely with a constant of variation c.
The variables u and x vary directly with a constant of variation d.
The variables x and z vary directly with a constant of variation k.
LESSON 12.1
Determine an equation relating the given variables and tell whether the given
variables vary directly or inversely.
1. x and v
2. v and y
3. u and w
4. u and z
5. v and z
6. y and u
7. w and x
8. v and w
9. w and y
10. y and z
In Exercises 11–15, use the following information.
The points (1, 2a) and (a 2 1, a 2 ) are two of the points that lie on the graph of an inverse
c
variation equation of the form y 5 }x .
11. Find the value of a.
13. Find the value of x when y is 6.
14. Find the value of y when x is 8.
15. Find the value of x when y is 1000.
312
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
12. Find the value of c.
Name ———————————————————————
Date ————————————
Practice A
LESSON
12.2
For use with pages 799–808
Match the function with its graph.
1
1. y 5 }
5x
1
2. y 5 }
x25
A.
B.
y
1
3. y 5 }
x15
C.
y
6
6
2
2
y
1
22
22
6
210
10 x
22
22
26
1
2 x
x
26
Identify the domain and range of the function from its graph.
4.
5.
y
y
6
6
3
23
23
6.
y
9
2
3
9
15 x
29
2
x
22
22
26
26
x
2
Graph the function and identify its domain and range. Then compare the
1
graph with the graph of y 5 }
.
x
Copyright © Holt McDougal. All rights reserved.
25
9. y 5 }
x
1
8. y 5 }
3x
y
26
y
1
6
2
1
3
2
2
6 x
21
2
1
3
1
3
26
21
1
10. y 5 } 1 4
x
1
11. y 5 } 2 2
x
y
26
22
22
2
6 x
26
1
12. y 5 }
x16
y
6
1
23
6
21
21
23
2
22
22
1 x
y
10
26
y
6
22
22
LESSON 12.2
4
7. y 5 }
x
2
6 x
25
1
3 x
2
210
26
22
22
x
26
Algebra 1
Chapter 12 Resource Book
313
Name ———————————————————————
LESSON
12.2
Practice A
For use with pages 799–808
Date ————————————
continued
Match the function with its asymptotes.
1
13. y 5 } 2 2
x13
1
14. y 5 } 1 3
x22
A. x 5 3, y 5 2
B. x 5 2, y 5 3
1
15. y 5 } 1 2
x23
C. x 5 23, y 5 22
Determine the asymptotes of the graph of the function.
23
16. y 5 }
x28
211
17. y 5 }
x 2 14
6
18. y 5 } 1 5
x26
24
19. y 5 } 1 1
x 1 13
10
20. y 5 } 2 2
x 1 10
8
21. y 5 } 2 7
x15
2
23. y 5 } 1 2
x
1
24. y 5 } 2 5
x13
Graph the function.
y
26
y
6
5
2
3
22
22
2
210
23
21
21
1
3 x
2 x
210
25. Football Hall of Fame Your football team is planning a bus trip
to the Pro Football Hall of Fame. The cost for renting a bus is
$500, and the cost will be divided equally among the people
who are going on the trip. One admission costs $13.
a. Write an equation that gives the cost C (in dollars per person)
of the trip as a function of the number p of people going on
the trip.
b. Graph the equation.
Average number of
flowers per person
400 boutonnieres and corsages. Currently, 3 people are
scheduled to put together the flowers. The florist hopes
to call in some extra workers to complete all of the flowers.
Write an equation that gives the average number f of
boutonnieres and corsages made per person as a function
of the number p of extra workers that can come in and help
complete the work. Then graph the equation.
Algebra 1
Chapter 12 Resource Book
22
22
26
26. Prom It’s prom season and a florist has orders for
314
26
1
Cost (dollars/person)
LESSON 12.2
26
6 x
y
2
C
175
150
125
100
75
50
25
0
f
175
150
125
100
75
50
25
0
0 10 20 30 40 50 60 70 p
Number of people
0 2 4 6 8 10 12 14 p
Number of extra workers
Copyright © Holt McDougal. All rights reserved.
4
22. y 5 } 2 1
x
Name ———————————————————————
Date ————————————
Practice B
LESSON
12.2
For use with pages 799–808
Identify the domain and range of the function from its graph.
1.
2.
y
6
2
2
x
6
22
22
26
y
2
215
2
22
22
2
6
29
x
26
210
5.
y
29
23
22
3 x
23
22
26
4.
215
3.
y
6
6.
y
y
3 x
2
26
2
2
26
210
2
22
x
6
x
26
Graph the function and identify its domain and range. Then compare the
1
graph with the graph of y 5 }
.
x
8
7. y 5 }
x
1
8. y 5 }
6x
23
9. y 5 }
2x
y
6
y
y
1
Copyright © Holt McDougal. All rights reserved.
2
22
22
2
6 x
1
x
23
21
21
x
1
23
1
10. y 5 } 2 7
x
1
11. y 5 } 1 10
x
y
23
21
22
1
12. y 5 }
x24
y
1
LESSON 12.2
1
y
3
3 x
1
26
21
6
10 x
2
23
21
1
3 x
23
Algebra 1
Chapter 12 Resource Book
315
Name ———————————————————————
LESSON
12.2
Practice B
For use with pages 799–808
Date ————————————
continued
Determine the asymptotes of the graph of the function.
10
13. y 5 } 1 4
x26
28
14. y 5 } 2 6
x15
14
15. y 5 } 2 8
x23
12
16. y 5 } 1 7
x17
24
17. y 5 } 1 12
x28
9
18. y 5 } 1 10
x15
14
19. y 5 } 1 1
x 2 14
212
20. y 5 } 2 3
x 1 12
7
21. y 5 } 2 14
x25
1
23. y 5 } 1 2
x24
23
24. y 5 } 2 1
x16
Graph the function.
2
22. y 5 } 1 5
x
y
y
y
5
6
10
3
22
6
2
x
1
21
1
3 x
22
21
2
6
10 x
26
b. Graph the equation.
C
140
120
100
80
60
40
20
0
c. What would the cost per person be if 20 people go on the trip?
26. Fundraiser A pizza shop makes pizzas that organizations sell for
fundraisers. One organization has placed an order for 450 pizzas.
Currently, 4 people are scheduled to put together the pizzas.
The owner of the shop hopes to call in some extra workers to
complete all of the pizzas.
a. Write an equation that gives the average number n of pizzas
made per person as a function of the number p of extra
workers that can come in and help complete the work.
b. Graph the equation.
c. If 2 people come in to help out, what is the average number
of pizzas made person?
316
Cost (dollars/person)
to the National Baseball Hall of Fame. The cost for renting a bus
is $515, and the cost will be divided equally among the people
who are going on the trip. One admission costs $14.50.
a. Write an equation that gives the cost C (in dollars per person)
of the trip as a function of the number p of people going on
the trip.
Algebra 1
Chapter 12 Resource Book
Average number of pizzas
LESSON 12.2
25. Baseball Hall of Fame Your baseball team is planning a bus trip
n
140
120
100
80
60
40
20
0
0 5 10 15 20 25 30 35 p
Number of people
0 1 2 3 4 5 6 7 p
Number of extra workers
Copyright © Holt McDougal. All rights reserved.
23
Name ———————————————————————
Date ————————————
Practice C
LESSON
12.2
For use with pages 799–808
Graph the function and identify its domain and range. Then compare the
1
graph with the graph of y 5 }
.
x
4
2. y 5 }
5x
21
1. y 5 }
8x
25
3. y 5 }
3x
y
y
y
3
3
1
1
1
21
1
x
23
21
21
1
3 x
23
21
21
1
3 x
2
6 x
21
23
7
5. y 5 }
2x
22
4. y 5 }
3x
1
6. y 5 } 2 9
x
y
23
23
y
3
3
1
1
21
21
1
3 x
23
21
21
y
26
1
22
22
26
3 x
210
23
23
1
7. y 5 } 1 5
x
1
8. y 5 }
x26
Copyright © Holt McDougal. All rights reserved.
26
y
y
10
6
6
6
2
2
2
22
22
22
22
2
6 x
2
6
10 x
210
26
22
22
26
x
LESSON 12.2
y
1
9. y 5 }
x18
26
Determine the asymptotes of the graph of the function.
22
10. y 5 } 2 10
x 1 13
4
11. y 5 } 1 2
4x 2 8
210
12. y 5 } 2 3
5x 1 5
Algebra 1
Chapter 12 Resource Book
317
Name ———————————————————————
Practice C
LESSON
12.2
For use with pages 799–808
Date ————————————
continued
Graph the function.
4
13. y 5 } 1 5
x23
5
15. y 5 } 1 2
x14
22
14. y 5 } 2 1
x12
y
y
y
10
6
1
6
25
23
1 x
21
21
2
22
22
2
6
210
23
10 x
3
17. y 5 } 2 2
x16
22
16. y 5 } 2 4
x24
2
26
24
18. y 5 } 2 4
x12
y
y
y
2
2
22
22
x
22
22
2
6
2
10 x
210
26
22
22
26
x
26
22
22
2
x
26
26
20. Video Games You rent games from a web site for $17.25 per
month. You can rent any number of games per month, but you
usually rent at least 4 games per month.
a. Write an equation that gives the average cost C per rental
as a function of the number r of additional rentals beyond
4 rentals.
b. Graph the equation from part (a). Then use the graph to
approximate the number of additional rentals needed per
month so that the average cost is $2.25.
318
Algebra 1
Chapter 12 Resource Book
Average number of
box lunches per person
group of 6 people is responsible for putting together 225 box
lunches for the trip. The group hopes to recruit extra people for
the task. Write an equation that gives the average number n
of box lunches made per person as a function of the number p
of parents that can come in and help complete the task. Then
graph the equation. How many people need to come in so
that the average number of box lunches made per person is
15 box lunches?
n
35
30
25
20
15
10
5
0
Average cost per
rental (dollars)
LESSON 12.2
19. Zoo Trip A grade school is taking a trip to the zoo. A parent
C
4
3
2
1
0
0 2 4 6 8 10 12 14 p
Number of extra parents
0 1 2 3 4 5 6 7 r
Number of additional
rentals
Copyright © Holt McDougal. All rights reserved.
210
210
Name ———————————————————————
LESSON
12.2
Date ————————————
Review for Mastery
For use with pages 799–808
GOAL
Graph rational functions.
Vocabulary
A rational function has a rule given by a fraction whose numerator
and denominator are polynomials and whose denominator is not 0.
EXAMPLE 1
1
Graph y 5 }
1k
x
1
Graph y 5 }
2 2 and identify its domain and range. Compare the graph
x
1
with the graph of y 5 }
.
x
Graph the function using
a table of values.
The domain is all real
numbers except 0. The range
is all real numbers except 22.
1
The graph of y 5 }x 2 2 is a
vertical translation (of 2 units
1
EXAMPLE 2
y
22
22.5
21
23
20.5
24
0
undefined
0.5
0
1
21
2
21.5
y
3
1
y 5x
1
21
1
3
x
1
y 5 x 22
LESSON 12.2
Copyright © Holt McDougal. All rights reserved.
down) of the graph of y 5 }x .
x
1
Graph y 5 }
x2h
1
Graph y 5 }
and identify its domain and range. Compare the graph
x24
1
with the graph of y 5 }
.
x
Graph the function using
a table of values.
The domain is all real
numbers except 4. The range
is all real numbers except 0.
1
is a
The graph of y 5 }
x24
horizontal translation (of 4 units
1
up) of the graph of y 5 }x .
y
x
y
2
20.5
6
3
21
2
3.5
22
4
undefined
4.5
2
5
1
6
0.5
1
y 5x
1
y 5x 2 4
6
x
26
Algebra 1
Chapter 12 Resource Book
319
Name ———————————————————————
LESSON
12.2
Review for Mastery
Date ————————————
continued
For use with pages 799–808
Exercises for Examples 1 and 2
Graph the function and identify its domain and range. Compare the
1
graph with the graph of y 5 }
.
x
8
1. y 5 }
x
EXAMPLE 3
2.
1
y 5 }x 1 5
1
3. y 5 }
x 1 10
a
Graph y 5 }
1 k.
x 2h
5
Graph y 5 }
2 2.
x13
Solution
STEP 2
y
Identify the asymptotes of the graph.
The vertical asymptote is x 5 23.
The horizontal asymptote is y 5 22.
Plot several points on each side of the
vertical asymptote.
6
2
x
26
y5
LESSON 12.2
STEP 3
320
Graph two branches that pass through
the plotted points and approach the
asymptotes.
Exercise for Example 3
2
4. Graph y 5 } 11.
x22
Algebra 1
Chapter 12 Resource Book
26
5
22
x13
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
LESSON
12.2
Date ————————————
Challenge Practice
For use with pages 799–808
In Exercises 1–5, find the asymptotes of the graph of the function,
then graph the function.
3
1. f (x) 5 } 1 4
22x
5
2. f(x) 5 } 2 3
2x 1 1
21
3. f(x) 5 } 1 1
x11
22
4. f(x) 5 } 1 2
3 2 4x
1
6
5. f(x) 5 } 2 }
2
1
4
}x 1 }
3
2
In Exercises 6–10, find a function whose graph satisfies the
given conditions.
a
6. f has the form f(x) 5 } 1 d; f has a vertical asymptote at x 5 3;
bx 1 d
3
f has a horizontal asymptote at y 5 2; f (6) 5 }2.
a
1
7. f has the form f(x) 5 } 1 c; f has a vertical asymptote at x 5 };
7
ax 1 b
13
Copyright © Holt McDougal. All rights reserved.
LESSON 12.2
f has a horizontal asymptote at y 5 1; f (1) 5 }
.
6
6
8. f has the form f(x) 5 } 1 c; f has a vertical asymptote at x 5 22;
cx 1 b
f has a horizontal asymptote at y 5 21; f (1) 5 23.
6
1
9. f has the form f(x) 5 }; f has a vertical asymptote at x 5 2};
2
ax 1 b
f has a horizontal asymptote at y 5 0; f (0) 5 6 and f (1) 5 2.
3
24
10. f has the form f(x) 5 } 1 c; f has a vertical asymptote at x 5 };
2
ax 1 b
f has a horizontal asymptote at y 5 2; f (1) 5 22.
Algebra 1
Chapter 12 Resource Book
321
Name ———————————————————————
Date ————————————
Graphing Calculator Activity Keystrokes
LESSON
12.3
For use with pages 818 and 819
TI-83 Plus
Casio CFX-9850GC Plus
Example 1
Example 1
Y=
2
(
x
X,T,�,n
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2
ENTER
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ENTER
)
ENTER
5
�
10
ENTER
(�)
1
ENTER
3
(
�
X,T,�,n
(�)
ENTER
1
�
4
WINDOW
ENTER
10
X,T,�,n
2
)
1
TRACE
From the main menu, choose GRAPH.
( 2 X,,T
� 1 )
�
( 3 X,,T
2
x
� 4 X,,T
� 5 )
EXE SHIFT
(�) 10 EXE 10 EXE 2 EXE (�) 1 EXE
1 EXE .2 EXE EXIT F6 SHIFT F1
Use the arrow keys to identify the asymptotes.
Use the arrow keys to identify the asymptotes.
Example 2
Example 2
Y=
CLEAR
�
ZOOM
(
(
X,T,�,n
6
2
X,T,�,n
x
2
�
x2
9
�
)
1
)
ENTER
TRACE
Use the arrow keys to identify the asymptotes.
From the main menu, choose GRAPH.
( 2 X,,T
x2
� 1 )
�
(
2
x
� 9 )
EXE SHIFT F3 F3
F6
SHIFT
F3
X,,T
EXIT
F1
LESSON 12.3
Copyright © Holt McDougal. All rights reserved.
Use the arrow keys to identify the asymptotes.
322
Algebra 1
Chapter 12 Resource Book
Name ———————————————————————
LESSON
12.3
Date ————————————
Practice A
For use with pages 810–817
Simplify the expression.
18x 3
1. }
6x
215x 2
2. }
5x
210x
3. }
10x
Divide.
4.
(9x 3 2 6x 2 1 18x) 4 3x
5.
(14x 3 1 21x 2 2 28x) 4 7x
6.
(16x 4 2 16x 3 2 24x 2) 4 8x
7.
(20x 4 2 5x 2 1 10x) 4 5x
8.
(22x 3 1 6x 2 1 4x) 4 (22x)
9.
(4x 3 2 16x 2 1 20x) 4 (24x)
Match the equivalent expressions.
10.
(x 2 1 3x 2 10) 4 (x 1 5)
A. x 2 2
11.
(x 2 2 3x 2 10) 4 (x 1 5)
B. x 1 5
12.
(x 2 1 3x 2 10) 4 (x 2 2)
30
C. x 2 8 1 }
x15
Divide.
13.
(x 2 1 10x 1 24) 4 (x 1 6)
14.
(x 2 2 2x 2 15) 4 (x 1 3)
15.
(x 2 2 7x 1 6) 4 (x 2 1)
16. (x3y2 1 3x2y 1 2xy) 4 xy
18. Car Dealer The number of sports cars that a dealer sells per
year between 1995 and 2004 can be modeled by S 5 4t 1 21
where t is the number of years since 1995. The total number
of cars sold by the dealer can be modeled by C 5 24t 1 120.
a. Use long division to find a model for the ratio R of
the number of sports cars sold to the total number
of cars sold.
b. Graph the model.
Average cost per
hour (dollars)
Ratio of sports cars
sold to total cars sold
to see the sights. A local rental store offers mopeds for $20 an
hour plus a $5 gasoline fill-up fee.
a. Write an equation that gives the average cost C per hour as
a function of the number h of hours you rent the moped.
b. Graph the equation.
C
50
40
30
20
0
0 1 2 3 4 5 6 7 h
Number of hours rented
R
0.178
0.176
0.174
0.172
0.170
0.168
0
0 1 2 3 4 5 6 7 8 t
Years since 1995
Algebra 1
Chapter 12 Resource Book
LESSON 12.3
Copyright © Holt McDougal. All rights reserved.
17. Moped Rental While on vacation, you decide to rent a moped
323
Name ———————————————————————
LESSON
12.3
Date ————————————
Practice B
For use with pages 810–817
Divide.
1.
(18x 3 2 24x 2 1 12x) 4 6x
2.
(25x 3 1 15x 2 2 30x) 4 (25x)
3.
(22x 4 2 18x 2 1 6x) 4 (22x)
4.
(x 2 1 6x 1 5) 4 (x 1 5)
5.
(5x 2 1 7x 2 6) 4 (x 1 2)
6.
(4x 2 1 x 2 5) 4 (x 2 1)
7.
(6x 2 1 22x 2 8) 4 (x 1 4)
8.
(4x 2 1 x 2 8) 4 (x 2 2)
9. (10x3y4 1 4x2y 2 2xy) 4 2xy
10. (24a5b 1 16a4b2 2 8a3b) 4 8a3b
Graph the function.
x18
11. y 5 }
x
3x 2 5
12. y 5 }
x
y
x15
13. y 5 }
x22
y
y
12
6
6
2
2
4
212
4
12 x
26
22
22
2
6 x
6
x
(a cross between a scooter and a small car) for $40 per hour
plus a $4.50 gasoline fill-up fee.
a. Write an equation that gives the average cost C per hour as
a function of the number h of hours the scootcar is rented.
b. Graph the equation.
Average cost per
hour (dollars)
14. Scootcar Rental A resort area offers rentals of scootcars
C
80
70
60
50
40
30
0
0 1 2 3 4 5 6 7 8 h
LESSON 12.3
15. Juice Bar Between 1995 and 2004, the number D of drinks
324
(in thousands) sold at a juice bar can be modeled by
D 5 4t 1 18 where t is the number of years since 1995.
The number F of drinks (in thousands) made from fruit juice
rather than vegetable juice can be modeled by F 5 2t 1 32.
a. Use long division to find a model for the ratio R of
the number of fruit drinks sold to the total number
of drinks sold.
b. Graph the model.
Algebra 1
Chapter 12 Resource Book
Ratio of fruit drinks sold
to total drinks sold
Time (hours)
R
1.8
1.5
1.2
0.9
0.6
0.3
0
0 1 2 3 4 5 6 7 8 t
Years since 1995
Copyright © Holt McDougal. All rights reserved.
212
Name ———————————————————————
LESSON
12.3
Date ————————————
Practice C
For use with pages 810–817
Divide.
1.
(45x 4 2 60x 2 1 30x) 4 15x
2.
(96x 3 2 64x 2 2 24x) 4 (28x)
3.
(7x 2 1 2x 2 5) 4 (x 2 2)
4.
(9 2 3x 2 x 2) 4 (1 2 x)
5.
(22 2 4x 1 3x 2) 4 (x 2 4)
6.
(6x 1 x 2 1 5) 4 (3 1 x)
7.
(8x 1 x 2 2 3) 4 (2 2 x)
8.
(9x 2 2 4) 4 (3x 1 1)
9. (15x8y5 2 3x6y4 2 2x2y2) 4 3x2y
10. (56a5b4 1 14a3b3 2 9a4b2) 4 7a3b2
Graph the function.
52x
11. y 5 }
x17
3 1 6x
12. y 5 }
x22
8 2 5x
13. y 5 }
x14
y
y
y
9
15
5
3
215
29
23
23
3 x
215
5
25
25
5
15
25
25
x
x
5
215
29
Average cost per
mile (dollars)
car rental for $24 per day plus $.06 per mile. You want to
rent the car for three days.
a. Write an equation that gives the average cost C per mile as
a function of the number m of miles you drive the rental.
b. Graph the equation.
C
28
24
20
16
12
8
4
0
0 1 2 3 4 5 6 7 8 m
15. Athletic Shoes Between 1999 and 2002, the sales S of
athletic and sport footwear (in millions of dollars) can be
modeled by S 5 546t 1 12,552 where t is the number of
years since 1999. The sales W of walking shoes (in millions
of dollars) can be modeled by W 5 91t 1 3141.
a. Use long division to find a model for the ratio R of
walking shoe sales to all athletic shoe sales.
b. Graph the model.
Ratio of walking shoes
sold to total shoes sold
Number of miles
R
0.255
0.250
0.245
0.240
0.235
0.230
0
0
1
2
3
4 t
Years since 1999
Algebra 1
Chapter 12 Resource Book
LESSON 12.3
Copyright © Holt McDougal. All rights reserved.
14. Car Rental A local car rental company offers an economy
325
Name ———————————————————————
LESSON
12.3
Date ————————————
Review for Mastery
For use with pages 810–817
GOAL
EXAMPLE 1
Divide polynomials.
Divide a polynomial by a monomial
Divide 15x 3 2 10x 2 2 20x by 25x.
Solution
METHOD 1:
Write the division as a fraction.
3
2
15x 2 10x 220x
(15x 3 2 10x 2 2 20x) 4 (25x) 5 }}
25x
15x3
METHOD 2:
220x
1}
1}
5}
25x
25x
25x
Divide each term by 25x.
5 23x2 1 2x 1 4
Simplify.
Use long division.
Think:
3
210x2
Write as fraction.
15x 4 (25x)
Think:
210x2 4 (25x)
Think:
220x 4 (25x)
23x2 1 2x 1 4
25x qww
15x3 2 10x2 2 20x
(15x3 2 10x2 2 20x) 4 (25x) 5 23x2 1 2x 1 4
25x(23x2 1 2x 1 4) 0 15x3 2 10x2 2 20x
25x(23x2) 1 (25x)(2x) 1 (25x)(4) 0 15x3 2 10x2 2 20x
15x3 2 10x2 2 20x 5 15x3 2 10x2 2 20x ✓
Exercises for Example 1
LESSON 12.3
Divide.
326
1.
(14p3 2 35p2 1 42p) 4 7p
2.
(12r 3 1 8r 2 2 22r) 4 2r
3.
(15t 3 1 6t 2 2 18t) 4 (23t)
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
CHECK
Name ———————————————————————
LESSON
12.3
Review for Mastery
For use with pages 810–817
EXAMPLE 2
Date ————————————
continued
Divide a polyomial by a binomial
Divide 6x 2 2 13x 1 2 by 2x 2 5.
Solution
3x 1 1
2x 2 5 q 6x 2 13x 1 2
6x2 2 15x
2x 1 2
2x 2 5
7
ww
2
Multiply 3x and 2x 2 5.
Subtract 6x2 2 15x. Bring down 2.
Multiply 1 and 2x 2 5.
Subtract 2x 2 5.
7
(6x2 2 13x 1 2) 4 (2x 2 5) 5 3x 1 1 1 }
2x 2 5
EXAMPLE 3
Insert missing terms
Divide 16y 2 2 7 by 3 1 4y.
Solution
4y 2 3
4y 1 3 q16y 1 0y 2 7
16y2 1 12y
212y 2 7
212y 2 9
2
Rewrite polynomials. Insert missing term.
Multiply 4y and 4y 1 3.
Subtract 16y2 1 12y. Bring down 27.
Multiply 23 and 4y 1 3.
Subtract 212y 2 9.
2
(16y 2 2 7) 4 (3 1 4y) 5 4y 2 3 1 }
4y 1 3
Exercises for Examples 2 and 3
Divide.
4.
(8x 2 2 22x 2 21) 4 (2x 2 7)
5.
(24x 2 2 19x 1 6) 4 (8x 2 1)
6.
(4x 2 2 25) 4 (25 1 2x)
7.
(16x 2 2 46) 4 (4x 1 7)
LESSON 12.3
Copyright © Holt McDougal. All rights reserved.
ww
2
Algebra 1
Chapter 12 Resource Book
327
Name ———————————————————————
LESSON
12.3
Date ————————————
Challenge Practice
For use with pages 810–817
In Exercises 1–5, divide.
1.
(x 3 1 3x 2 2 4x 2 12) 4 (x 2 2 4)
2.
(x 4 1 2x 3 2 10x 2 2 23x 2 6) 4 (x 2 2 3x 2 1)
3.
(x 2 1 1) 4 (x 2 2 1)
4.
(x 3 1 3x 2 1 3x 1 1) 4 (x 1 1)
5.
(5x 4 2 3x 2 1 6) 4 (x 2 1 3x 1 1)
In Exercises 6–10, find the polynomial p(x) that satisfies the
given equation.
6. p(x) 4 (6x 1 1) 5 3x 2 1 5
7. p(x) 4 (x 2 1 3x 2 5) 5 x 2 1 6x 1 1
5
8. p(x) 4 (2x 2 1 1) 5 3x 1 1 1 }
2x 2 11
2x 2 1
9. p(x) 4 (x 3 1 x 1 1) 5 x 2 1 5 1 }
x3 1 x 1 1
x3 1 x2 1 x 1 1
10. p(x) 4 (x 4 1 1) 5 1 1 }}
x4 1 1
11.
(x 2 1 8x 1 15) 4 q(x) 5 x 1 5
12.
(x 3 2 2x 2 2 8x 2 3) 4 q(x) 5 x 2 1 2x 1 1
13.
(x 4 2 5x 3 1 4x 2 2 5x 1 3) 4 q(x) 5 x 2 1 1
14.
(2x 7 1 5x 5 2 x 4 1 2x 3 1 5x 2 1) 4 q(x) 5 x 4 1 1
15.
25x 1 14
(x 5 1 3x 2 2 1) 4 q(x) 5 x 3 1 5x 1 3 1 }
2
LESSON 12.3
2
328
Algebra 1
Chapter 12 Resource Book
x 25
Copyright © Holt McDougal. All rights reserved.
In Exercises 11–15, find the polynomial q(x) that satisfies the
given equation.
Name ———————————————————————
FOCUS ON
12.3
Date ————————————
Practice
For use with pages 820–821
Divide using synthetic division.
1. (x2 1 2x 2 3) 4 (x 2 1)
2. (x2 1 5x 1 4) 4 (x 1 4)
3. (x3 2 4x2 1 4x 2 2) 4 (x 2 2)
4. (2x4 2 x2 1 2x 2 4) 4 (x 1 1)
5. (x3 2 6x2 1 4x 1 5) 4 (x 2 5)
6. (2x4 1 6x3 2 x2 2 5x 2 6) 4 (x 1 3)
7. (x3 1 6x2 1 6x 1 4) 4 (x 1 5)
8. (x3 2 3x2 – 3x 1 1) 4 (x 2 3)
9. (x4 2 3x2 2 3) 4 (x 2 1)
11.
1 x3 2 }12 x2 1 x 2 }32 2 4 1 x 2 }12 2
10. (x3 2 3x2 1 2x 2 24) 4 (x 2 4)
12.
1 x3 2 }13 x2 1 2x 2 }23 2 4 1 x 2 }13 2
13. Application Can you use synthetic division to divide x4 2 2x2 1 1 by x2 2 1?
Explain why or why not.
14. Challenge What value of a makes the remainder of (5x3 1 52x2 1 15x 2 a) 4 (x 1 10)
FOCUS ON 12.3
Copyright © Holt McDougal. All rights reserved.
equal to zero?
Algebra 1
Chapter 12 Resource Book
329
Name ———————————————————————
FOCUS ON
12.3
Date ————————————
Review for Mastery
For use with pages 820–821
GOAL
Use synthetic division to divide polynomials.
Vocabulary
Synthetic division is a convenient method to use when dividing a
polynomial by a binomial of the form x 2 k where k is a constant.
Synthetic division is derived from polynomial long division but uses
only the value of k and the coefficients of the dividend.
EXAMPLE 1
Use synthetic division
Divide 2x3 1 6x2 2 8x 2 12 by x 2 2 using synthetic division.
Solution
Write the value of k from the
divisor and coefficients of
the dividend in order of
descending exponents.
STEP 2
STEP 3
STEP 4
Bring down the leading coefficient. Multiply the leading coefficient by
the k-value. Write the product under
the second coefficient. Add.
Multiply the previous sum by the
k-value, and write the product under
the next coefficient. Add. Repeat for
all of the remaining coefficients.
K VALUE
Identify the quotient and remainder.
The bottom row gives the coefficients
of the quotient and the remainder.
COEFFICIENTS
COEFFICIENTS
OF QUOTIENT
REMAINDER
FOCUS ON 12.3
12
330
(2x3 1 6x2 2 8x 2 12) 4 (x 2 2) 5 2x2 1 10x 1 12 1 }
x22
Exercises for Example 1
Divide using synthetic division.
1. (x3 2 3x2 1 2x 2 3) 4 (x 2 1)
Algebra 1
Chapter 12 Resource Book
2. (2x4 2 4x2 2 6x 2 7) 4 (x 2 2)
Copyright © Holt McDougal. All rights reserved.
STEP 1
Name ———————————————————————
FOCUS ON
12.3
Review for Mastery
For use with pages 820–821
EXAMPLE 2
Date ————————————
continued
Use synthetic division
Divide x3 1 6x2 1 6x 2 9 by x 1 3 using synthetic division.
Solution
STEP 1
Write the value of k from the divisor and the coefficients of the dividend in
order of descending exponents.
K VALUE
STEP 2
COEFFICIENTS
Bring down the leading coefficient. Multiply the leading coefficient by the
k-value. Write the product under the second coefficient. Add.
STEP 3
Multiply the previous sum by the k-value, and write the product under the
next coefficient. Add. Repeat for all of the remaining coefficients.
Identify the quotient and remainder from the bottom row. The quotient is
x2 1 3x 2 3, and the remainder is 0.
(x3 1 6x2 1 6x 2 9) 4 (x 1 3) 5 x2 1 3x 2 3
Exercises for Example 2
Divide using synthetic division.
3. (x3 2 2x2 1 4x 1 2) 4 (x 1 1)
4. (x3 1 4x2 1 6x 1 4) 4 (x 1 2)
Algebra 1
Chapter 12 Resource Book
FOCUS ON 12.3
Copyright © Holt McDougal. All rights reserved.
STEP 4
331
Name ———————————————————————
LESSON
LESSON 12.4
12.4
Date ————————————
Practice A
For use with pages 822–828
Find the excluded values, if any, of the expression.
8x
1. }
24
15
2. }
4x
10
3. }
x26
24
4. }
x13
1
5. }
2x 2 2
5
6. }
8x 2 16
8
7. }
3x 1 6
5
8. }
2x 2 1
21
9. }
3x 1 2
Determine whether the expression is in simplest form.
x21
10. }
3x 2 3
x11
11. }
x2 2 1
x 1 10
12. }
x2 2 4
x13
13. }
x 2 2 4x
x15
14. }
x 2 1 5x
x
15. }
x 2 2 4x 1 4
14
16. }
21x
42
17. }
12x
2x 1 4
18. }
x12
x15
19. }
x25
x26
20. }
x 2 2 36
10x
21. }
2
x 2 100
22. Deck You have drawn up preliminary plans for a rectangular
deck that will be attached to the back of your house. You have
decided that the length of the deck should be twice the width
as shown.
a. Write a rational expression for the ratio of the perimeter to
the area of the deck.
b. Simplify your expression from part (a).
23. School Enrollment The total enrollment (in thousands) of students in public
schools from kindergarten through college from 1996 to 1999 can be modeled
by E 5 465t 1 56,780 where t is the number of years since 1996. The total
enrollment (in thousands) of students in public schools from kindergarten
through grade 8 can be modeled by K 5 245t 1 32,800.
a. Write a model for the ratio R of the number of enrollments in kindergarten
through grade 8 to the total number of enrollments.
b. Simplify your model from part (a).
332
Algebra 1
Chapter 12 Resource Book
x
2x
Copyright © Holt McDougal. All rights reserved.
Simplify the rational expression, if possible. Find the excluded values.
Name ———————————————————————
Date ————————————
Practice B
LESSON
12.4
For use with pages 822–828
14
1. }
3x
28
2. }
x25
5x
3. }
x 1 10
2x
4. }
4x 2 8
3x
5. }
7x 1 21
x11
6. }
3x 1 7
x16
7. }
2
x 2 2x 1 1
8
8. }}
x 2 1 4x 2 12
7x
9. }
2
x 2 25
LESSON 12.4
Find the excluded values, if any, of the expression.
Simplify the rational expression, if possible. Find the excluded values.
236x 2
10. }
18x
6x 2 24
11. }
x24
4x 2 12
12. }
32x
x 1 11
13. }
x 2 2 121
x13
14. }}
2
x 1 10x 1 21
x24
15. }}
x 2 1 11x 1 24
Write and simplify a rational expression for the ratio of the perimeter to
the area of the given figure.
16. Square
17. Rectangle
2x
8x
Copyright © Holt McDougal. All rights reserved.
8x
18. Triangle
x15
2x 1 1
2x
2x 1 1
2x 1 2
19. Zoo Exhibit The directors of a zoo have drawn up
preliminary plans for a rectangular exhibit. They have
decided on dimensions that are related as shown.
a. Write a rational expression for the ratio of the
perimeter to the area of the exhibit.
b. Simplify your expression from part (a).
4x 2 2
4x 1 3
20. Materials Used The material consumed M (in thousands of pounds) by a plastic
injection molding machine per year between 1995 and 2004 can be modeled by
8t 2 1 66t 1 70
(3 2 0.2t 1 0.1t )(t 1 7)
M 5 }}
2
where t is the number of years since 1995. Simplify the model and approximate the
number of pounds consumed in 2000.
Algebra 1
Chapter 12 Resource Book
333
Name ———————————————————————
LESSON
LESSON 12.4
12.4
Date ————————————
Practice C
For use with pages 822–828
Find the excluded values, if any, of the expression.
2x
1. }}
3x 2 1 11x 2 4
12
2. }
2
8x 2 3x 2 5
5x 2
3. }}
2
x 2 14x 1 49
Simplify the rational expression, if possible. Find the excluded values.
x27
4. }
x 2 2 6x 2 7
28x 3
5. }
12x 2 2 20x
9x 2 2 36x
6. }2
12x 2 24x
15x 4
7. }
15x 2 1 20x
2x 2 4
8. }
x 2 1 8x 2 20
4x 2 2 12x
9. }}
2x 2 2 5x 2 3
x 2 1 4x 2 60
10. }}
2x 2 1 23x 1 30
x24
11. }}
x 3 2 8x 2 1 16x
x 2 1 7x 1 10
12. }
2x 3 2 8x
5x 1 1
a
13. The expression }}
simplifies to }
. What is the value of a?
3x 1 2
15x 2 1 13x 1 2
Explain how you got your answer.
3x 2 1
14. Find two polynomials whose ratio simplifies to } and whose sum is
5x 1 1
8x 2 1 24x. Describe your steps.
15. Gazebo You have drawn up a preliminary plan for a gazebo that
x13
you want to build in your backyard. Your plan for the base is to
use two identical trapezoids as shown at the right.
a. Write a rational expression for the ratio of the perimeter to
the area of the floor of the gazebo.
b. Simplify your expression from part (a).
x
x21
x
Copyright © Holt McDougal. All rights reserved.
2x 1 4
16. Advertisement Flyers The number A (in hundreds of thousands) of advertising
flyers sent out by a department store between 1995 and 2004 can be modeled by
6t 2 1 102t 1 312
(18 2 0.5t 1 0.01t )(t 1 13)
where t is the number of years since 1995.
a. Simplify the model.
b. Use the model to approximate how many flyers were sent
out in 2001.
c. Graph the model. Describe how the number of flyers sent
out changed over time.
Number of flyers
(hundreds of thousands)
A 5 }}}
2
A
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 t
Years since 1995
334
Algebra 1
Chapter 12 Resource Book
Name ———————————————————————
LESSON
12.4
Date ————————————
Review for Mastery
For use with pages 822–828
Simplify rational expressions.
LESSON 12.4
GOAL
Vocabulary
A rational expression is an expression that can be written as a
ratio of two polynomials.
A rational expression is undefined when the denominator is 0.
A number that makes a rational expression undefined is called
an excluded value.
A rational expression is in simplest form if the numerator and
denominator have no factors in common other than 1.
EXAMPLE 1
Find excluded values
Find the excluded values, if any, of the expression.
8
a. }
22x
x
b. }
3x 2 9
x12
c. }
x2 1 2x 2 15
12
d. }
5x2 1 2x 1 7
Solution
8
a. The expression } is undefined when 22x 5 0, or x 5 0. The
22x
excluded value is 0.
x
b. The expression } is undefined when 3x 2 9 5 0, or x 5 3.
3x 2 9
Copyright © Holt McDougal. All rights reserved.
The excluded value is 3.
x12
c. The expression }
is undefined when x2 1 2x 2 15 5 0,
x2 1 2x 2 15
or (x 2 3)(x 1 5) 5 0.
The solutions of the equation are 3 and 25. The excluded values
are 3 and 25.
12
d. The expression }
is undefined when 5x2 1 2x 1 7 5 0.
5x2 1 2x 1 7
The discriminant is b2 2 4ac 5 22 2 (4)(5)(7) < 0. So, the quadratic equation has no real roots. There are no excluded values.
Exercises for Example 1
Find the excluded values, if any, of the expression.
9x
1. }
5x 2 15
x 21
2. }
x2 2 16
7
3. }
2x2 2 5x 1 6
x16
4. }
x2 1 4x 2 12
Algebra 1
Chapter 12 Resource Book
335
Name ———————————————————————
LESSON
LESSON 12.4
12.4
Review for Mastery
For use with pages 822–828
EXAMPLE 2
Date ————————————
continued
Simplify expressions by dividing out monomials
Simplify the rational expression, if possible. State the excluded values.
2m
2pm
a. } 5 }}
8m(m 2 1)
2 p 4 p m p (m 2 1)
1
4(m 2 1)
5}
Divide out common factor.
Simplify.
The excluded values are 0 and 1.
11
b. The expression } is already in simplest form. The excluded
y16
value is 26.
7 p q p (q 2 2)
7q2 2 14q
c. }
5 }}
3
14q
7 p 2 p q p q2
q22
5}
2q2
Divide out common factors.
Simplify.
The excluded value is 0.
EXAMPLE 3
Simplify an expression by dividing out binomials
x 2 1 4x 2 21
Simplify }}
. State the excluded values.
2
x 2 5x 1 6
x17
5}
x22
Factor numerator and denominator.
Divide out common factors.
Simplify.
The excluded values are 2 and 3.
Exercises for Examples 2 and 3
Simplify the expression. State the excluded value(s).
5.
3x3
15x
}5
3x2
7. }
2x 1 6
9.
336
Algebra 1
Chapter 12 Resource Book
x2 1 13x 1 42
x 2 2x 2 63
}}
2
28x
7x 2 21
6.
}
8.
}
2
10.
5x 1 10
3x 1 6x
4x2 1 20x 1 25
4x 2 25
}}
2
Copyright © Holt McDougal. All rights reserved.
Solution
(x 1 7)(x 2 3)
x2 1 4x 2 21
}
5 }}
2
(x 2 2)(x 2 3)
x 2 5x 1 6
(x 1 7)(x 2 3)
5 }}
(x 2 2)(x 2 3)
Name ———————————————————————
LESSONS
12.1–12.4
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 791–828
(in thousands) of people attending private
colleges in the United States during the
period 1995–2001 can be modeled by
3176 2 124x
5. Short Response The table shows
the relationship between the density
D (in kilograms per cubic meter) and
the volume V (in liters) of a substance
in a rectangular prism container.
LESSON 12.4
1. Multi-Step Problem The number N
N5}
1 2 0.056x
where x is the number of years since 1995.
a. Rewrite the model so that it has only
whole number coefficients. Then
simplify the model.
b. Approximate the number of people
attending private colleges in 2000.
2. Multi-Step Problem The amount S
(in millions of dollars) of federal budget
outlays for Social Security and the amount
O (in millions of dollars) of federal budget
outlays in the United States during the
period 1994–2001 can be modeled by
S 5 15x 1 320 and O 5 55x 1 1455
where x is the number of years since 1994.
Copyright © Holt McDougal. All rights reserved.
a. Write and simplify a rational model
for the percent p (in decimal form) of
federal budget outlays that were for
Social Security as a function of x.
b. Make a table for the percent of federal
budget outlays that were for Social
Security for the years 1994–2001.
c. Was the percent increasing or
decreasing from 1994–2001?
Density D (kg/m3)
Volume V (L)
1.4
10
2
7
3.5
4
14
1
a. Explain why the density and the
volume are inversely related. Then
write an equation that relates the
density and the volume.
b. Suppose that only the height of the
container can be changed. Describe
how the density changes as the height
increases.
6. Gridded Response A rectangular garden
has an area of 6x 2 1 7x 2 20 and a width
of 2x 1 5. What is the length of the garden
when x 5 5?
7. Extended Response The number N
(in millions) of new trucks sold in the
United States during the period 1993–2002
can be modeled by
5.684 1 0.674x
3. Gridded Response You pay $75 for an
annual membership to an aerobics club and
pay $2 per aerobics class. How much less
(in dollars) will the average cost per class be
if you go to 30 aerobics classes than if you
go to 10 aerobics classes?
4. Open-Ended Write an equation whose
graph is a hyperbola that has a vertical
asymptote of x 5 2 and a horizontal
asymptote of y 5 21.
N 5 }}
1 1 0.028x
where x is the number of years since 1993.
a. Rewrite the model so that it has only
whole number coefficients. Then
simplify the model and approximate
the number of new trucks sold
in 2001.
b. Graph the model. Describe how the
number of new trucks sold changed
during the period.
c. Can you use the model to conclude
that the revenue of new trucks sold
increased over time? Explain.
Algebra 1
Chapter 12 Resource Book
337
Name ———————————————————————
LESSON
LESSON 12.4
12.4
Date ————————————
Challenge Practice
For use with pages 822–828
6x 2 2 ax 2 5
2x 1 1
1. Find the value of a so that }}
5}
.
3x 2 2
9x 2 2 3ax 1 10
2
26x 1 (b 1 10)x 1 20
2x 1 4
2. Find the value of b so that }}
5}
.
2
3x 2 1
18x 2 bx 2 5
x 3 2 5x 2 1 cx 2 5
x2 1 1
3. Find the value of c so that }}
5}
.
3
2
x 2 5x 2 cx 1 5
x2 2 1
x2 1 x 1 1
x 4 1 x 3 1 dx 2 1 x 1 1
4. Find the value of d so that }}}
5}
.
4
3
2
x 2 5x 1 (d 1 1)x 2 5x 1 2
x 2 2 5x 1 2
x 3 1 2x 2 2 x 2 e
x11
5. Find the value of e so that }}
5}
.
x21
x 3 2 3x 1 e
p(x)
2x 2 1
6. Find the expressions for p(x) and q(x) so that } 5 } and
4x 2 1
q(x)
p(x) 2 q(x) 5 22x 2 1 5x.
p(x)
2x 1 5
7. Find the expressions for p(x) and q(x) so that } 5 } and
x23
q(x)
p(x) 1 q(x) 5 3x 2 1 5x 1 2.
p(x)
x11
8. Find the expressions for p(x) and q(x) so that } 5 } and
x15
q(x)
p(x) 2 q(x) 5 28x 1 4.
p(x)
x2 2 1
9. Find the expressions for p(x) and q(x) so that } 5 }
and
q(x)
2x 2 2 1
p(x)
2x 1 3
10. Find the expressions for p(x) and q(x) so that } 5 } and
x21
q(x)
p(x) 2 q(x) 5 3x 3 1 14x 2 1 9x 1 4.
338
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
p(x) 1 q(x) 5 3x 4 1 x 2 2 2.
Name ———————————————————————
LESSON
12.5
Date ————————————
Practice A
For use with pages 830–837
Match the equivalent expressions.
5
4x 2
1. } p }
10 22x
5
4x 2
2. } 4 }
22x
10
2x 10
3. } p }2
5 4x
1
A. }
x
24x 3
B. }
25
C. 2x
14x 2 9
4. } p }
3
2x
3x 2
7
5. } p }
4
2
9x
6x 2
10
6. } p }3
5
12x
x13
2x 2
7. } p }
4x
4x 1 12
3x 2 6 10x 4
8. } p }
x22
5x 2
x15
15x
9. }
p}
3
2x
1 10
6x
Find the product.
x 1 2 x 2 2 4x 1 3
12. } p }
x 2 3 x 2 1 6x 1 8
8x
4x 2
13. } 4 }
15
5
22
11
14. } 4 }2
6x
9x
x14
x14
15. } 4 }
5x
9x 2
x11
2x 1 2
16. }
4}
4
3x 2
4x 2 8
8x 2 16
17. } 4 }
10x
5x 2
x 2 1 3x 1 2
x11
18. } 4 }
14x
27x 2
LESSON 12.5
5x 1 5 x 2 1 5x 1 6
11. } p }
x11
x13
x13
x22
10. }
p}
x 2 2 2x x 2 1 4x 1 3
Find the quotient.
Copyright © Holt McDougal. All rights reserved.
19. Model Cars You want to create a display box that will hold
your model cars. You want each section of the box to be 5 inches
by 3 inches and you want the box’s dimensions to be related
as shown. Write and simplify an expression that you can use
to determine the number of sections you can have in the
display box.
5x
4x
3 in.
5 in.
20. Total Cost The cost C (in dollars) of producing a product from 1995 to 2004
can be modeled by
10 1 3t
C5}
80 2 t
where t is the number of years since 1995. The number N (in hundreds of
thousands) of units made each year from 1995 to 2004 can be modeled by
160 2 2t
N5}
11 2 t
where t is the number of years since 1995.
a. Write a model that gives the total production cost T.
b. Approximate the total production cost in 2000.
Algebra 1
Chapter 12 Resource Book
339
Name ———————————————————————
LESSON
12.5
Date ————————————
Practice B
For use with pages 830–837
Find the product.
5
4x 2
1. } p }5
15 8x
24 14x 6
2. } p }
40
7x 2
4x 1 24
21
3. } p }
2x 1 12
15
5x 1 10
x23
4. } p }
2x 2 6 10x 1 20
x14
x23
5. } p }}
2x 1 8 x 2 1 2x 2 15
x 2 1 4x 2 12 x 1 5
6. }}
p}
x 2 1 7x 1 10 2x 2 4
6x
2x 2 1 7x 1 3
7. }
p }}
2
18
4x 2 1
x4
8. } p (x 1 5)
4
x 1 5x 3
3x 2 6
9. }
p (x 2 1 6x 1 5)
2
x 2x22
Find the quotient.
22
11x 4
11. } 4 }2
18
9x
12.
7x 1 21
30
21x 1 63
20
}4}
12x 2 72
4x 2 24
13. } 4 }
x15
3x 1 15
x 2 1 11x 1 18
x12
14. } 4 }}
x21
3x 2 3
x 2 1 x 2 12
x 2 1 4x
15. } 4 }
x23
4x
4x 2
2x 1 10
16. }
4}
2
2
2x 2 10x
x 2 25
2x 2 14
17. }} 4 (x 1 3)
x 2 2 4x 2 21
18. Wall Art You want to create a rectangular picture from
2-inch by 3-inch tiles. You want the picture’s dimensions
to be related as shown.
a. Write and simplify an expression that you can use to
determine the number of 2-inch by 3-inch tiles that
will be needed for the picture.
b. If x 5 5, how many tiles will you need?
6x
4x
19. Profit The total profit P (in millions of dollars) earned by a company from 1995 to
2004 can be modeled by
3500 1 500t
P5}
98 2 t
where t is the number of years since 1995. The number N (in hundreds of thousands)
of units sold can be modeled by
(t 1 7)(3000 2 20t)
N 5 }}
490 2 5t
where t is the number of years since 1995. Write a model that gives the profit earned
per unit per year. Then approximate the profit per unit in 2002.
340
Algebra 1
Chapter 12 Resource Book
2 in.
3 in.
Copyright © Holt McDougal. All rights reserved.
LESSON 12.5
6
24
10. }3 4 }2
25x
5x
Name ———————————————————————
LESSON
12.5
Date ————————————
Practice C
For use with pages 830–837
Find the product.
8x
4x 2 1 2x 2 6
1. }
p }}
2
16
2x 1 x 2 3
14x 2
x2 2 x 2 2
2. } p }
x2 1 x 2 6
18x 3
2x 2 3
10x 2 1 20x
3. }
p}
2
4x 1 12
5x 1 10x
x 2 1 8x 1 15 x 2 2 2x 2 8
4. }
p}
x 2 1 7x 1 10
3x 2 1 9x
x6
5. } p (x 2 1 7)
3
9x 1 63x
4x 2 12
6. }
p (2x 2 1 11x 2 40)
2
x 1 5x 2 24
Find the quotient.
x 2 2 8x 2 9
x 2 2 5x 2 36
8. } 4 }
x11
5x 2 1 16x
x28
x 2 2 2x 2 48
7. }}
4}
2
8x
1 24
4x 1 24
x 2 2 16
x 2 1 4x
}
10. }
4
10x 2 40
5x 3 1 20x 2
16x 2 2 112
4x 4 2 20x 2
11. } 4 }
x17
x 2 2 49
6x 2 1 x 2 2
3x 2 2 10x 2 8
12. }}
4}
2
30x 2 2 120x
5x 2 20x
3x 2 1 21x
x 2 1 2x 2 35
13. }}
4}
2
9x 1 18
x 2 3x 2 10
x 2 1 7x 2 8
x 3 2 x 2 1 4x 2 4
14. }} 4 }
3
5x 2 1 40x
10x
LESSON 12.5
2x 2 1 7x 1 3
2x 2 2 9x 2 5
9. } 4 }}
x13
52x
Let a be a polynomial in the given equation. Find a.
2x 2 1 11x 1 5
a
15. } p }} 5 2x 2 2 11x 2 6
x15
x16
x13
4x 2 1 7x 2 15
16. }} 4 } 5 4x 2 2 33x 1 35
a
2x 1 1
Copyright © Holt McDougal. All rights reserved.
17. Snow Tires The average amount C (in dollars) of money spent per snow tire and
the number N of snow tires bought by an auto body shop from 2000 to 2004, can
be modeled by
t 1 80
C5}
1 2 0.05t
500(t 1 20)
and N 5 }
t 1 80
where t is the number of years since 2000. Write a model that gives the total
amount A spent by the shop each year on snow tires. Then approximate the
amount spent in 2003.
18. Drive-in Movies The average monthly revenue R (in dollars) from admissions
at a drive-in theater and the average price p (in dollars) per car from 1988 to 2000
can be modeled by
13,124 1 3122t
R 5 }}
and
26 2 t
294 1 7t
p5}
130 2 5t
where t is the number of years since 1988.
a. Write a model that gives the average number x of cars admitted per month
to the theater.
b. Graph the model on a graphing calculator and describe how the number of
cars admitted changed over time.
Algebra 1
Chapter 12 Resource Book
341
Name ———————————————————————
LESSON
12.5
Date ————————————
Review for Mastery
For use with pages 830–837
GOAL
EXAMPLE 1
Multiply and divide rational expressions.
Multiply rational expressions involving polynomials
x2 1 x 2 6
5x 2 1 15x
Find the product }}
p }}
.
2
2
10x 2 20x
x 2 2x 2 15
5x2 1 15x
x 1x26 }
p
}
10x2 2 20x x2 2 2x 2 15
2
(x2 1 x 2 6)(5x2 1 15x)
(10x 2 20x)(x 2 2x 2 15)
Multiply numerators and denominators.
(x 2 2)(x 1 3)5x(x 1 3)
2 p 5x(x 2 2)(x 2 5)(x 1 3)
Factor and divide out common factors.
x13
2(x 2 5)
Simplify.
5 }}}
2
2
LESSON 12.5
5 }}}
5}
EXAMPLE 2
Multiply a rational expression by a polynomial
2
4x
Find the product }}
p (x 1 8).
3
2
2x 1 10x 2 48x
2
4x
2x 1 10x 2 48x
p (x 1 8)
}}
3
2
x18
4x2(x 1 8)
5 }}
3
2
2x 1 10x 2 48x
2x(2x)(x 1 8)
2x(x 1 8)(x 2 3)
5 }}
2x
Rewrite polynomial as a fraction.
Multiply numerators and denominators.
Factor and divide out common factors.
Simplify.
5}
x23
Exercises for Examples 1 and 2
Find the product.
342
x2 2 1
2x 2 3x 1 1
4x 2 2
3x 1 18
1.
}
p}
2
2.
}}
p (x 1 5)
2
Algebra 1
Chapter 12 Resource Book
9x
3x 1 9x 2 30
Copyright © Holt McDougal. All rights reserved.
4x2
2x 1 10x 2 48x
5 }}
p}
3
2
1
Name ———————————————————————
LESSON
12.5
Review for Mastery
Date ————————————
continued
For use with pages 830–837
EXAMPLE 3
Divide rational expressions involving polynomials
8x 2 1 24x
x 2 1 7x 112
Find the quotient }
4 }}
.
2
2
x 2 5x
2
8x 1 24x
x 2 5x
x 27x 110
2
x 1 7x 1 12
x 2 7x 1 10
}
4}
2
2
8x2 1 24x
x 2 5x
x2 2 7x 1 10
x 1 7x 1 12
5}
p }
2
2
(8x2 1 24x)(x2 2 7x 1 10)
(x 2 5x)(x 1 7x 1 12)
Multiply numerators and denominators.
8x(x 1 3)(x 2 2)(x 2 5)
x(x 2 5)(x 1 4)(x 1 3)
Factor and divide out common factors.
8(x 2 2)
Simplify.
5}
x14
Divide a rational expression by a polynomial
LESSON 12.5
5 }}
2
2
5 }}
EXAMPLE 4
Multiply by multiplicative inverse.
2
5x 2 10x
Find the quotient }
4 (x 2 2).
2
4x 1 12
2
5x 2 10x
4x 1 12
}
4 (x 2 2)
2
5x2 2 10x
4x 1 12
x22
5}
4}
2
1
5x2 2 10x
4x 1 12
1
Copyright © Holt McDougal. All rights reserved.
5}
p}
2
x22
5x2 2 10x
(4x 1 12)(x 2 2)
5 }}
2
5x(x 2 2)
5 }}
2
4(x 1 3)(x 2 2)
5x
4(x 1 3)
5}
2
Rewrite polynomial as a fraction.
Multiply by multiplicative inverse.
Multiply numerators and denominators.
Factor and divide out common factors.
Simplify.
Exercises for Examples 3 and 4
Find the quotient.
x2 1 3x 2 10
x2 2 8x 1 12
3. }
4}
2
x21
3x 2 3x
2x4 2 6x3 2 56x2
4. }}
4 (x 2 7)
x3 2 5x2
Algebra 1
Chapter 12 Resource Book
343
Name ———————————————————————
LESSON
12.5
Date ————————————
Challenge Practice
For use with pages 830–837
Find the missing polynomial p(x) in the equation.
(x 1 1)(2x 1 3) (x 2 1)(3x 2 1)p(x)
1. }} p }}
5 8x 2 1 10x 2 3
(3x 2 1)
(x 2 2 1)
(3x 2 5)(24x 1 3) (3x 2 5)(24x 1 3)
2. }} 4 }}
5 x2 1 1
p(x)
(x 2 1 1)(x 1 2)
(27x 1 1)(22x 1 3) (4x 2 5)(2x 1 7)p(x)
3. }} p }}
5 22x 2 1 3x 2 1
2
2
(
)
28x
1
22x
2
15
(214x 2 47x 1 7)
LESSON 12.5
(22x 1 3)
(4x 1 5)(22x 1 3)
4. }} 4 }
5 24x 3 2 5x 2 1 12x 1 15
2
(
3x 2 2 1)p(x)
(3x 2 1)
(x 2 1 1)(x 1 1) (x 2 1)(26x 1 7)p(x)
5. }} p }}
5 16x 3 2 3x 2 1 16x 2 3
(x 2 2 1)
(26x 1 7)
(5x 2 4)(26x 1 1) (4x 1 1)(26x 1 1)p(x)
6. }} 4 }}
5 5x 2 4
(4x 2 1)
(16x 2 2 1)
2
(x 1 1)(x 1 2)(x 1 3) (x 2 3x 1 2)p(x)
7. }} p }}
5 x2 2 9
(x 2 1)(x 2 2)(x 2 3)
(x 2 1 3x 1 2)
Copyright © Holt McDougal. All rights reserved.
(23x 1 2)(2x 1 1)
(28x 1 5)(23x 1 2)
8. }} 4 }}
5 216x 2 2 14x 1 15
(7x 1 1)
(14x 2 1 9x 1 1)p(x)
344
Algebra 1
Chapter 12 Resource Book
Name ———————————————————————
FOCUS ON
12.5
Date ————————————
Practice
For use with pages 838–839
Find the product.
5x4
3
}4
}
1.
2. }
2
}3
22x8
3
}
5
}
2x
4. }
8x
5
8. }
2
x2 2 9
9. }
5.
2x
}
4
4x2
5
}
}
}
x2 1 4x
3x 2 9
}
2
11.
x 14
x –4
4x2 2 x
x 29
}
4x3 2 4x
}
x13
12.
x 2 16
}
2
3x 1 6
2x 1 6
}}
2
3x 1 12x 1 12
}}
x13
}
2
}
14.
1
16. Are the complex fractions } and
2
3
}
1
2
}
x 29
2x3 1 10x2
2x 1 2
}
3
x 1 4x2 2 5x
}}
x27
}
15.
}
3
equivalent? Explain your answer.
a2 2 b2
a2b
}
}
17. Challenge Are the complex fractions
answer.
x3 1 3x2
x 2 3x
}
2
}
2
}
x 2 16
10. }
2
Copyright © Holt McDougal. All rights reserved.
x23
x13
215x
x
}
3
4
13.
2x
4x
}
FOCUS ON 12.5
22x
23x
3x4
6. }
3
2
7.
3.
8x
3x
10x
9x5
24
}
}
2x
a+b
a2 2 b2
a+b
}
}
and
a2b
equivalent? Explain your
Algebra 1
Chapter 12 Resource Book
345
Name ———————————————————————
FOCUS ON
12.5
Date ————————————
Review for Mastery
For use with pages 838–839
GOAL
Simplify complex fractions
Vocabulary
A complex fraction is a fraction that contains a fraction in its
numerator, denominator, or both. To simplify a complex fraction,
divide its numerator by its denominator.
Key Concept
Simplifying a Complex Fraction
FOCUS ON 12.5
Let a, b, c and d be polynomials where b Þ 0, c Þ 0, and d Þ 0.
a
b
}
c
}
d
x
}
2
}
x
}
3
}
Algebra
Example
EXAMPLE 1
a
b
c
d
a
b
d
x
x
x
3
5 } 4 } 5 } p }c
3x
3
5 }2 4 }3 5 }2 p }x 5 }
5 }2
2x
Simplify a complex fraction
Simplify the complex fraction.
a.
23x
9x
5}
4 (23x3)
4
9x
Write the fraction as quotient.
1
23x
• }3
5}
4
Multiply by multiplicative inverse.
9x
212x
3
5 2}2
4x
Multiply numerators and denominators.
5 }3
Simplify.
x2 2 4
x12
b. } 5 (x2 2 4) 4 }
x12
x22
}
Write fraction as quotient.
x22
x22
5 (x2 2 4) 3 }
x12
(x2 2 4) • (x 2 2)
5 }}
x12
(x 1 2)(x 2 2)(x 2 2)
346
Algebra 1
Chapter 12 Resource Book
Multiply by multiplicative inverse.
Multiply numerators and denominators.
5 }}
x12
Factor and divide out common factor.
5 (x 2 2)2
Simplify.
Copyright © Holt McDougal. All rights reserved.
9x
4
}3
}
Name ———————————————————————
FOCUS ON
12.5
Review for Mastery
For use with pages 838–839
Date ————————————
continued
Exercises for Example 1
Simplify the complex fraction.
2
9
}
}
1.
EXAMPLE 2
3
2. }
4
6
5
3.
}
4x
3x3
4. }
2
6x
7
}
Simplify a complex fraction
Simplify
3x2 2 6x
x 2 4x 1 4
.
}}
x3 2 9x
}
x13
}}
2
3x2 2 6x
x 2 4x 1 4
x3 2 9x
3x2 2 6x
x 2 4x 1 4
x13
x 2 9x
÷}
5}
2
x13
•}
5}
2
3
(3x2 2 6x)(x 1 3)
5 }}
2
3
(x 2 4x 1 4)(x 2 9x)
Write fraction as quotient.
Multiply by multiplicative inverse.
Multiply numerators and denominators.
3x(x 2 2)(x 1 3)
(x 2 2)(x 2 2)x(x 1 3)(x 2 3)
Factor and divide out common factors.
3
(x 2 2)(x 2 3)
Simplify.
5 }}}
5 }}
FOCUS ON 12.5
3x2 2 6x
x 2 4x 1 4
}
x3 2 9x
}
x13
}
2
Copyright © Holt McDougal. All rights reserved.
2x2
5
}
}
Exercises for Example 2
Simplify the complex fraction.
5.
x2 1 4x
3x 2 75
}
x2 2 16
}
x15
}
2
6.
4x 1 24
2x 2 24x 1 72
}}
x2 1 12x 1 36
}}
x26
}}
2
Algebra 1
Chapter 12 Resource Book
347
Name ———————————————————————
LESSON
12.6
Date ————————————
Practice A
For use with pages 840–847
Find the sum or difference.
2
1
1. } 1 }
4x
4x
6
4
2. } 1 }
5x
5x
7
8
3. }2 2 }2
3x
3x
6
20
4. }3 2 }3
7x
7x
7
x23
5. } 1 }
2x
2x
17
x 2 10
6. } 2 }
9x
9x
6
2x 1 1
7. } 1 }
5x
5x
x
x14
8. }
2 }2
2
2x
2x
x22
x16
9. } 1 }
x21
x21
Find the LCD of the rational expressions.
2 4
10. }, }
5x 10x
1 x11
11. }, }
12x 4x 3
3
1
12. }, }
x11 x
5
3
13. }, }
x24 x
6x
5
14. }, }
x12 x14
9
8x
15. }, }
x23 x17
1
8x
16. } 1 }
5x
3
4
7x
17. } 2 }
8x
2
7
5
18. } 1 }
9x
4x
8
2
19. }2 2 }
5x
3x
3
4
20. } 1 }
x14
x
5
4
21. } 1 }
x17
x22
22. Cabin Cruiser A cabin cruiser travels 48 miles upstream (against the current) and
48 miles downstream (with the current). The speed of the current is 4 miles per hour.
a. Write an expression for the time it takes the cruiser going upstream and write an
expression for the time it takes the cruiser going downstream.
b. Use your answers from part (a) to write an equation that gives the total travel
time t (in hours) as a function of the boat’s average speed r (in miles per hour)
in still water.
c. Find the total travel time if the cabin cruiser’s average speed in still water is
12 miles per hour.
23. Driving You drive 40 miles to visit a friend. On the drive back home, your average
speed decreases by 4 miles per hour. Write an equation that gives the total driving
time t (in hours) as a function of your average speed r (in miles per hour) when
driving to visit your friend. Then find the total driving time if you drive to your
friend’s house at an average speed of 52 miles per hour. Round your answer to the
nearest tenth.
348
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 12.6
Find the sum or difference.
Name ———————————————————————
LESSON
12.6
Date ————————————
Practice B
For use with pages 840–847
Find the sum or difference.
x
8
1. } 1 }
x15
x15
6x
10x
2. } 2 }
x24
x24
5x
x13
3. } 1 }
x29
x29
x16
x25
4. } 2 }
x12
x12
7x 2 3
3x 2 4
5. } 1 }
2
x2 2 9
x 29
x21
2x 1 4
6. }
2}
2
3x 2
3x
Find the LCD of the rational expressions.
6
7
7. }3 , }
15x
5x
10 9x
8. } , }
x x17
3x 1 1 x 2 4
9. } , }
x24 x16
x15
4x
10. } , }
2x 2 4 x 2 2
8
1
11. } , }
2
2
x
2
3x
2 10
x 2 5x
3
4x
12. }
,}
x 2 1 5x 1 4 x 2 1 2x 1 1
4
11
13. } 1 }
7x
2x
5
8
14. }3 2 }
12x
3x
15.
}2}
1
x
16. } 1 }
5x 2 3
6x 2 5
3
4
17. }
2 }x
x 2 2 7x
18.
}2 1 }
x12
x13
19. } 1 }
x11
x21
x14
2x
20. }
1}
2
x23
x 2 3x
21.
}
2}
2
2
8x
x25
5
x
3x
x12
x13
x21
1
x 1 5x 1 4
1
x 2 16
22. Paddle Boat You paddle boat 8 miles upstream (against the current) and 8 miles
downstream (with the current). The speed of the current is 1 mile per hour.
a. Write an equation that gives the total travel time t (in hours) as a function of
your average speed r (in miles per hour) in still water.
b. Find your total travel time if your average speed in still water is 3 miles per hour.
c. How much faster is your total travel time if you increased your average speed in
still water to 3.5 miles per hour? Round your answer to the nearest tenth.
LESSON 12.6
Copyright © Holt McDougal. All rights reserved.
Find the sum or difference.
23. Bike Ride You bike 50 miles from home. On your way back home, your average
speed increases by 3 miles per hour.
a. Write an equation that gives the total biking time t (in hours) as a function of
your average speed r (in miles per hour) when you are biking away from home
b. Find the total biking time if you bike away from your home at an average speed
of 15 miles per hour. Round your answer to the nearest tenth.
c. How much longer is your total biking time if you bike away from your home at
an average speed of 12 miles per hour?
Algebra 1
Chapter 12 Resource Book
349
Name ———————————————————————
LESSON
12.6
Date ————————————
Practice C
For use with pages 840–847
Find the sum or difference.
2x 1 3
x29
1. } 1 }
x13
x13
x14
2x 2 4
2. } 2 }
x25
x25
6x 2 2
3x
3. } 2 }
2x 2 5
2x 2 5
x14
10x
4. } 1 }
x12
x25
3x
x19
5. } 2 }
x21
x 1 10
4x 1 3
6x 2 5
6. } 2 }
x15
2x 2 3
x21
3x 2 5
7. } 2 }
x22
3x 2
x24
x16
8. }
1}
2
x12
5x
2x
x25
9. } 2 }
x16
8x
x11
4x
10. }
2}
x 2 1 8x 1 7
x2 2 1
x11
x22
11. }
2}
x 2 1 2x 2 15
x 2 2 6x 1 9
x21
x16
12. }}
1}
x 2 1 3x 1 2
x 2 2 4x 2 12
Use the order of operations to write the expression as a single
rational expression.
x25
x
13. 4 } 2 5 }
x12
x11
1
2
1
2
4x
5
x22
15. }}
1}
p}
x11 x16
x 2 1 10x 1 24
1
7
4x
14. 6 } 1 }
x23
x 2 1 5x 2 24
2
x 2 1 3x 1 2
2x 2 1 3x 1 1
x13
16. } 2 }} 4 }
x23
x27
x2 2 9
18. Inline Skating You inline skate 10 miles from the beginning of a trail. On your way
back, your average speed decreases by 2.75 miles per hour.
a. Write an equation that gives the total skating time t (in hours) as a function of
your average speed r (in miles per hour) when you are skating away from the
beginning of the trail.
b. Find the total skating time if you skate away from the beginning of the trail at
an average speed of 10 miles per hour. Round your answer to the nearest tenth.
c. How much faster is your total skating time if you skate away from the beginning
of the trail at an average speed of 10.75 miles per hour?
19. Advertisement Delivery You and your friend plan to spend 45 minutes delivering
pizza shop advertisements to houses in the shop’s delivery area. You can deliver all
of the advertisements on your own in two and a half hours.
a. Write an equation that gives the fraction y of advertisements that your friend can
deliver alone as a function of the time t (in minutes).
b. Suppose that your friend can deliver the advertisements alone in two hours and
fifteen minutes. Can you deliver all of the advertisements if you and your friend
work together for 45 minutes? Explain.
350
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 12.6
c25
17. Suppose that a 5 4b 2 b 2 and b 5 }. Write a in terms of c.
3c 1 4
Name ———————————————————————
LESSON
12.6
Date ————————————
Review for Mastery
For use with pages 840–847
GOAL
Add and subtract rational expressions.
Vocabulary
The least common denominator (LCD) of two or more rational
expressions is the product of the factors of the denominators of the
rational expressions with each common factor used only once.
EXAMPLE 1
Add and subtract with the same denominator
Find the sum or difference.
a.
2
5x
8
5x
10
5x
5p2
5}
5x
Add numerators.
}1}5}
Factor and divide out
common factors.
2
Simplify.
5 }x
b.
11r
r27
3r 2 5
r27
11r 2 (3r 2 5)
r27
8r 1 5
5}
r27
} 2 } 5 }}
Subtract numerators.
Simplify.
Exercises for Example 1
Copyright © Holt McDougal. All rights reserved.
1.
EXAMPLE 2
x13
7x
x22
7x
}1}
5x 1 7
3x 2 4
2x 2 9
3x 2 4
}2}
2.
Find the LCD of rational expressions
LESSON 12.6
Find the sum or difference.
Find the LCD of the rational expression.
3x
x12
a. }
,}
2
2
x 2 5x 1 6 x 2 7x 1 10
b.
7
1
2x 2 1 4x 2 5
}, }
Solution
a. Find the least common multiple of
b. Find the least common multiple of
x2 2 5x 1 6 and x2 2 7x 1 10.
x2 2 5x 1 6 5 (x 2 2) p (x 2 3)
x2 2 7x + 10 5 (x 2 2) p (x 2 5)
The LCD of x2 2 5x 1 6 and
x2 2 7x 1 10 is (x 2 2)(x 2 3)(x 2 5).
2x 2 1 and 4x 2 5.
Because 2x 2 1 and 4x 2 5 cannot be
factored, they don’t have any factors in
common. The LCD is their product,
(2x 2 1)(4x 2 5).
Algebra 1
Chapter 12 Resource Book
351
Name ———————————————————————
LESSON
12.6
Review for Mastery
For use with pages 840–847
Date ————————————
continued
Exercises for Example 2
Find the LCD of the rational expression.
3.
EXAMPLE 3
3 x17
10x 15x
}2 , }
5
4.
9
3x 2 1 x 1 6
2x
}, }
5.
4x 1 1
8x
(x 1 5) x 1 8x 1 15
}2 , }
2
Add expressions with different denominators
15
11
Find the sum }
1 }5.
2
16x
12x
11
12x
11 p 4x3
12x p 4x
15
16x
15 p 3
16x p 3
}2 1 }5 5 }
1}
5
2
3
44x3
48x
45
48x
Simplify numerator and
denominator.
5 }5 1 }5
44x3 1 45
48x
Add fractions.
5}
5
EXAMPLE 4
Rewrite fraction using LCD,
48x5.
Subtract expressions with different denominators
12
4x
Find the difference }
2}
.
x23
LESSON 12.6
12(x 2 3) 2 4x(x 1 2)
(x 1 2)(x 2 3)
Subtract fractions.
24x2 1 4x 2 36
(x 1 2)(x 2 3)
Simplify numerator.
5 }}
5 }}
Exercises for Examples 3 and 4
Find the sum or difference.
12
7
6. }2 1 }3
9r
18r
7.
x
x 2 2x 2 15
352
Algebra 1
Chapter 12 Resource Book
3
x 29
}
1}
2
2
t22
t11
8. } 2 }
t13
t27
Rewrite fraction using LCD,
(x + 2)(x 2 3).
Copyright © Holt McDougal. All rights reserved.
x12
12(x 2 3)
4x(x 1 2)
4x
12
} 2 } 5 }} 2 }}
x23
(x 1 2)(x 2 3)
(x 2 3)(x 1 2)
x12
Name ———————————————————————
LESSON
12.6
Date ————————————
Problem Solving Workshop:
Worked Out Example
For use with pages 840–847
PROBLEM
Driving Beth drives 135 miles to another city. On the drive back home, her average
speed decreases by 9 miles per hour. Write an equation that gives the total driving
time t (in hours) as a function of her average speed r (in miles per hour) when driving
to the city. Then find the total driving time if she drives to the city at an average speed
of 45 miles per hour.
STEP 1
Read and Understand
What do you know?
The distance that Beth drives and the decrease of her average speed on the
way back.
What do you want to find out?
The total driving time.
STEP 2
Make a Plan Use what you know to write and solve an equation.
STEP 3
Solve the Problem An equation to represent the situation is t 5 }
1}
r
r29
135
135
135
135
where }
is the time to drive
r is the time to drive to the other city and }
r29
back home. Find the sum of the expressions.
135(r 2 9)
135r
t5}
1}
r(r 2 9)
r(r 2 9)
270r 2 1215
LESSON 12.6
Add fractions and simplify.
5}
r(r 2 9)
Copyright © Holt McDougal. All rights reserved.
Rewrite fractions using the LCD, r(r 2 9).
Calculate the value of t when r 5 45.
270(45) 2 1215
10,935
5}
5 6.75 hours
t 5 }}
1620
45(45 2 9)
The total travel time is 6.75 hours.
STEP 4
135
Look Back The time of the trip to the city is }
5 3 hours and the time of
45
135
the trip back home is }
5 3.75 hours. The total time of the trip is
45 2 9
6.75 hours.
PRACTICE
1. What If? Suppose in the example that
on the drive back home, Beth’s average
speed decreases by 15 miles per hour
because of construction. Write an
equation that gives the total driving
time t (in hours) as a function of her
average speed r (in miles per hour)
when driving to the city. Find the total
travel time if her average driving speed
to the city is 45 miles per hour.
2. Boat Travel A boat travels 25 kilome-
ters against the current and 25 kilometers with the current. The speed of the
current is 5 kilometers per hour. Write
an equation that gives the total travel
time t (in hours) as a function of the
boat’s average speed r (in kilometers
per hour) in still water. The boat’s speed
in still water is 15 kilometers per hour.
Find the total travel time.
Algebra 1
Chapter 12 Resource Book
353
Name ———————————————————————
LESSON
12.6
Date ————————————
Challenge Practice
For use with pages 840–847
In Exercises 1–5, write w in terms of x.
x25
1. w 5 3u 1 3u 2 and u 5 }
3x 1 1
2u 2 1
2. w 5 }
and u 5 x 2 1 1
u2 1 5
u11
x11
3. w 5 } and u 5 }
u12
x21
u2 1 1
x23
4. w 5 }
and u 5 }
x12
u2 2 1
2x 1 1
5. w 5 u 2 2u 2, u 5 3 1 5v, and v 5 }
x21
In Exercises 6–10, use the following information.
Billy and Mark are painting a fence together. Working alone, it would take Billy 60 hours
to paint the fence. Working alone, it would take Mark x hours to paint the fence.
6. Let y represent the fraction of the fence that is painted after t hours by Billy and
Mark working together. Write y as a function of t and x.
7. If Mark working alone can paint the fence in 45 hours, then how long would it take
Billy and Mark working together to paint the fence?
354
would it take Mark to paint the fence alone?
9. If working together Billy and Mark can paint the fence in 20 hours, then how long
would it take Mark to paint the fence alone?
10. Suppose Tom, who paints as fast as Mark, also helps paint. If working together,
Billy, Mark, and Tom can paint the fence in 20 hours, then how long would it take
Mark to paint the fence alone?
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
LESSON 12.6
8. If working together Billy and Mark can paint the fence in 30 hours, then how long
Name ———————————————————————
LESSON
12.7
Date ————————————
Practice A
For use with pages 848–854
Identify the excluded values for the rational expressions in the equation.
5x
1. } 5 0
x26
1
x14
2. } 5 }
x14
x 1 10
1
x12
3. }
5}
x23
x2 2 9
Solve the equation. Check your solution.
x
4
4. } 5 }
9
x
32
x
5. } 5 }
x
2
4
5
6. } 5 }
x23
x
12
10
7. } 5 }
x
x14
2
1
8. } 5 }
x26
x15
x
5
9. } 5 }
3
x12
Find the LCD of the rational expressions in the equation.
1
7
10. } 1 } 5 8
x
x14
1
4
11. } 1 3 5 }
x
x23
3
1
12. 7 2 } 5 }
x25
x12
4
1
1
13. } 1 } 5 }
x
x
3
6
1
1
14. } 2 } 5 }
5x
x
5
2x
1
15. } 1 2 5 }
x24
x24
5
2x
16. } 1 1 5 }
x25
x25
21
x
17. } 2 4 5 }
x16
x16
3
x
18. 3 1 } 5 }
x22
x22
19. Rain It has rained 3 of the last 8 days. How many consecutive days does it have to
rain in order for the percent of the number of rainy days to be raised to 75%?
20. Field Goal Average A field goal kicker has made 25 out of 40 attempted field goals
so far this season. How many consecutive field goals must he make to increase his
average to at least 0.680?
21. Paint Mixing You have a 4-pint mixture of paint that is made up of equal amounts
of blue paint and red paint. To create a certain shade of purple, you need a paint
mixture that is 60% blue.
a. Let p represent the number of pints of blue paint that you have to add. Write an
expression for the number of pints of blue paint that will be in the new mixture.
Write an expression for the total number of pints of blue and red paint that will
be in the new mixture.
b. Use your expressions from part (a) to write an equation that represents a paint
mixture that is 60% blue.
c. How many pints of blue paint do you need to add?
d. How many total pints of paint are there in the new mixture?
Algebra 1
Chapter 12 Resource Book
LESSON 12.7
Copyright © Holt McDougal. All rights reserved.
Solve the equation. Check your solution.
355
Name ———————————————————————
Date ————————————
Practice B
LESSON
12.7
For use with pages 848–854
Solve the equation. Check your solution.
3
x
1. } 5 }
x
27
2
3
2. } 5 }
x14
x
2
4
3. } 5 }
x
x27
7
10
4. } 5 }
x24
x12
x
25
5. } 5 }
x14
x14
x
8
6. } 5 }
x12
x18
x
21
7. } 5 }
x12
x12
x13
2
8. } 5 }
2x 2 5
3x
22
6x
9. } 5 }
x12
x12
Find the LCD of the rational expressions in the equation.
7x
3
11. } 1 4 5 }
x21
2x 2 2
x11
7x
10. } 1 4 5 }
x23
x23
4
7
12. } 1 1 5 }
x23
x22
Solve the equation. Check your solution.
212
3x
13. } 2 2 5 }
x14
x14
4
3
14. } 1 5 5 }
x12
x12
10
2x
15. } 1 2 5 }
x12
x21
22
x21
16. } 1 6 5 }
x12
x15
9
4x
17. } 1 1 5 }
x21
x25
5x
218
x
18. } 2 } 5 }
x22
x22
x24
19. Stain Mixing You are staining a coffee table you just made. After testing some
sample pieces of wood, you decide that you want a mix of a yellow stain and a red
stain. You estimate that you want a mix that contains 75% of the yellow stain. You
only have 1 pint that is made up of equal parts of the stain. How many pints of the
yellow stain do you have to add to the current mixture?
hang the wallpaper in a room in 3 hours. The assistant can hang the wallpaper in
one and one-half times the time it takes the expert wallpaper hanger to hang the
wallpaper alone. Let x represent the time (in hours) that the assistant can hang
the wallpaper alone.
a. Copy and complete the table.
LESSON 12.7
Person
356
Fraction of room
papered each hour
Time
(hours)
Fraction of
room papered
Assistant
}
1
x
3
?
Expert
?
3
?
b. Explain why the sum of the expressions in the last column must be 1.
c. Write a rational equation that you can use to find the amount of time it takes
the assistant to wallpaper the room alone. Then solve the equation.
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
20. Wallpaper Working together, an expert wallpaper hanger and an assistant can
Name ———————————————————————
LESSON
12.7
Date ————————————
Practice C
For use with pages 848–854
Solve the equation. Check your solution.
2
14
1. } 5 }
x
22x
26
x12
2. } 2 x 5 }
x11
x11
6
2
3. } 1 2 5 }
x24
x24
4
10
4. } 5 }
x21
x12
x
3x 1 2
5. } 5 }
x21
3x 2 5
2
8
6. } 5 }
x22
x24
5
1
1
7. } 1 } 5 }
x11
4
x
4x 2 3
x
8. } 1 1 5 }
x24
x25
3
4
7
9. } 2 } 5 }
2
x12
x22
x 24
22
2x 1 3
10. } 1 3x 5 }
x12
x12
6
x
2
11. } 2 } 5 }
2x 1 6
2
x13
3
5
x13
12. } 5 } 1 }
x21
2x 2 2
8
x21
1
13. 1 2 }2 5 }
x11
(x 1 1)
3x 2 4
2
14. 2x 1 } 5 }
x22
x22
18
2x
x12
15. } 2 } 5 }
x21
x24
x 2 2 5x 1 4
24
16. Let a and b be real numbers. The solutions of the equation ax 1 b 5 } 2 1 are
x13
29 and 9. What are the values of a and b?
17. Paint Mixing You have a 6-pint mixture of paint that is made up of equal amounts
of red paint and yellow paint. To create a certain shade of orange, you need a paint
mixture that is 30% red. How many pints of yellow paint do you need to add to
the mixture?
interest at annual rates of r% and (r 1 1)%. After 1 year, she earned $50 in interest
on the first account, and $180 in interest on the second account. How much did Mrs.
Jackson invest in each account?
19. Roofing Working together, an expert roofer and an assistant can complete the roof
on a certain building in 24 hours. The expert roofer can roof the building alone in
about three fifths of the time it takes the assistant to roof the building alone. Let x
represent the time (in hours) that the expert can roof the building alone.
a. Copy and complete the table.
Person
Fraction of roof
completed each hour
Time
(hours)
Fraction of
roof completed
Expert
}
1
x
24
?
Assistant
?
24
?
b. Explain why the sum of the expressions in the last column must be 1.
c. Write a rational equation that you can use to find the time that the expert can roof
the building alone. Then solve the equation.
d. How long does it take the assistant to roof the building alone?
Algebra 1
Chapter 12 Resource Book
LESSON 12.7
Copyright © Holt McDougal. All rights reserved.
18. Investing Mrs. Jackson invested a total of $4000 in two accounts earning simple
357
Name ———————————————————————
LESSON
12.7
Date ————————————
Review for Mastery
For use with pages 848–854
GOAL
Solve rational equations.
Vocabulary
A rational equation is an equation that contains rational expressions.
EXAMPLE 1
Use the cross products property
x
6
Solve }
5 }. Check your solution.
6
x1 5
Solution
x
6
Write original equation.
36 5 x2 1 5x
Cross products property
6
x15
}5}
0 5 x2 1 5x 2 36
Subtract 36 from each side.
0 5 (x 1 9)(x 2 4)
Factor polynomial.
x1950
or x 2 4 5 0
Zero-product property
x54
x 5 29 or
Solve for x.
CHECK
If x 5 29:
If x 5 4:
6
29
}0}
29 1 5
6
}0}
6
415
4
6
2
3
2
3
21.5 5 21.5 ✓
EXAMPLE 2
}5} ✓
Multiply by the LCD
3x
5
5
Solve }
2}
5}
.
x 15
x15
2
3x
x15
5
2
5
x15
}2}5}
5
2
3x
x15
5
x15
} p 2(x 1 5) 2 } p 2(x 1 5) 5 } p 2(x 1 5)
5 p 2(x 1 5)
2
LESSON 12.7
3x p 2(x 1 5)
x15
6x 2 5x 2 25 5 10
The solution is 35.
358
5 p 2(x 1 5)
x15
}2}5}
Algebra 1
Chapter 12 Resource Book
Write original equation.
Multiply by LCD.
Multiply. Divide out common
factors.
Simplify.
x 2 25 5 10
Combine like terms.
x 5 35
Add 25 to each side.
Copyright © Holt McDougal. All rights reserved.
The solutions are 29 and 4.
Name ———————————————————————
LESSON
12.7
Review for Mastery
For use with pages 848–854
EXAMPLE 3
Date ————————————
continued
Factor to find the LCD
16
4
Solve }
1 2 5 }}
. Check your solution.
2
x24
x 1 x 2 20
Solution
Write each denominator in factored form. The LCD is (x 2 4)(x 1 5).
16
x 1 x 2 20
4
x24
}125}
2
16
(x 2 4)(x 1 5)
4
x24
} p (x 2 4)(x 1 5) 1 2 p (x 2 4)(x 1 5) 5 }} p (x 2 4)(x 1 5)
4(x 2 4)(x 1 5)
x24
16(x 2 4)(x 1 5)
(x 2 4)(x 1 5)
}} 1 2(x 2 4)(x 1 5) 5 }}
4(x 1 5) 1 2(x2 1 x 2 20) 5 16
2x2 1 6x 2 20 5 16
2x2 1 6x 2 36 5 0
2(x2 1 3x 2 18) 5 0
2(x 2 3)(x 1 6) 5 0
x 2 3 5 0 or x 1 6 5 0
x 5 3 or
x 5 26
The solutions are 26 and 3.
If x 5 3:
If x 5 26:
16
3 1 3 2 20
4
324
16
(26) 2 6 2 20
4
26 2 4
}120}
2
} 1 2 0 }}
2
22 5 22 ✓
1.6 5 1.6 ✓
Exercises for Examples 1, 2, and 3
Solve the equation. Check your solution.
x 2 14
212
1. } 5 }
x
4
6
x23
x
18
x
x 1 10
1
5
2.
}5}
3.
}1}5}
4.
}2} 5 }
2
x
x14
4
x22
LESSON 12.7
Copyright © Holt McDougal. All rights reserved.
CHECK
27
x 1 10
11
x 1 2x 2 8
Algebra 1
Chapter 12 Resource Book
359
Name ———————————————————————
LESSONS
12.5–12.7
Date ————————————
Problem Solving Workshop:
Mixed Problem Solving
For use with pages 830–854
on steep trails and 8 miles on flat trails. Your
average speed on steep trails is 2 miles per
hour slower than your average speed on
flat trails.
a. Write an equation that gives the total
time t (in hours) of the hike as a
function of your average speed x
(in miles per hour) on flat trails.
b. Your average speed on flat trails is
4 miles per hour. Find the total time of
the hike.
2. Short Response Baseball player Roberto
Clemente’s career number B of times at
bat and career number H of hits during the
period 1955–1972 can be modeled by
355 1 555x
67 1 168x
and H 5 }
B5}
1 1 0.001x
1 2 0.003x
where x is the number of years since 1955.
a. A baseball player’s batting average
is the number of hits divided by the
number of times at bat. Write a model
that gives Roberto Clemente’s career
batting average A as a function of x.
b. The table shows Clemente’s actual
career number of times at bat and
actual career number of hits for three
different years. For which year does
the model give the best approximation
of A? Explain your choice.
4. Multi-Step Problem The number M
(in thousands) of males and the number F
(in thousands) of females participating in
high school athletic programs during the
period 1996–2003 can be modeled by
3634 1 332x
2369 1 355x
and F 5 }
M5}
1 1 0.07x
1 1 0.1x
where x is the number of years since 1996.
a. Write a model that gives the total
number S of high school students who
participated in high school athletic
programs as a function of x.
b. Approximate the total number of high
school students who participated in
high school athletic programs in 2001.
5. Gridded Response After 25 times at bat,
a major league baseball player has a batting
average of 0.160. How many consecutive
hits must the player get to raise his batting
average to 0.300?
6. Extended Response The amount A
(in millions of dollars) of passenger fares
by all commuter rails and the number P
(in millions) of passengers who traveled by
commuter rails in the United States during
the period 1997–2002 can be modeled by
1175.8 1 213.3x
and
A 5 }}
1 1 0.107x
P 5 356.2 1 28.1x 2 3.2x 2
LESSON 12.7
where x is the number of years since 1997.
360
Year
1955
1964
1972
Career
times at bat
474
5321
9454
Career hits
121
1633
3000
3. Open-Ended Describe a real-world
situation that can be modeled by a rational
equation and can be solved using the cross
products property. Explain what the solution
means in this situation.
Algebra 1
Chapter 12 Resource Book
a. Write a model that gives the average
cost C (in dollars) per passenger as a
function of x.
b. Approximate the average cost in 2000.
c. Graph the equation in part (a) on a
graphing calculator. Describe how
the average cost changed during the
period. Can you use the graph to
describe how the amount of
passenger fares changed during the
period? Explain your answer.
Copyright © Holt McDougal. All rights reserved.
1. Multi-Step Problem You hike 6 miles
Name ———————————————————————
LESSON
12.7
Date ————————————
Challenge Practice
For use with pages 848–854
In Exercises 1– 5, let a and b be real numbers. Find the values of a and b
that satisfy the equation.
216
1. The solutions to the equation ax 1 b 5 } 1 7 are x 5 22 and x 5 2.
x16
215
1
2. The solutions to the equation } x 2 2 5 } 1 1 are x 5 26 and x 5 6.
3
ax 1 b
5
2a
5
3. The solutions to the equation 2x 1 3 5 } 2 b are x 5 2} and x 5 }.
2
2x 2 7
2
3
240
3
4. The solutions to the equation ax 1 b 5 } 2 44 are x 5 2} and x 5 }.
7
7
x21
5
2a
5
5. The solutions to the equation 32x 1 1 5 } 2 b are x 5 2} and x 5 }.
8
2x 2 3
8
In Exercises 6 –9, use the following information.
The octane rating of a gasoline, which is a measure of the gasoline’s tendency to cause
“engine knock” is regulated by many states in the United States. Typically a refinery will
manufacture gasoline in two octane ratings, 87 and 93, and then mix these two octane
levels to make a variety of grades. For example, the mixing of one gallon of 87 octane
gasoline with one gallon of 93 octane gasoline, results in two gallons of 90 octane
gasoline. Suppose a refinery has 100,000 barrels of 87 octane gasoline available and
50,000 barrels of 93 octane gasoline available.
6. If an order comes in for 60,000 barrels of 91 octane gasoline, can the refinery fill
this order? If so, how many barrels of each octane must they mix?
this order? If so, how many barrels of each octane must they mix?
8. If an order comes in for 160,000 barrels of 89 octane gasoline, can the refinery fill
this order? If so, how many barrels of each octane must they mix?
9. If an order comes in for 20,000 barrels of 88 octane gasoline and 40,000 barrels of
91 octane gasoline, can the refinery fill this order? If so, how many barrels of each
octane must they mix?
LESSON 12.7
Copyright © Holt McDougal. All rights reserved.
7. If an order comes in for 90,000 barrels of 89 octane gasoline, can the refinery fill
Algebra 1
Chapter 12 Resource Book
361
Name ———————————————————————
CHAPTER
12
Date ————————————
Chapter Review Game
For use after Chapter 12
Mathematics Terminology
Solve the following exercises. Find the answer at the right of the page.
Place the letter associated with the correct answer on the line with the
exercise number to answer the following question.
a
b
What is the correct term for the division bar symbol in the expression }?
Exercises
Answers
4
1. What is the horizontal asymptote of y 5 } 1 2?
x25
(U) x 5 9
(S) x 5 2
2. Divide: (12x 2 1 7x 2 10) 4 (3x 2 2)
(T) 4x 2 5
(N) }
2x 1 1
x2 2 9
3. Simplify: }
2x 2 2 5x 2 3
(V) y 5 2
(I) 4x 1 5
x 2 1 3x 2 4
2x 2 1 14x
4. Multiply: }
p}
2
4x 1 28
x 2 3x 1 2
(E) x 5 5
(C) }
3x 2 1 12x
x 2 1 10x 1 24
5. Divide: }}
4 }}
2
3x 2 1
3x 1 14x 2 5
(M) x 5 0 or x 5 2
x15
x21
6. Subtract: }}
2}
2
x 2 1 3x
x 1 10x 1 21
(K) }}
2
x23
x11
7. Solve: } 5 }
x24
x26
(U) }}
3
1
8. Solve: } 1 } 5 2
x11
x21
(L) }}
362
Algebra 1
Chapter 12 Resource Book
x(x 1 4)
2(x 2 2)
3x(x 1 4)2(x 1 6)
(x 1 5)(3x 2 1)
3x
(x 1 5)(x 1 6)
213x 2 35
x(x 1 3)(x 1 7)
2
3
4
5
6
7
8
Copyright © Holt McDougal. All rights reserved.
CHAPTER REVIEW GAME
1
x13
Answers
Lesson 7.1
13. (1, 3)
14. (24, 4)
y
y
Practice Level A
6
3
1. yes 2. no 3. yes 4. no 5. no 6. yes
21
23
1
3
26
x
2 x
13. (2, 21) 14. (3, 4) 15. (21, 21)
16. (4, 2)
17. (2, 23)
y
y
5
1
3
21
21
1
1
3
15. (2, 25)
16. (23, 0)
y
y
1
5 x
3
1
25
23
21
21
1
15
x
23
x
21
21
26
23
5
3
25
18. (23, 0)
215
19. (3, 3)
y
y
17. (25, 3)
9
29
3
y
35
3
3
29
23
3
9
1
x
23
9 x
3
23
x
18. (22, 24)
y
29
ANSWERS
2
1
7. B 8. A 9. C 10. F 11. E 12. D
21
21
1
x
6
10 x
29
29
20. (21, 4)
21. (3, 22)
215
y
y
5
19. (23, 6)
20. (4, 25)
6
y
23
21
21
26
1
x
22
22
2
6
22
22
x
29
23
people
9 x
26
29
22. between 1995 and 1996; about 1175 thousand
21. (2, 7)
y
16
14
12
10
8
6
4
2
0
y
21
x 1 y 5 15
15
1.5x 1 2y 5 26
9
3
29
0 2 4 6 8 10 12 14 16 x
Bottles of apple juice
8 bottles of apple juice and 7 bottles of orange
juice
Practice Level B
1. no 2. yes 3. yes 4. no 5. no 6. yes
7. (3, 25) 8. (21, 4) 9. (22, 2) 10. (4, 22)
11. (25, 3) 12. (0, 4)
22.
Non-blooming annuals
Bottles of orange juice
Copyright © Holt McDougal. All rights reserved.
26
23.
y
9
2
1
25
x
y
35
30
25
20
15
10
5
0
23
9 x
3.2x 1 1.5y 5 49.6
x 1 y 5 24
0 5 10 15 20 25 30 35 x
Blooming annuals
8 blooming annuals and 16 non-blooming annuals
Algebra 1
Chapter 7 Resource Book
A1
Lesson 7.1, continued
23. a. x 1 y 5 27 and 0.25x 1 y 5 12
y
32
28
24
20
16
12
8
4
0
y
6
2
x 1 y 5 27
22
22
0.25x 1 y 5 12
0 4 8 12 16 20 24 28 32 x
Outs made by infielders
17.
y
7
5
Practice Level C
1. no 2. yes 3. no 4. yes 5. no 6. yes
7. (6, 1)
12
1
4
24
x
19.
10. (23, 5)
y
y
3
1
5
x
23
6
22
22
x
2
12. (21, 7)
y
y
21
3
15
1
(4, 21)
y 5 2t 1 15
y 5 22t 1 25
0 1 2 3 4 5 6 7 t
Years since 2000
y 5 x 2 25
150
y 5 x 2 0.2x
100
(125, 100)
50
x
1
21
3
0
5
23
29
13. (5, 6)
23
3 x
14. (29, 2)
y
0
50
100 150 200 x
Amount of purchase
(dollars)
For purchases greater than $125, 20% off is the
better deal.
y
10
5
6
3
2
210
1
21
y
35
30
25
20
15
10
5
0
y
200
Amount you pay
(dollars)
11. (2, 2)
x
1
mid-2002
20. 20% off;
25
210
21
21
18. 5.5%: $20,000; 6.5%: $25,000
4
Number of umbrellas
212
7x
9. (5, 25)
21
23
y
3
21
3
8. (28, 4)
5
(1, 5)
(22, 2)
y
3
10 x
16. Sample answer: m 5 1, b 5 24
c. infielders: 20 outs; outfielders: 7 outs
1
6
1
3
5 x
26
22
22
Review for Mastery
1. (21, 1) 2. (2, 4) 3. (24, 22) 4. 60 mi
x
2
Problem Solving Workshop:
Worked Out Example
1. 10 square feet 2. 115 student tickets,
98 general admission tickets 3. 124 student
tickets, 117 general admission tickets 4. 4
Challenge Practice
1. yes 2. yes 3. no 4. Yes, if a2 1 b2 Þ 0.
A2
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
Outs made by outfielders
ANSWERS
b.
15. (4, 4)
Lesson 7.1, continued
y
110,000
105,000
100,000
95,000
90,000
85,000
0
1. Equation 1. 2. Equation 2.
3. Equation 2. 4. (6, 10) 5. (28, 24)
6. (3, 7) 7. (29, 5) 8. (2, 10) 9. (212, 8)
10. (27, 6) 11. (8, 1) 12. (23, 23)
3
3 1
2 1
13. 1, } 14. 2}, } 15. 2}, }
4
2 2
3 3
16. a 5 25, b 5 22 17. cleanups: 250 hr;
painting: 150 hr 18. x 5 16, y 5 4 19. yes;
1
0 2 4 6 8 10 t
Years since 1990
7. Hockey: y 5 1200t 1 20,000; Soccer:
y 5 2000t; Baseball: y 5 21000t 1 90,000
8. 1975
9. 1980
y
120,000
100,000
80,000
60,000
40,000
20,000
0
Spectators
Spectators
Practice Level C
y
120,000
100,000
80,000
60,000
40,000
20,000
0
0 10 20 30 40 50 t
Years since 1950
2
1
2
1
2
ANSWERS
Number of households
5. Bayside: y 5 500t 1 100,000; Coal Flats:
y 522000t 1 105,000
6. 1992
The linear system x 1 y 5 8 and x 1 0.5y 5 6.4
where x is the amount of soil and y is the amount
of the half and half mix has a solution of x 5 4.8
and y 5 3.2. So 3.2 buckets are needed and there
are 4 buckets.
Review for Mastery
1. (2, 24) 2. (23, 6) 3. (6, 2) 4. (3, 8)
5. (27, 6) 6. (4, 2)
0 10 20 30 40 50 t
Years since 1950
Lesson 7.2
Practice Level A
1. y 5 7 2 9x 2. y 5 3x 2 10 3. x 5 4y 1 1
Challenge Practice
15
3
1. (2, 3) 2. }, 2}
16
2
1
2
Î
}
Î2
}
23
5
3. 2 } , 2 } ,
6
6
1
1 Î 23 Î 5 2 1 Î 236 , 2Î 56 2, 1 Î 236 , Î 56 2
}
}
2 }
, }6 ,
6
}
}
}
}
}
}
}
}
}
}
4. (214, 2Ï10 ), (214, Ï10 )
4. x 5 3 2 2y 5. y 5 x 2 4 6. x 5 6y 1 14
7. Equation 1. 8. Equation 2.
9. Equation 1. 10. Equation 1.
Copyright © Holt McDougal. All rights reserved.
11. Equation 2. 12. Equation 1.
Lesson 7.3
Practice Level A
17. (4, 1) 18. (2, 2) 19. (6, 5) 20. (1, 5)
1. 3x 2 y 5 23 and 8x 1 y 5 11
2. 8x 2 y 5 1 and 8x 1 3y 5 7
3. 7x 2 4y 5 8 and 7x 1 4y 5 9
21. (23, 21) 22. (21, 4) 23. (3, 3)
4. 7x 2 y 5 13 and 214x 1 y 5 23
24. (5, 22) 25. brother: 6 hr; sister: 5 hr
5. x 2 3y 5 14 and x 1 10y 5 23
26. a. x 5 2y b. 3x 1 4.5y 5 252
c. popcorn: 48 boxes; nuts: 24 cans
6. 8x 2 4y 5 21 and 214x 1 4y 5 23
13. (1, 0) 14. (2, 23) 15. (21, 2) 16. (3, 22)
Practice Level B
3
1. y 5 22x 1 3 2. y 5 } x 2 3
4
2
4
}
}
3. x 5 y 1
4. Equation 1.
3
3
5. Equation 2. 6. Equation 2.
7. (2, 1) 8. (23, 4) 9. (4, 21) 10. (25, 5)
11. (3, 22) 12. (24, 22) 13. (6, 23) 14. (7, 4)
15. (3, 8) 16. (1, 1) 17. (4, 24) 18. (1, 2)
19. 4 pairs of sticks and 2 pairs of brushes
20. a. x 1 y 5 12; 225x 1 200y 5 2600
b. households mowed: 8; households shoveled: 4
21. length of hole: 16 cm; length of sheet: 17 cm
7. Add the equations. 8. Arrange the terms.
9. Subtract the equations. 10. Arrange the
terms. 11. Add the equations. 12. Arrange the
terms.
4
13. (1, 1) 14. (215, 6) 15. 22, }
3
1
2
16. (6, 25) 17. (3, 2) 18. (24, 1) 19. (2, 1)
20. (23, 4) 21. (21, 5) 22. (6, 0) 23. (8, 5)
19
1
24. 2}, 2}
3
2
1
2
25. Your speed with no wind:
5.5 mi/h; Wind speed: 2.5 mi/h 26. Car wash:
$6; One gallon of regular gasoline: $2.10
Algebra 1
Chapter 7 Resource Book
A3
Lesson 7.3, continued
1. 8x 2 y 5 19 and 3x 1 y 5 7
ANSWERS
2. 4x 2 y 5 211 and 4x 1 6y 5 23
3. 9x 2 2y 5 5 and 11x 1 2y 5 8 4. Arrange
the terms. 5. Arrange the terms. 6. Arrange the
terms. 7. Add the equations. 8. Arrange the
terms. 9. Subtract the equations. 10. (3, 5)
Challenge Practice
4 1
4
1. 2}, } 2. 1 2}, 2
11 2
15
1
2
}
3}
}
2
3}
3. (Ï 7 , Ï 23 ), (2Ï 7 , Ï 23 )
1
37 23
5. }, }
6a 6b
2
1
13b
5b
6. 2}, 2}
3a
3
1
1 1
4. }, }
a 2b
2
2
11. (22, 4) 12. (7, 23) 13. (26, 2)
Lesson 7.4
14. (10, 5) 15. (29, 25) 16. (3, 11)
Practice Level A
17. (10, 9) 18. (15, 8) 19. (21, 21)
1. C 2. B 3. A 4. Sample answer: Multiply
the first equation by 7. 5. Sample answer:
Multiply the first equation by 2. 6. Sample
1
37
20. (24, 3) 21. 8, }
3
2
22. Speed of barge in
still water: 5.9 mi/h; Speed of current: 2.1 mi/h
23. a. Flat fee: $15; Hourly fee: $12 b. $147
Practice Level C
1. (24, 5) 2. (8, 6) 3. (210, 3) 4. (26, 25)
5. (9, 14) 6. (21, 7) 7. (18, 18) 8. (26, 24)
9. (15, 20) 10. (3, 5) 11. (28, 24)
12. (11, 12) 13. (23, 8) 14. (9, 16)
1
12
15. (28, 27) 16. 5, }
b
2
17. (1, 2, 1);
Answers will vary. 18. a. 5x 1 30y 5 207.5 and
5x 1 50y 5 212.5; Let x be the cost of one day of
rental and let y be the cost per mile over
150 miles. Because a person is only charged for
miles over 150, subtract the number of miles
traveled from 150 to get the number of miles a
person is charged for. b. Daily rental fee: $40;
Per mile fee: $.25 19. $24.72; Use the table
to set up a linear system to find the cost of one
stamp and one package of cards. Then use this
information to find the total cost of 3 stamps and
3 packages of cards.
Review for Mastery
1. (4, 2) 2. (23, 4) 3. (21, 22) 4. (6, 5)
5. (2, 26) 6. (3, 3)
Problem Solving Workshop:
Using Alternative Methods
1. speed of Calvin in still air: 7.95 miles per hour,
speed of wind: 0.45 miles per hour 2. speed of
Calvin in still air: 7.2 miles per hour, speed of
wind: 1.2 miles per hour 3. speed of boat in still
water: 25 miles per hour, speed of current: 5 miles
per hour
answer: Multiply the second equation by 6.
7. Sample answer: Multiply the first equation
by 22. 8. Sample answer: Multiply the second
equation by 3. 9. Sample answer: Multiply the
1
second equation by }2 . 10. (1, 2) 11. (23, 4)
12. (5, 5) 13. (6, 23) 14. (22, 22)
15. (8, 10) 16. (25, 7) 17. (8, 21) 18. (4, 4)
19. (10, 12) 20. (22, 24) 21. (1, 5)
22. a. Adult: $9; Youth: $5 b. $43
23. a. y 5 30 1 45x and y 5 45 1 40x
b. x 5 3, y 5 165 c. 3 h
Practice Level B
1. Sample answer: Multiply the first equation
by 2. 2. Sample answer: Multiply the second
equation by 23. 3. Sample answer: Multiply
the first equation by 23. 4. Sample answer:
Multiply the second equation by 22. 5. Sample
answer: Multiply the first equation by 25.
6. Sample answer: Multiply the first equation
by 2. 7. (4, 21) 8. (3, 6) 9. (22, 25)
10. (26, 7) 11. (9, 5) 12. (2, 22) 13. (10, 8)
14. (21, 12) 15. (5, 4) 16. (25, 23)
17. (15, 24) 18. (8, 8) 19. a. 2x 1 4y 5 28
and 4x 1 5y 5 45.5 b. Adult: $7; Youth: $3.50
c. $31.50 20. a. 3x 1 2y 5 557 and
5x 1 4y 5 974 b. Hotel: $140/night;
Tickets: $68.50/pair 21. x 1 y 5 15 and
180x 1 155y 5 2400; $180/day: 3 workers;
$155/day: 12 workers
Practice Level C
1. (4, 8) 2. (23, 21) 3. (5, 29)
4. (210, 10)
5. (22, 25) 6. (6, 7)
7. (0, 3) 8. (8, 14) 9. (6, 4) 10. (1, 9)
A4
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
Practice Level B
Lesson 7.4, continued
11. (2, 7) 12. (23, 23) 13. (2, 21) 14. (4, 3)
1 1
15. (25, 22) 16. (6, 1) 17. 2}, }
2 4
18. (24, 4) 19. a 5 2, b 5 1
1
2
21. Thai: 5 people; Szechwan: 3 people
22. To school: 2.72 mi/h; Home: 2.04 mi/h
Review for Mastery
x 1 30y 5 63
5. 3 mi 6. 2 mi
Lesson 7.5
Practice Level A
1. 3 2. 22 3. 3 4. A; infinitely many
solutions 5. C; one solution 6. B; no solution
7. no solution
8. one solution
1. (3, 25) 2. (6, 24) 3. (7, 2) 4. (2, 3)
y
ANSWERS
20. a. 2x 1 4y 5 166 and 4x 1 5y 5 263
b. Adult: $37; Youth: $23 c. $189
4. x 1 20y 5 43
y
3
5. (9, 21) 6. (5, 6)
3
1
Problem Solving Workshop:
Mixed Problem Solving
21
21
1
1
3 x
23
21
21
Copyright © Holt McDougal. All rights reserved.
1. a. x 5 student tickets, y 5 general admission
tickets; x 1 y 5 556, 5x 1 8y 5 3797 b. 217
student tickets, 339 general admission tickets.
2. a. 6 miles per hour into the wind, 10 miles per
hour with the wind b. x 5 speed of bicyclist,
y 5 speed of wind; x 2 y 5 6, x 1 y 5 10
c. The bicyclist’s speed in still air is 8 miles per
hour. The speed of the wind is 2 miles per hour.
3. a. x 5 amount in the 3% annual interest account, y 5 amount in the 4% annual interest
account; x 1 y 5 30,000, 0.03x 1 0.04y 5 1020
b. $18,000 at 3%, $12,000 at 4% 4. 11
5. Answer will vary. 6. By solving the linear
system, 1 pound of chicken costs $2.25 and
1 pound of fish costs $3.75. So, 2 pounds of
chicken and 2 pounds of fish costs $12.
7. Answer will vary. Sample answer: m 5 2,
b 5 25 8. 5 9. a. x 5 amount of 20% acid
solution, y 5 amount of 70% solution;
x 1 y 5 900, 0.2x 1 0.7y 5 360
b. 540 milliliters of 20% acid solution,
360 milliliters of 70% acid solution
c. No; The chemist needs 450 milliliters of both
acid solutions.
Challenge Practice
1. x 1 2y 5 5
3x 1 3y 5 8
1 7
1
2. }, } ; it takes Terry } hour to mow a small
3 3
3
7
lawn and }3 hours to mow a large lawn.
1
2
3. 13 small lawns
1
3 x
23
9. infinitely many solutions
y
3
1
21
21
23
1
3
x
23
10. one solution
11. one solution
y
y
3
3
1
23
1
1
3 x
23
21
21
3
x
23
12. no solution
y
3
21
3 x
23
13. no solution 14. no solution 15. (0, 0)
16. (21, 21) 17. no solution 18. infinitely
many solutions 19. one solution 20. one
solution 21. infinitely many solutions 22. one
solution 23. no solution 24. one solution
25. one solution 26. one solution 27. one
solution 28. Yes; the system 15x 1 8y 5 263.25
and 20x 1 13y 5 358 can be used to model the
situation, and this system has one solution.
Algebra 1
Chapter 7 Resource Book
A5
Lesson 7.5, continued
29. a. 45x 1 10y 5 425 and 225x 1 50y 5 2125
b. infinitely many solutions c. No, because one
ANSWERS
equation in the system is a multiple of the other,
so specific values for neither x nor y can be found.
Practice Level C
1. C; infinitely many solutions
2. B; one solution 3. A; no solution
4. one solution
Practice Level B
5. one solution
y
1. C; infinitely many solutions 2. A; no solution
y
6
3. B; one solution
2
4. no solution
5. one solution
3
22
22
6
10 x
1
y
y
3
21
21
26
x
1
3
1
23
3 x
1
6. infinitely many solutions
21
21
5 x
1
y
3
23
1
6. one solution
7. one solution
y
21
21
23
1
3 x
y
23
3
1
23
7. no solution
1
21
21
x
3
1
21
3
5
8. one solution
y
x
y
3
2
23
8. no solution
22
22
2
x
9. infinitely many
1
23
21
3 x
1
solutions
y
210
y
23
1
23
21
21
1
23
1
3
x
9. no solution
y
1
3
3
5 x
23
1
3 x
10. (8, 0) 11. infinitely many solutions
12. (20, 30) 13. (21, 21) 14. (3, 4) 15. no
solution 16. no solution 17. no solution
18. one solution 19. one solution 20. one
solution 21. one solution 22. infinitely many
solutions 23. no solution 24. infinitely many
solutions 25. Yes; the system 2x 1 12y 5 1859.3
and 2x 1 22y 5 3158.8 can be used to model the
situation, and this system has one solution, (about
$153, about $130).
26. a. 30x 1 20y 5 910 and 45x 1 30y 5 1365
b. infinitely many solutions c. No, because one
equation in the system is a multiple of the other,
no specific values for x or y can be found.
A6
Algebra 1
Chapter 7 Resource Book
23
10. (212, 28) 11. infinitely many solutions
5
1
4 8
12. 6, } 13. no solution 14. 1, } 15. }, }
4
2
3 3
1
2
1
2
1
2
16. one solution 17. no solution
18. one solution 19. infinitely many solutions
20. one solution 21. no solution
22. one solution 23. one solution 24. infinitely
many solutions 25. a. 28x 1 44y 5 964.4 and
21x 1 33y 5 723.30 b. infinitely many solutions
c. No, because one equation in the system is a
multiple of the other, no specific values for
x or y can be found. 26. y 5 10x and
y 5 8(x 2 10) b. x 5 240, y 5 2400
c. No, because x and y both represent quantities
that are never negative.
Copyright © Holt McDougal. All rights reserved.
21
3
Lesson 7.5, continued
19. a. x 1 y ≤ 10 and 15x 1 18y ≤ 90
b.
c. Answers will
y
1. infinitely many solutions 2. no solution
5. one solution
Challenge Practice
1
1. a 5 } 2. No value of a gives infinitely many
2
1
solutions. 3. a Þ }2 4. The number of solutions
depends only on the value of c. 5. When c 5 4
there are an infinite number of solutions. When
c Þ 4 there are no solutions.
c1b2 2 c2b1
vary.
0 1 2 3 4 5 6 7 8 9 x
Boxes of 5-ounce cups
20. a. 6 h
b.
a1c2 2 a2c1
6. x 5 }; y 5 }
a1b2 2 a2b1
a1b2 2 a2b1
b2
7. a1b2 Þ a2b1 8. a1b2 5 a2b1 and c2 Þ }c1
b1
b2
9. a1b2 5 a2b1 and c2 5 }c1
b1
9
8
7
6
5
4
3
2
1
0
ANSWERS
Boxes of 8-ounce cups
3. infinitely many solutions 4. one solution
Hours spent on history
Review for Mastery
y
6
5
4
3
2
1
0
x1y<6
x1y>4
0 1 2 3 4 5 6 x
Hours spent on science
Practice Level B
Lesson 7.6
1. yes
Practice Level A
2. yes 3. no 4. B 5. A 6. C
y
7.
y
8.
3
1. yes 2. no 3. yes 4. no 5. yes 6. yes
1
7. D 8. B 9. A 10. C 11. F 12. E
13.
14.
y
23
1
21
y
23
3 x
21
21
3 x
1
3
23
3
23
1
1
Copyright © Holt McDougal. All rights reserved.
21
21
23
23
1
3
x
1
3
x
y
9.
3
23
y
15.
1
3
11.
21
21
23
x
1
3
y
3
y
1
21
23
1
1
3
x
x
y
1
23
21
21
1
3 x
23
23
x
21
21
y
18.
3
23
3
1
25
17.
12.
x
23
23
1
23
1
21
21
1
3 x
1
21
21
3
1
23
21
y
16.
3
y
10.
3
23
21
21
23
1
3 x
13. y ≥ 24 and y < 1 14. x ≥ 24 and y < 23
15. y ≥ x 1 1 and x ≤ 0 16. y ≤ 4 2 x and y > 2
17. y ≤ x and y < 1 2 x
18. x ≥ 0, y ≥ 0, and y ≤ x 1 2
Algebra 1
Chapter 7 Resource Book
A7
Lesson 7.6, continued
1.9x 1 5.2y < 20
x1y>5
0
0 1 2 3 4 5 6 x
Packages of hot dogs
20. a. 5.5 h
Hours weeding
b.
y
6
5
4
3
2
1
0
x 1 y < 5.5
x1y>4
0 1 2 3 4 5 6 x
Hours cleaning
b.
Swimming
Practice Level C
1. no 2. no 3. yes 4. C 5. A 6. B
y
y
8.
3
3
1
23
1
23
21
21
9.
1
21
x
y
3
23
21
21
y
14
12
10
8
6
4
2
0
0 2 4 6 8 10 12 14 x
Running
1
3 x
1
200 400 600 x
Adult tickets
c. The solution of the system is the portion of the
graph x 1 y 5 15 for which 0 ≤ x ≤ 5. This means
that if you run for no more than 5 hours, you can
spend the remaining time swimming.
23
10.
y
3
3 x
1
0
c. Yes. If there are twice as many student
tickets sold, then 175 adult tickets are sold and
350 student tickets are sold, which is a solution
of the system.
20. a. x 1 y 5 15 and y ≥ 2x, where x is the
number of hours you run and y is the number of
hours you swim.
c. Answers will vary.
7.
600
500
400
300
200
100
21
21
1
x
3
Review for Mastery
1.
2.
y
y
3
3
11.
12.
y
20
y
3
12
23
4
24
1
23
1
12
x
21
1
21
21
1
3
x
1
3
x
27
25
21
21
1 x
3 x
23
13. x ≤ 21 and y > 5 14. y ≤ 2x and y < 21
3.
y
15. y ≤ 4 2 x and y ≥ 2x 16. y ≥ 1 2 x and x ≥ 0
7
17. x ≥ 0, y ≥ 0, and y ≤ x 1 2
5
18. x < 3, x > 21, and y < 2x 2 1
3
1
23
21
4. y > 3x 1 1; y ≤ x 1 2 5. y < 23; 2x 1 3y > 6
A8
Algebra 1
Chapter 7 Resource Book
Copyright © Holt McDougal. All rights reserved.
6
5
4
3
2
1
0
19. a. x 1 y ≤ 525 and 8x 1 5y ≥ 3000
b.
y
Student tickets
Packages of hamburgers
ANSWERS
19. a. x 1 y ≥ 5 and 1.9x 1 5.2y ≤ 20
b.
c. Answers will vary.
y
Lesson 7.6, continued
3. 0 ≤ 2x 1 y ≤ 160
Problem Solving Workshop:
Mixed Problem Solving
Chicken orders
b.
y
360
320
280
240
200
160
120
80
40
0
4.
Pounds of Premium Mix
system does not make sense because you do not
consider negative reading times.
2. a. x 1 y ≥ 260, y ≥ x, 6x 1 4y ≤ 1600
ANSWERS
0 ≤ x 1 y ≤ 88
0 ≤ x 1 2y ≤ 140
0 ≤ x ≤ 80
0 ≤ y ≤ 70
1. a. y 5 32x, y 5 28(x 2 0.25)
b. (21.75, 256) c. The solution of the linear
y
70
60
50
40
30
20
10
0
0
20
40
60
80 x
Pounds of Country Blend
5. 36 bags of Country Blend and 52 bags of Pre
ium Mix 6. $544
0
80
160 x
Salmon orders
c. Yes, 120 orders of salmon and 160 orders of
chicken can be ordered. 3. Answers will vary.
4. 24 5. No; Solving the linear system produces
Chapter Review Game
Row 1: 8, 1, 6; Row 2: 3, 5, 7; Row 3: 4, 9, 2
Each row, column, and diagonal has a sum of 15.
Copyright © Holt McDougal. All rights reserved.
Grocery Store
infinitely many solutions, so you need more
information.
6. a. x 1 y ≤ 25, and 9x 1 6y ≥ 120
y
30
25
20
15
10
5
0
x 1 y < 25
9x 1 6y > 120
0 5 10 15 20 25 30 x
Babysitting
b. No, you will earn $111. c. You can work
between 2 and 13 hours at the grocery store. 7.
Yes, just pick any value that is not equal to 22.
8. a. 6x 1 8y 5 94, and 12x 1 16y 5 188 b. No,
solving the linear system produces infinitely many
solutions. c. A large brick costs $9 and a small
brick costs $5.
Challenge Practice
1.
2.
y
y
3
5
1
3
23
21
21
1
3 x
1
23
21
1
3 x
23
Algebra 1
Chapter 7 Resource Book
A9
Answers
Review for Mastery
Lesson 8.1
1. 814 2. 64 3. y11 4. (210)8 5. 1330
1. power of a product property 2. product of
powers property 3. power of a power property
4. (z3)5 5 z3p5 5 z15 5. (5x)4 5 54x4 5 625x4
6. 3 p 3 5 3
3
1
311
4
53
2
3
7. (24y ) 5 (24)3( y2)3 5 264y6
8. (x2y4)3 5 (x2)3( y4)3 5 x6y12
9. x2(x3y)2 5 x2(x3)2y 2 5 x2x6y 2 5 x8y 2
10. 87 11. 56 12. 79 13. 220 14. 621
15. 418 16. 132 p 182 17. 215 p 255 18. 76 p 1546
19. x4 20. y8 21. z13 22. m28 23. b18
24. p15 25. 27n3 26. 32x5 27. x6y6
28. Wisconsin: 106; Nebraska: 105; New Jersey:
106; Oregon: 105 29. 103 mi2
30. 106 metric tons
10. 21331p3 11. 9x4y10 12. 8m23
Challenge Practice
1. a (x 1 9)/3 2. a12yb9y 3. xy1/2 4. x 8y 12
1
2
5. (x 1 2)5a 2 4 6. a3 cubic feet 7. } 8. }
4
3
9. (a 1 1)3 cubic feet 10. 210610 11. 220310
Lesson 8.2
Practice Level A
1. quotient of powers 2. power of a quotient
11. 486 p 276 12. 1355 p 85 13. x7 14. y8
38
3. quotient of powers 4. }5 5 3825 5 33
3
3 4
34
86
86
5. } 5 }4 6. }
5 }6 5 80 7. 44 8. 93
4
2
4
4
8 p8
8
57
15
9. 35 10. (25)1 11. (27)4 12. }5 13. }7
4
3
29
x3
3
6
1
4
14. }9 15. 4 16. y 17. z 18. m 19. }
7
y3
9
13
1
a
20. } 21. }9 22. a. 103 b. 10 c. 102 d. 102
z
b13
15. a18 16. z25 17. b14 18. (b 1 1)6
Practice Level B
19. 81x4 20. 281x4 21. 32a5b5 22. 64x18y 6
1. 66 2. 141 3. (25)5 4. 124 5. 87
(21)6
35
6. }5 7. }
8. 37 9. 48 10. y6 11. z9
4
56
216
243x20
a8
a12
12. }8 13. 2}
14. }
15. }
3
20
z
b
16b
y30
12
28
25
m
81x
32m
16. }
17. } 18. } 19. 103
125n27
16y 48
243n45
3087π
20. a. 104 b. 103 21. } in.3
2
Practice Level B
1. 512 2. (24)10 3. (210)7 4. 87 5. 210
6. 310 7. 921 8. 158 9. (24)45 10. 134 p 194
23. 81m31 24. 36p12 25. 6 26. 3 27. 6
28. 106 newspapers 29. a. 104 tons b. 1012 tons
c. 1018 tons 30. a. x4 square units
b. 625 square units c. 10,000 square units
Practice Level C
1. (29)14 2. 109 3. (27)8 4. 456 5. 1127
6. (26)18 7. 205 p 315 8. 1258 p 88
9. (216)6 p 266 10. x12 11. (c 1 5)18
12. 264c21 13. 264c21 14. 625x32y 20
15. 2100,000a35b5 16. 250p13 17. 640m34
18. 22304x21 19. 768n17 20. 3z20
21. 32,000c13 22. 4 23. 5 24. 3 25. Answers
will vary. 26. 105 computers; First find the
number of computers in use in Bahrain by finding
103 p 101. Then find the number of computers in
use in Australia by finding (103 p 101) p 101.
27. 104 metric tons; Solve the equation
10? p 102 5 106. 28. a. x6 cubic units
b. 15,625 cubic units c. 1,000,000 cubic units
A10
6. (28)21 7. f 16 8. (w 1 8)18 9. 56 p 186
Algebra 1
Chapter 8 Resource Book
1 2
Practice Level C
87
1. 155 2. 64 3. 2}7 4. 87 5. 510 6. 2105
9
81x24
64a6
a7
m42
7. 2}7 8. }
9. }
10. }9
36
60
b
y
64n
125b
49x6
27x11
8x5
20x14
}
}
}
11. }
12.
13.
14.
64y14
200y6
y15
27y13
17
8x
15. 2}
16. x 5 8, y 5 3; Use the properties
9y35
of exponents to write two equations in x and y.
Then solve the system of equations.
π
9π
9π
17. 102 18. 1012 19. } ft3; } ft3; } ft3
6
16
2
Copyright © Holt McDougal. All rights reserved.
ANSWERS
Practice Level A
Lesson 8.2, continued
Review for Mastery
Challenge Practice
1
1. a 5 1 2. b 5 3, or b 5 } 3. x 5 1, y 5 4
3
4. 4 5. 0 6. $65,155.79 7. $63,814.08
8. $265,329.77
Lesson 8.3
Practice Level A
Copyright © Holt McDougal. All rights reserved.
1
1
1
1
1. C 2. A 3. B 4. } 5. } 6. } 7. }
125
64
32
81
1
4
8. 2} 9. 1 10. 1 11. 1 12. 36 13. }
9
3
125
6
1
1
14. } 15. undefined 16. }5 17. }9 18. }3
8
x
m
y
y3
8
x3
1
1
19. }
20. }4 21. }2 22. }4 23. }2
a10
81b
y
x
ab
2y
3
1
1
24. } 25. } in. 26. } in.; } in.
4
4
2
x3
π
3π
27. } in.2; } in.2
16
16
Practice Level B
64
1
1
1
1. } 2. } 3. } 4. 1 5. 1 6. 1 7. }
243
1000
64
25
343
1
1
8. } 9. undefined 10. } 11. }
64
100,000
64
6
1
1
1
12. 625 13. }7 14. }4 15. }5 16. }
x
32b
y
81m4
12
5
2
3
x
d
a
17. } 18. }
19. }6 20. 1 21. }
2 5
3
4
xy
64y
c
b
x3y7
1
1
2 4
22. x y 23. } 24. } 25. a. } in.
3
20
4x6y5
4π
2
b. } in. 26. } cm3 27. a. 1026 m
25
375
b. 10215 m c. 1022 m
Practice Level C
1
1
1. } 2. 6561 3. } 4. 100,000 5. 125
243
625
x4y8
x6
1
1
6. } 7. 25 8. } 9. 210 10. }8 11. }
64
4
9
16y
11
y
y10
16
12. }4 13. }
14. 128d 8 15. 2}9
16x
6x
x10
A knitting needle narrows at one end.
Review for Mastery
8
1
1
1. 1 2. } 3. 36 4. } 5. 1 6. 81 7. }
625
125
16
n6p2
625x8z4
s4t
}
8. 100,000 9. }
10.
11. }
12
4
y
3m
48r11
ANSWERS
81
b7
1. 129 2. (28)4 3. 135 4. w 5 5. }7 6. }4
c
w
27s15
9m2
7. }
8. }
9. 10
12
t
n3
2y12
5x11
}
16. 81x2y5 17. }
18.
19. false; a 5 2;
y2
3x2
225
1
}
5 2 Þ }2 20. true 21. false; a 5 1, b 5 1;
226
1
1
} 5 } Þ 2 5 1 1 1 22. 106 23. a. 2
111
2
7π
7π
b. } cm3; } cm3 c. 4 d. overestimated;
2
8
Problem Solving Workshop:
Mixed Problem Solving
3375
1. a. } cubic inches
64
b. power of a quotient property
2. a.
Blood (cubic
millimeters)
Number of white
corpuscles
10
104
100
105
1000
106
10,000
107
100,000
108
b. 103 p 105 5 108 3. No, the mass of a sweet
corn seed is 0.1 gram.
4. a. Answers will vary. b. Answers will vary.
5. 2800 6. a. 1022 in. b. 1 cubic inch
c. Assuming the same thickness, the amount of oil
needed to cover a container of water with a surface
area of 10x square inches is 10x 2 2 cubic inches.
Challenge Practice
1. Always true 2. Never true 3. Never true
4. Always true 5. Sometimes true; true when
a 5 1 and b 5 1, false when a 5 2 and b 5 2.
9
6. True if a > 1. 7. 1 8. 1 9. 21 10. }
256
Focus On 8.3
Practice
1
1
1
1. 25 2. } 3. 32 4. } 5. } 6. 343
13
27
14
1
1
7. 5 8. } 9. 9 10. } 11. 24 12. 16
7
256
Algebra 1
Chapter 8 Resource Book
A11
Lesson 8.3, continued
1
13. 625 14. } 15. 6 16. 24 17. 9 18. 16
216
19. b3 5 a, definition of cube root; (ak)3 5 a,
1
property; 3k 5 1, set exponents equal; k 5 }3,
solve for k; a1/3 • a1 5 a4/3, substitute value of
k into equation; a1/3 ? a3/3 5 a4/3, find LCD for
exponents; a4/3 5 a4/3, sum of powers property
1
20. }
8
Review for Mastery
1
1
1. } 2. 27 3. } 4. 5 5.
12
512
1
8. } 9. 64 10. 3 11. 32
625
Practice Level C
1
6
12. 2
} 6. 81 7. 16
1. 1.5 3 1023 2. 3.04 3 104 3. 4.6 3 1026
4. 9.120006 3 106
5. 2.45 3 101
6. 1.256 3 1021 7. 7.05 3 102
8. 1.00456 3 105 9. 5.01 3 1027
Lesson 8.4
10. 132,500 11. 705,123,000
Practice Level A
12. 0.0000000815 13. 0.09044 14. 5100
1. C 2. A 3. B 4. 6.4 3 100 5. 8.52 3 101
6. 2.5 3 1021 7. 1.04 3 1021 8. 5.4 3 102
15. 31,112,000,000 16. 0.000081101
29. 7.8 3 1026; 0.0006; 0.0012; 2.15 3 102
17. 0.00000077 18. 62,500,000 19. 758.4;
7.208 3 103; 7.914 3 103; 72,164 20. 0.000526;
1.305 3 1023; 2.018 3 1023; 0.00205
21. 3.016 3 1024; 0.000316; 3.28 3 1024;
0.003028 22. 1.254 3 1022 23. 5 3 1024
24. 2.43 3 10243 25. about 82.48 people/km2
26. about 6.68 3 10224 g 27. a. Dione, Tethys,
Mimas, Phoebe, Calypso b. 275,000
c. Mimas: 8.25 3 1019 lb; Calypso: 8.8 3 1015 lb;
Tethys: 1.38 3 1021 lb; Dione: 2.42 3 1021 lb;
Phoebe: 8.8 3 1017 lb
30. 0.0125; 1.3 3 1022; 6.15 3 1021; 1.765
Review for Mastery
9. 9.1245 3 103 10. 9.5 3 1023 11. 6.3 3 105
12. 3 3 1022 13. 2.396 3 104 14. 4.57 3 1022
15. 4.5 3 1025 16. 52,000 17. 910,000,000
18. 625,000 19. 605 20. 8,125,000
21. 11,130,000,000 22. 0.0047
23. 0.000000016 24. 0.00000445
25. 0.000924 26. 0.0071123 27. 0.000020123
28. 4.5 3 103; 15,625; 21,000; 3 3 104
1
31. Oxygen: 9.75 3 10 lb; Chlorine: 0.3 lb;
24
Cobalt: 2.4 3 10 lb; Magnesium: 0.06 lb;
Sodium: 1.65 3 1021 lb; Hydrogen: 15 lb
32. about 28.5%
1. 1.04 3 101 2. 6.751 3 103 3. 5.4 3 1021
4. 1.03 3 1024 5. 4.1562 3 105
6. 8.104 3 1022 7. 3.412 3 106
8. 5.255 3 102 9. 1.0425 3 102
25
27
10. 4.56 3 10
12. 2.3551 3 10
11. 2.07 3 10
4
1. 7.9 3 1026 2. 1.356 3 106 3. 1012
4. 0.000037 5. 2.8 3 105; 361,000; 2.1 3 106
6. 4.0 3 1029 7. 2.093 3 103 8. 8.41 3 1012
Challenge Practice
Practice Level B
13. 158,000 14. 321,000,000
15. 4,502,100,000 16. 810,450 17. 17,220,000
18. 101.2 19. 0.000812 20. 0.0000004014
21. 0.0081025 22. 0.00000000312056
23. 0.01211 24. 0.0000700135
A12
3.0214 3 104
26. 1.04 3 1023; 2.5 3 1023; 0.0985; 0.16
27. 8.79 3 102; 1.0085 3 103; 1023; 1146
28. 1.2 3 1025; 0.001023; 1.045 3 1023;
0.01036 29. 3 3 107 30. 5 3 1024
31. 3.2 3 10224 32. 5.4 3 105 pixels
33. about 13.57 people/km2 34. a. Titania,
Oberon, Ariel, Umbriel, Miranda b. about 53
Algebra 1
Chapter 8 Resource Book
1. 1.44 3 102 2. 5 3 1010 3. 3.5 3 106
4. 4.24 3 100 5. 1 3 106 6. 5% 7. 2%
8. about 2.4% 9. 1 10. 20.00000005
Lesson 8.5
Practice Level A
1. y 5 3x 2. y 5 5x 3. C 4. B 5. A
Copyright © Holt McDougal. All rights reserved.
ANSWERS
substitute ak for b; a3k 5 a1, product of powers
25. 9.287 3 103; 1.3759 3 104; 14,205;
Lesson 8.5, continued
6.
domain: all real numbers;
range: all positive real
numbers
y
5
3
domain: all real numbers;
range: all positive real
numbers
y
13.
5
3
21
21
1
3
domain: all real numbers;
range: all positive real
numbers
y
7.
23
x
10
6
21
21
1
3
x
domain: all real numbers;
range: all positive real
numbers
y
14.
5
3
ANSWERS
1
23
2
23
21
22
8.
1
3
23
x
domain: all real numbers;
range: all positive real
numbers
y
5
3
21
21
1
3
x
y
15.
y
16.
3
3
1
23
1
21
3
1
x
23
1
23
21
21
9.
1
3
3
3
x
23
domain: all real numbers;
range: all positive real
numbers
5
1
23
x
y
21
21
reflection in x-axis
y
17.
vertical stretch
vertical shrink
3
1
1
23
23
21
21
1
3
x
21
21
1
3
x
23
10.
domain: all real numbers;
range: all positive real
numbers
y
Copyright © Holt McDougal. All rights reserved.
5
3
23
21
21
1
3
x
domain: all real numbers;
range: all positive real
numbers
y
11.
5
3
18. initial amount: 3; growth rate: 0.05;
growth factor: 1.05 19. initial amount: 2;
growth rate: 0.25; growth factor: 1.25
20. initial amount: 0.1; growth rate: 0.75;
growth factor: 1.75 21. a. $206 b. $212.18
c. $231.85 22. Freshmen: 2; Sophomore: 2.5;
Junior: 3.125; Senior: about 3.906
Practice Level B
1. y 5 11x 2. y 5 0.25(2) x
y
3.
23
21
21
1
3
domain: all real numbers;
range: all positive real
numbers
10
x
6
12.
domain: all real numbers;
range: all positive real
numbers
y
5
3
2
23
21
22
21
21
1
3
3
x
domain: all real numbers;
range: all positive real
numbers
y
4.
23
1
3
x
23
21
21
1
3
x
23
Algebra 1
Chapter 8 Resource Book
A13
Lesson 8.5, continued
5.
domain: all real numbers;
range: all positive real
numbers
y
3
ANSWERS
23
21
21
10
3
21
22
14.
x
25
x
reflection in x-axis
15.
y
5
y
3
23
21
23
1
3
x
29
x
3
1
1
23
y
1
23
3
3
21
21
7.
3
1
vertical stretch
1
23
1
21
23
domain: all real numbers;
range: all positive real
numbers
y
23
2
23
5
y
1
6
23
6.
13.
y
x
3
1
12.
x
3
1
21
domain: all real numbers;
range: all negative real
numbers
21
21
3
1
215
x
vertical shrink
16.
vertical stretch and
reflection in x-axis
17.
y
1
23
23
25
y
2
23
21
21
3
1
x
21
22
1
3
x
26
1
x
3
1
domain: all real numbers;
range: all negative real
numbers
23
25
9.
domain: all real numbers;
range: all positive real
numbers
y
10
6
210
25
vertical shrink and
vertical stretch and
reflection in x-axis
reflection in x-axis
18. a. $512.50 b. $565.70 c. $819.31
19. y 5 8000(1.07)t 20. a. y 5 10,000(1.08)t
b. $19,990.05
Practice Level C
1. y 5 24x 2. y 5 5 p 2x
3.
2
23
23
21
22
15
3 x
1
domain: all real numbers;
range: all positive real
numbers
y
9
10.
domain: all real numbers;
range: all positive real
numbers
y
5
3
11.
21
21
1
3
1
y
1
3
23
x
3
x
domain: all real numbers;
range: all positive real
numbers
y
3
x
1
21
21
21
23
4.
1
23
3
23
domain: all real numbers;
range: all negative real
numbers
23
21
21
3 x
1
23
5.
domain: all real numbers;
range: all positive real
numbers
y
5
25
3
1
23
A14
Algebra 1
Chapter 8 Resource Book
21
21
1
3
x
Copyright © Holt McDougal. All rights reserved.
8.
y
Lesson 8.5, continued
6.
domain: all real numbers;
range: all positive real
numbers
y
3
21
21
1
3
3
21
21
3
1
x
23
21
21
domain: all real numbers;
range: all negative real
numbers
16.
domain: all real numbers;
range: all negative real
numbers
y
1
23
1
3
x
23
9.
domain: all real numbers;
range: all positive real
numbers
y
15
9
x
vertical stretch and
reflection in x-axis
y
17.
y
3
1
21
21
1
23
x
3
21
1
3
x
23
23
vertical shrink and
vertical shrink and
reflection in x-axis
reflection in x-axis
18. Subtract the amount deposited from the
balance. a. $10.31 b. $54.48 c. $270.16
19. a. y 5 65,000(1.025)t b. about 71,748 people
20. a. 100% b. y 5 10(2)t c. 160 students
Review for Mastery
1. y 5 9 p 3x
3
2.
21
23
domain: all real
numbers; range: all
positive real numbers
y
3 x
1
7
domain: all real numbers;
range: all positive real
numbers
y
10
6
Copyright © Holt McDougal. All rights reserved.
3
215
3
23
3
10.
1
23
29
x
3
1
27
23
1
vertical shrink
25
8.
23
23
y
23
3
1
23
x
23
7.
y
15.
y
ANSWERS
23
14.
5
y 5 4(3)x
1
2
23
21
22
1
3
23
x
21
3.
11.
2
23
22
1
3 x
domain: all real numbers;
range: all negative real
numbers
26
3
x
1
3
x
y
23
y
1
y5
25(6)x
25
27
210
12.
13.
y
10
y
1
6
23
21
1
3 x
2
23
21
22
1
3 x
23
25
vertical stretch
reflection in x-axis
Because the y-values for y 5 25 p 6x are
25 times the corresponding y-values for y 5 6x,
the graph of y 5 25 p 6 x is a vertical stretch and
reflection in the x-axis of the graph of y 5 6x.
Problem Solving Workshop:
Worked Out Example
1. $389.78 2. The value raised to the x power
should have been 1 1 0.36; and the final calculation
of 0.10 is also incorrect. The spending per person
per year on the Internet in 2007 is $389.78.
Algebra 1
Chapter 8 Resource Book
A15
Lesson 8.5, continued
3. y 5 179,323,175(1.011)x; 309,880,465
5
4. 7.59 feet 5. 16.41 feet
3
ANSWERS
Challenge Practice
1
1. y 5 3x 2. y 5 3 p 2x 3. y 5 } p 5x
2
3
1 x
x
2
2
4. y 5 } 3 or y 5 3
5. y = } 2x or
9
2
1 2
1 2
y53p2
6. f (x) 5 3 p 28x and g(x) 5 3 p 212x, so
g(1) > f (1)
x21
23
so g(1) > f (1)
8. f (x) 5 25 p 52x and g(x) 5 52x, so f (1) > g(1)
21
21
11.
23
21
21
12.
1
3
x
13.
y
y
3
5
1
23
21
21
3
x
1
23
Lesson 8.6
21
21
1
3
x
vertical stretch
14.
reflection in x-axis
vertical shrink
y
3
domain: all real numbers;
range: all positive real
numbers
5
domain: all real numbers;
range: all positive real
numbers
1
g(x) 5 2000 p (1.5) , so f (1) > g(1)
y
x
3
3x
1 2
3
y
1
9. f (x) 5 6 p 42x and g(x) 5 } p 43x, so
2
f(1) > g(1) 10. f (x) 5 1000 p (1.5)10x and
Practice Level A
1 x
1. yes; y 5 }
2. no 3. C 4. A 5. B
10
1
5
1
7. f (x) 5 } p 16x and g(x) 5 1280 p 16x,
2
6.
domain: all real numbers;
range: all positive real
numbers
y
10.
1
23
21
21
3 x
1
23
1
15. exponential decay 16. exponential growth
21
21
7.
1
3
x
17. exponential decay 18. exponential decay
domain: all real numbers;
range: all positive real
numbers
y
5
3
21
21
1
3
x
8.
5
3
22. a. y 5 4000(0.98)t b. 3689 employees
1 2
domain: all real numbers;
range: all positive real
numbers
y
21. a. $10,200 b. $7369.50 c. $5324.46
Practice Level B
1 x
1. yes; y 5 }
2. no
5
1
23
19. exponential growth 20. exponential growth
domain: all real numbers;
range: all positive real
numbers
y
3.
10
6
1
23
21
21
1
3
domain: all real numbers;
range: all positive real
numbers
y
9.
5
3
1
23
A16
21
21
23
x
21
22
4.
3
Algebra 1
Chapter 8 Resource Book
x
x
domain: all real numbers;
range: all positive real
numbers
y
5
3
26
1
3
1
22
21
2
6 x
Copyright © Holt McDougal. All rights reserved.
23
Lesson 8.6, continued
5.
y
2
23
21
1
3
x
domain: all real numbers;
range: all negative real
numbers
17. exponential growth; y 5 4(2)x
18. a. $2400 b. $1536 c. $983.04
19. a. y 5 7(0.979)t b. about 5.4%
6.
domain: all real numbers;
range: all positive real
numbers
y
10
6
Practice Level C
9 x
1. yes; y 5 }
2. no
10
1 2
3.
domain: all real numbers;
range: all positive real
numbers
y
15
2
23
21
22
7.
1
3
9
x
domain: all real numbers;
range: all positive real
numbers
y
7
5
23
21
23
3 x
1
4.
domain: all real numbers;
range: all positive real
numbers
y
3
1
1
23
21
1
8.
3
x
y
0.5
23
21
20.5
3
x
23
domain: all real numbers;
range: all negative real
numbers
21
21
3 x
1
23
5.
domain: all real numbers;
range: all negative real
numbers
y
3
1
23
9.
10.
y
20
ANSWERS
20. a. y 5 18,000(0.945)t
b. about 13,565 people
21
21
3
x
y
2
Copyright © Holt McDougal. All rights reserved.
23
23
21
24
1
3
1
3
x
reflection in x-axis
vertical shrink
y
5
21
21
1
3
domain: all real numbers;
range: all positive real
numbers
y
23
21
26
7.
3
23
6.
x
vertical stretch
11.
21
1
3
x
domain: all real numbers;
range: all positive real
numbers
y
x
2
23
12. never; The graphs are reflections in the x-axis.
13. always; At x 5 0, the graphs will intersect at
(0, 1).
14. sometimes; Sample answer: If a 5 1, then the
graphs are identical. If a 5 2, then the graphs are
not identical.
15. exponential decay; y 5 3(0.75)x
16. exponential decay; y 5 2(0.7)x
8.
21
22
1
3
x
y
0.1
23
21
20.1
3
x
domain: all real numbers;
range: all negative real
numbers
Algebra 1
Chapter 8 Resource Book
A17
Lesson 8.6, continued
Problem Solving Workshop:
Mixed Problem Solving
y
1
23
21
21
3
x
ANSWERS
5
23
21
25
1
x
3
vertical stretch
11.
reflection in x-axis
vertical shrink and
reflection in x-axis
y
3
1
23
21
1
x
3
23
12. always; The graphs are reflections in the
x-axis.
13. always; The graphs are reflections in the
y-axis, so they have the same range.
14. never; The function is an exponential decay
function.
15. exponential decay; y 5 5(0.4)x
16. exponential growth; y 5 3(1.25)x
17. exponential decay; y 5 6(0.8)x 18. a. $2700
b. $6729.48 c. $9438.90 19. a. y 5 8(0.982)t
b. about 7.6 h 20. no; At the beginning of the
second 5-year period, there was more money being
lost than there was being gained at the beginning
of the first 5-year period.
Review for Mastery
1. yes; y 5 9 p 3
2.
x
domain: all real
numbers;
range: all positive real
numbers
y
7
5
3
y 5 (0.7)x
23
21
1
3.
area of Jupiter is about 6.423 3 1010 km2. The
surface area of Callisto is about 7.238 3 107 km2.
c. about 8.874 3 102, The surface area of Jupiter
is about 887.4 times larger than the surface area of
Callisto.
2. a. y 5 20(0.5)x, where x is the number of
45-day periods b. 1.25 ounces 3. a. exponential
growth b. y 5 91(1.59)x c. 365.79 million
4. Yes; After two years of depreciation, the value
of the boat is $5057.50. The family is getting more
for the boat than it is worth. 5. 0.14 6. Answers
will vary.
7. a. y 5 20,000(0.94)x b. 6%; The decay rate
for the car is 0.06, or 6%.
8. a. y 5 200(1.04)x
b.
y
250
200
150
100
50
0
0 1 2 3 4 5 6 7 8 9 x
Time (years)
c. No; After 3 years there is $224.97 in the
account.
Challenge Practice
1 x
1 x
1. f (x) 5 3 p }
2. f (x) 5 2 p }
2
3
5
3 x
2 x
3. f (x) 5 4 p }
4. f (x) 5 } p }
2
5
5
x
7
3
1 5x
5. f (x) 5 } p }
6. f (x) 5 3 p } and
3
7
9
1 3x
g(x) 5 4 p }9 , so g(1) > f (1)
5
1 x
1 x
7. f (x) 5 8 p } and g(x) 5 } p } , so
256
16
16
1
1 x
1 4x
f (1) > g(1) 8. f (x) 5 }5 p }5 and g(x) 5 }5 ,
9 x
and
so f (1) > g(1) 9. f (x) 5 6 p }
16
1
9 x
, so f (1) > g(1)
g(x) 5 }2 p }
16
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
x
3
1. a. 7.1492 3 104, 2.4 3 103 b. The surface
1 2
1 2
y
1 2
3
y54
1 2
x
( 18 )
1
23
21
1
3
x
1 x
exponential decay; y 5 4 p 1 }8 2
A18
Algebra 1
Chapter 8 Resource Book
1 2
Copyright © Holt McDougal. All rights reserved.
10.
y
25
Value (dollars)
9.
Lesson 8.6, continued
1 n21
9. an 5 281 }
;
3
1 2
Focus On 8.6
Y
10. an 5 6n 2 1;
/
.UMBER OF .EW 0EA 0LANTS A
N
X
Y
/
3. arithmetic;
X
2. geometric;
ANSWERS
D: 1, 2, 3, ...
R: 281, 227,
29, 23, ...
Practice
1. arithmetic;
Y
/
Y
X
Y
/
X
.UMBER OF 'ENERATIONS N
Review for Mastery
1. geometric; 2500 2. arithmetic; 3.5
X
3. geometric; 2
4.
1 n21
4. an 5 64 }
;
4
1 2
5. an 5 (26)
;
6.
X
Y
D: 1, 2, 3, ...
R: 1, –6, 36, 2216, ...
Copyright © Holt McDougal. All rights reserved.
X
X
X
X
Y
/
Y
/
n21
Y
D: 1, 2, 3, ...
R: 64, 16, 4, 1, ...
5.
Y
n21
6. an 5 3(2)
;
7. an 5 (22)n 2 1; a10 5 2512
1 n21
1
8. an 5 }
; a10 5 }
3
19683
Y
1 2
D: 1, 2, 3, ...
R: 3, 6, 12, 24, ...
9. an 5 10(2)n 2 1; a10 5 5120
Chapter Review Game
/
1 n21
1 2
7. an 5 }
4
;
1
R: 1, }4 , }
, }, ...
16 64
/
1
8. an 5 21 ? 2}
2
n21
1 2
D: 1, 2, 3, ...
1 1
1
R: 21, }2, 2}4 , }8 , ...
x6
1. x 8 2. 3.10091 3 107 3. 1 4. }3 5. 0.891
y
10
7
6. 2324x
7. 0.0000987 8. 2x y
y 14
500
9. 3.0 3 1025 10. }6 11. 2}
12. 0.055
x 4y 6
16x
13. 1.495 3 1011
Y
D: 1, 2, 3, ...
1 1
X
;
X
RENE DESCARTES
Y
X
/
Algebra 1
Chapter 8 Resource Book
A19
Answers
Lesson 9.1
11. 24a3b2 1 15a2b2 2 10a2b 1 5
Practice Level A
12. 3m2n 2 11mn2 2 8n 1 2m
2. 29z 1 1; degree: 1; leading coefficient: 29
3. 2x 5 1 4; degree: 5; leading coefficient: 2
4. 2x 2 1 18x 1 2; degree: 2; leading coefficient:
21 5. 3y 3 1 4y 2 1 8; degree: 3; leading
coefficient: 3 6. 220m3 1 m 1 5; degree: 3;
leading coefficient: 220 7. 23a7 1 10a 4 2 8;
degree: 7; leading coefficient: 23
8. 6z 4 1 z 3 2 5z 2 1 4z; degree: 4; leading
coefficient: 6 9. h7 2 6h4 1 8h3; degree: 7;
leading coefficient: 1 10. polynomial; degree: 2;
monomial 11. not a polynomial; variable
exponent 12. not a polynomial; negative
exponent 13. polynomial; degree: 2; binomial
14. polynomial; degree: 2; trinomial
15. polynomial: degree: 3; binomial 16. 7x 1 9
17. 7m2 2 7 18. 9y 2 1 5y 2 4 19. 2x 2 1 3
20. 7a2 1 2a 2 6 21. 2m2 2 8m 1 3 22. 4x 1 4
23. 4x 1 9 24. B 5 0.014t 2 1 0.13t 1 12
25. Area: 4x 2 2 12πx 1 6π
Practice Level B
1. 4n5; degree: 5; leading coefficient: 4
2. 22x 2 1 4x 1 3; degree: 2; leading
coefficient: 22 3. 4y 4 1 6y 3 2 2y 2 2 5; degree:
4; leading coefficient: 4 4. not a polynomial;
variable exponent 5. polynomial; degree: 3;
trinomial 6. not a polynomial; negative exponent
7. 5z 2 1 3z 2 7 8. 5c 2 2 3c 1 6
9. 3x 2 1 6 10. 6b2 2 8b 1 1
11. 24m2 1 2m 2 3 12. 22m2 1 9m 2 1
13. 10x 1 2 14. 9x 2 1
17
15. Area: } x 2 1 8x 2 32
4
1 2
16. P 5 } t 1 2t 1 200
6
13. a. T 5 4.93t 4 2 56.78t 3 1 177.65t 2 2
126.42t 1 1367.51 b. In 1997, 1367.51 thousand
metric tons were produced and in 2003, 1129.19
thousand metric tons were produced. So more peat
and perlite were produced in 1997.
14. a. N 5 187,443 1 13,857t;
M 5 151,629 1 5457t b. 1997: $35,814; 2003:
$86,214; Northeast: $83,142; Midwest: $32,742
Review for Mastery
1. 22x2 1 9; degree: 2; coefficient: 22
2. 3y3 1 2y 1 16; degree: 3; coefficient: 3
3. 23z5 1 6z3 1 7z2; degree: 5; coefficient: 23
4. 9a2 1 4a 1 4 5. 13b2 2 2b 1 5
6. 22c3 1 5c2 2 5c 1 4 7. 13d 2 2 23d 1 11
Problem Solving Workshop:
Worked Out Example
1. 22,055,300 people 2. $1,115,940
Challenge Practice
1. x 1 x 1 4 5 2x 14 5 2(x 1 2); Because the
number of quarters and dimes is a multiple of 2,
it is even. 2. x 1 2x 1 1 5 3x 11; If x is even,
then 3x is even and 3x 1 1 is odd. If x is odd, then
3x is odd and 3x 1 1 is even. So, whether the
total number of coins is even or odd can’t be
determined. 3. x 1 3x 1 5 5 4x 1 5; Whether x
is even or odd, 4x is even, so 4x 1 5 is odd.
4. x 1 4 1 3x 1 5 5 4x 1 9; Whether x is even
or odd, 4x is even, so 4x 1 9 is odd.
5. x 1 4 1 2x 1 1 1 3x 1 5 5 6x 1 10 5
2(x 1 5); Because the number of dimes, nickels,
and pennies is a multiple of 2, it is even.
81
6. 0 7. 1 8. x 9. 3 10. } 11. 25 12. 19
4
Lesson 9.2
Practice Level C
1. polynomial; degree: 0; monomial 2. not a
polynomial; negative exponent 3. polynomial;
degree: 2; trinomial
4. 3m3 1 4m2 2 m 1 2 5. 25y 2 2 2y 1 9
6. c3 1 c 2 2 9c 1 5 7. 24z 2 1 4z 1 14
8. 14x 4 2 3x3 2 7x 2 2 3
9. 2x 4 2 2x 3 1 6x 2 2 5x
10. f(x) 1 g(x) 5 6x 3 2 3x 2 1 2x 2 6;
f(x) 2 g(x) 5 26x 3 2 7x 2 1 2x 1 4
A20
Algebra 1
Chapter 9 Resource Book
Practice Level A
1. 3x 3 2 2x 2 1 x 2. 6y 4 1 2y 3 2 8y
3. 23m3 2 12m2 1 3m 4. 4d 4 2 3d 3 1 d 2
5. 2w 5 2 3w 4 6. 2a4 2 3a3 1 a2
7. x 2 2 3x 2 4 8. y 2 1 8y 1 12
9. a2 2 8a 1 15 10. 2m2 1 7m 1 3
11. 3z 2 2 11z 2 20 12. 3d 2 1 17d 2 6
13. y 2 1 5y 2 24 14. n2 1 11n 1 30
Copyright © Holt McDougal. All rights reserved.
ANSWERS
1. 8n6; degree: 6; leading coefficient: 8
Lesson 9.2, continued
15. 5x 3y 2 20x 2y 2 1 5xy 3 16. 33n2 1 36n 1 3
17. w 3 1 3w 2 1 3w 1 1
17. w6 1 13w 5 1 3w 4 2 10w 3 1 5w 2
7
1
18. } x 2 1 } x 1 6 19. 22x 2 2 2x 1 96
2
2
20. a. A 5 2330.6934t 5 1 14,967.1039t 4 2
18. m3 2 4m2 1 7m 2 6 19. 8y 2 2 23y 2 3
20. 15b2 1 7b 2 2 21. 6d 2 2 14d 1 4
22. 6x 2 1 8x 1 2 23. 6x 2 1 22x 2 8
2
2
24. 2s 1 s 2 15 25. 40c 2 46c 2 14
26. 16p2 2 46p 1 15 27. 14t 2 1 26t 2 4
28. a. V 5 288x 2 1 1152x 1 1152 b. 41,472 in.3
29. a. A: 76,226; P: 0.6; A p P indicates the
number of acres (in thousands) that are parks.
b. A p P 5 20.1688t 3 2 59.0818t 2 1 812.634t 1
45,735.6
Practice Level B
1. 6x 4 2 3x 3 2 x 2 2. 220a7 1 15a4 2 5a3
3. 28d 5 1 20d 4 2 24d 3 1 8d 2
Review for Mastery
1. 21x4 2 6x3 1 9x2
2. 12x8 2 8x7 2 32x6 1 36x5
4. 6x 2 2 13x 2 5 5. 2y 2 2 7y 2 15
6. 24a2 2 18a 1 3 7. 5b2 2 42b 1 16
8. 16m2 1 38m 1 21 9. 23p3 1 6p2 2 p 1 2
10. 22z2 1 13z 2 21 11. 26d 2 1 23d 2 10
12. n3 1 5n2 1 9n 1 5 13. w 3 1 5w2 2 23w 2 3
3. 3m3 1 17m2 1 6m 2 4 4. 6n2 1 29n 1 28
5. 2p3 1 13p2 2 p 1 42
6. 12q3 2 28q2 1 7q 1 12 7. 15t 2 2 13t 2 72
8. 72s2 2 119s 1 49 9. 2y21 15y 2 27
Challenge Practice
14. 2s3 1 11s2 1 13s 2 5
15. 5x 3y 2 20x 2y 2 1 5xy 3 16. 4a2 1 a 2 1
17. 23x 2 1 8x 1 10 18. 2m2 1 5m 2 41
19. 3x 2 1 15x 20. x 2 1 6x 1 8
1. x7 1 3x5 1 2x3 2. 2y7 1 3y5 2 y4 1 3y2
3. 2x7 1 4x3y3 1 2x4y 1 4y4
4. 2x12 1 11x10 1 12x8
5. x5 1 2x4 1 3x3 1 6x 2 1 2x 1 4
21. a. A 5 4x 2 1 22x 1 30 b. 72 ft 2
Copyright © Holt McDougal. All rights reserved.
149,699.734t 3 1 178,230.4684t 2 1 18,574.268t 1
106,563,461.4 b. $106,563,461,400
21. a. E: 14,439.09; P: 0.126; E p P indicates the
amount of money spent (in millions of dollars) on
exercise equipment.
b. E p P 5 0.0001112t 8 2 0.0002186t7 2
0.06424t 6 1 0.983634t 5 2 6.7188068t 4 1
22.667885t 3 2 120.819698t 2 1 568.42959t 1
1819.32534 c. $1,819,325,340
7. 4x 8. 4x
22. a. S: 66,939; P: 0.4; S p P
3
2
Practice Level C
Practice Level A
1. 216y 1 40y 2 24y
3
3
2
2. 3b 1 7b 2 5b 1 3
3. 218w 2 1 33w 2 12
6. 0
2
9. 2x 2 x 2 6x 1 1
10. 2x 1 8x 1 5
5
3
2
indicates the number of students (in thousands)
that were between 7 and 13 in 1995.
b. A p P 5 0.000163t7 2 0.01166225t 6 1
0.218856t 5 2 1.510115t 4 1 0.46605t 3 1
38.8676t 2 1 181.107t 1 26,775.6
c. about 26,775,600 students
7
ANSWERS
15. 3x 2 1 13x 2 10 16. 8a2 2 2a 2 1
11. V 5 9x(50x 1 150) (8x 1 16)
12. V 5 3600x3 1 18,000x 2 1 21,600x
13. 168 trailers
Lesson 9.3
1. 2ab
2. 2mn 3. 2x 4. 10x 5. y 2 6. 9
7. C 8. A 9. B 10. x 2 1 8x 1 16
4. 36m5 2 9m3 1 4m2 2 1
11. m2 2 16m 1 64 12. a2 1 20a 1 100
5. 2x 3 1 11x 2 1 13x 2 6
13. p2 2 24p 1 144 14. 4y 2 1 4y 1 1
6. 24n4 2 32n3 1 37n2 1 4n 2 5
15. 9y 2 2 6y 1 1 16. 100r 2 2 20r 1 1
7. 6p6 2 12p4 2 10p2 1 20
17. 16n2 1 16n 1 4 18. 9c 2 2 12c 1 4
8. 248r 5 1 8r 3 1 12r 2 2 2 9. 10z 4 2 39z 2 2 27
19. z 2 2 25 20. b2 2 4 21. n2 2 64
10. x 3y 1 2xy 2 11. 26x 2y 2 15xy
22. a2 2 100 23. 4x 2 2 1 24. 25m2 2 1
12. x 2y 3 1 xy 4 13. 5x 2 1 xy 2 6y 2
25. 16d 2 2 1 26. 9p2 2 4 27. 4r 2 2 9
14. 2xy 3 1 3x 2y 2 1 210x 1 140y
28. Find the product (10 2 3)(10 1 3).
Algebra 1
Chapter 9 Resource Book
A21
ANSWERS
Lesson 9.3, continued
29. Find the product (30 2 6)(30 1 6).
23. 324x 2 24. 1 25. (x 2 12)(x 1 12)
30. Find the product (60 1 9)(60 2 9).
26. (a 2 b)3 5 (a 2 b)2(a 2 b) 5
31. T 5 9t 2 2 4 32. a. 0.25B2 1 0.5Bb 1 0.25b2
b. 25%
(a2 2 2ab 1 b2)(a 2 b) 5 a3 2 3a2b 1 3ab2 2 b3
Practice Level B
model represent at least one goal being made.
b. The chance of making a goal is 50% and
the chance of not making a goal is 50%.
So the polynomial (0.5C 1 0.5I)2 5
0.25C 2 1 0.5CI 1 0.25I 2 represents this situation
where C represents a goal made and I represents
a goal missed. 28. The expression 8(122)
represents the original volume. If the side
lengths are changed as described, the expression
8(12 2 x)(12 1 x) 5 8(122) 2 8x 2 represents the
new volume. Because x is positive, subtracting 8x 2
will always decrease the original volume.
1. x 2 2 18x 1 81 2. m2 1 22m 1 121
3. 25s2 1 20s 1 4 4. 9m2 1 42m 1 49
5. 16p2 2 40p 1 25 6. 49a2 2 84a 1 36
7. 100z 2 2 60z 1 9 8. 4x 2 1 4xy 1 y 2
9. 9y 2 2 6xy 1 x 2 10. a 2 2 81
11. z 2 2 400 12. 25r 2 2 1 13. 36m2 2 100
14. 49p2 2 4 15. 81c 2 2 1 16. 16x 2 2 9
17. 2w 2 1 16 18. 24y 2 1 25 19. Find the
product (20 2 5)(20 1 5). 20. Find the product
(50 2 7)(50 1 7). 21. Find the product (20 2 2)2.
22. 16x 2 2 0.25 23. 16x 2 1 4x 1 0.25
2
27. a. 75%; Three of the four squares in the area
Review for Mastery
1. y2 1 18y 1 81 2. 9z2 1 42z 1 49
24. 16x 2 4x 1 0.25
3. 4w2 2 12w 1 9 4. 100r 2 2 60rs 1 9s2
25. a.
5. g2 2 121 6. 49f 2 2 1 7. 4h2 2 81
S
s
S
SS
Ss
s
sS
ss
8. 36k 2 2 64 9. Square of a binomial pattern;
(50 1 5)2 10. Sum and difference pattern;
(40 2 9)(40 1 9)
1. 8x2 1 18 2. 2x 4 1 2x 2 1 5
26. a. 75%; Three of the four squares in the area
3. 2a2x 2 1 2b2y2 4. 2a2x 4 1 2b2y 4 5. 34x 1 50
model represent at least one foul shot being made.
b. The chance of making a foul shot is 50% and
the chance of not making a foul shot is 50%.
So the polynomial (0.5C 1 0.5I)2 5 0.25C 2 1
0.5CI 1 0.25I 2 represents this situation where C
represents a foul shot made and I represents a foul
shot missed.
6. (a 2 b 1 c)2
Practice Level C
1. 64x 2 2 80x 1 25 2. 16p2 1 32p 1 16
3. 100 m2 2 220m 1 121 4. 121s2 2 220s 1 100
5. 400b2 2 600b 1 225 6. m2 1 8mn 1 16n2
7. r 2 2 16rs 1 64s2 8. 100a2 1 60ab 1 9b2
9. 4x 2 2 16xy 1 16y 2 10. 64p2 2 9
11. 121t 2 2 16 12. 49n2 2 25 13. 81z 2 2 144
14. 2w 2 1 225 15. 225p2 1 36
16. 29m2 1 400 17. 100a2 2 25b2
18. 16x 2 2 9y 2 19. Find the product (40 2 4)
(40 1 4). 20. Find the product (20 1 3)2.
21. Find the product (50 2 1)2. 22. 81x 2 2 0.25
A22
Algebra 1
Chapter 9 Resource Book
5 a(a 2 b 1 c) 2 b(a 2 b 1 c) 1 c(a 2 b 1 c)
5 a2 2 ab 1 ac 2 ab 1 b2 2 bc 1 ac 2 bc 1 c2
5 a2 1 b2 1 c2 2 2ab 1 2ac 2 2bc
7. 9x 2 1 4y 2 1 25z 2 2 12xy 1 30xz 2 20yz
8. a2x 2 1 b2y 2 1 c 2z 2 2 2abxy 1 2acxz 2 2bcyz
9. 8x3 1 24x 2 1 16x 10. Because
8x 3 1 24x 2 1 16x 5 2(4x3 1 12x2 1 8x), the
expression represents an even number.
11. 8x3 1 36x2 1 46x 1 15
12. Because 8x 3 1 36x 2 1 46x 1 15 5
2(4x 3 1 18x2 1 23x) 1 15, the expression
represents the sum of an even number and an
odd number, which gives an odd number.
Lesson 9.4
Practice Level A
1. B 2. A 3. C 4. 26, 22 5. 23, 5 6. 7, 10
7. 21, 8 8. 29, 9 9. 215, 212 10. 250, 25
Copyright © Holt McDougal. All rights reserved.
Challenge Practice
b. 0.25S 2 1 0.5Ss 1 0.25s2 c. 25%
Lesson 9.4, continued
11. 23, 1 12. 2, 3 13. 2(2m 2 1) 14. 5(x 2 2)
15. 3(2y 1 5) 16. 8(x 1 y) 17. 7(a 2 b)
b. Check student’s work; 40 ft
c. Check student’s work; (0, 60)
18. 2(a 1 5b) 19. 9(m 2 2n) 20. 3(5p 2 q)
Review for Mastery
21. 4(3x 1 y) 22. 2c(c 1 2) 23. m (9m 1 1)
1. 7, 9 2. 22, 23 3. 0, 216 4. 0, 2 5. 0, 3
24. 2w(w 1 2) 25. C 26. B 27. A 28. 28, 0
6. 0.625 sec
29. 0, 7 30. 21, 0 31. 0, 1 32. 0, 2
33. 22, 0 34. 9 sec 35. about 0.33 sec
1. a. 4x 2 1 24x 1 35 b. 99 square inches
Practice Level B
1. 214, 3 2. 25, 12 3. 224, 215 4. 8, 9
3
1
5. 28, } 6. 2} , 6 7. 25, 4 8. 22, 3
4
2
5 5
1 1
2
1
9. } , 8 10. 2}, } 11. 23, } 12. 2} , }
2 2
4 4
3
2
2. a. E 5 4.3791t 2 1 235.3518t 1 2944.308
b. $3,955,780,800 3. a– c. Answers will vary.
4. 2 5. 0, 1.125; The kangaroo jumped off
22. 0, 10 23. 214, 0 24. 0, 1 25. 21, 0
5
5
26. 0, 3 27. 22, 0 28. 0, } 29. 2} , 0
4
2
3
1
5
1
}
}
}
30. 0,
31. 0,
32. 2 , 0 33. 2}, 0
2
8
2
2
34. 1.5 sec; Yes. From the equation, you can see
the ground at 0 seconds and landed back on the
ground at 1.125 seconds. 6. Brian was in the air
longer during his first jump since he had a larger
initial velocity, which means that he landed on the
ground in the first jump later than he landed on the
second jump.
7. a. S 5 144.3t3 2 841.1t2 1 520.5t 1 6559.3
b. $6,559,300,000, $4,418,900,000 c. a loss of
$535,100,000 per year; Find the difference in the
sales figures from part (b) and divide by the
number of years.
that the factor t 2 1.5 will be zero when t 5 1.5.
Challenge Practice
13. 10(x 2 y) 14. 4(2x 2 1 5y) 15. 6(3a2 2 b)
16. 4x(x 2 1) 17. r(r 1 2s) 18. 2m(m 1 3n)
19. 5q( p2 1 2) 20. a3(9a2 1 1) 21. 2w2(3w 2 7)
7
35. a. h 5 216t 2 1 14t b. } sec
8
36. a. w(w 1 3) 5 w(7 2 w) b. 2 ft c. 20 ft 2
Copyright © Holt McDougal. All rights reserved.
Problem Solving Workshop:
Mixed Problem Solving
ANSWERS
2
Practice Level C
3 5
2
1. 23, } 2. 2}, } 3. 24, 6 4. 28, 2
2 2
5
3
2 8
1
9 5
2 3
5. 2}, 9 6. 2}, } 7. }, } 8. 2}, 4 9. }, }
5
2
3 5
9 7
8 2
10. 3(3x 2 2 7y) 11. 4m(m2 1 6)
12. 5pq(2p 2 q) 13. 3y(2x 1 3y)
3
14. 5ab(7ab 2 1) 15. 4mn(3m 2 2n)
16. w(w 3 2 2w 2 1 1) 17. 3p(2p3 1 5p 1 2)
2
3
18. 4r 2(2r 3 2 5r 2 2 3) 19. 0, } 20. 2} , 0
3
4
6
1
5
4
21. 0, } 22. 2} , 0 23. 2} , 0 24. 0, }
5
2
3
6
3
13
3
25. 2} , 0 26. 0, } 27. 2} , 0
50
17
10
3
1
4
28. 0, } 29. 0, } 30. 0, }
5
4
9
31. a. 0, 0.21875; These are the times at which
the fish leaves and enters the water. b. Sample
answer: Any value of t ≥ 0 because time should
be positive. 32. a. Locate the zeros and find
the horizontal distance between them.
1. (x 2 1)(x 2 2)(x 2 3); x3 2 6x2 1 11x 2 6
2. (x 1 1)x(x 2 1); x3 2 x
3. x p x(x 2 1)(x 2 1); x4 2 2x3 1 x2
5
1
4. x x 2 } (x 2 2); x3 2 } x 2 1 x
2
2
1
2
14
17
2
5. (x 1 1) x 1 } (x 1 3); x3 1 } x2 1 } x 1 2
3
3
3
1
2
6. (x 1 10)(2x 1 15) 5 1650 7. 20 feet wide
by 40 feet long 8. x 5 0, or y 5 x, or y 5 2x
9. x 5 0, or y 5 x, or y 5 2x
10. y 5 x, or y 5 2x
Lesson 9.5
Practice Level A
1. B 2. C 3. A 4. (x 1 1)(x 1 5)
5. (a 1 7)(a 1 3) 6. (w 1 5)(w 1 3)
7. (p 2 5)(p 1 2) 8. (c 2 1)(c 1 11)
9. (y 1 7)(y 2 2) 10. (n 2 1)(n 2 3)
11. (b 2 3)(b 2 2) 12. (r 2 7)(r 2 5)
13. (z 1 3)(z 1 4) 14. (s 2 6)(s 1 3)
15. (d 2 8)(d 1 3) 16. 24, 21 17. 25, 22
Algebra 1
Chapter 9 Resource Book
A23
Lesson 9.5, continued
1 1
8. x 52}, }
3 4
18. 27, 22 19. 1, 11 20. 22, 3 21. 5, 7
22. 21, 5 23. 25, 3 24. 27, 1 25. C 26. A
31. 27, 3 32. 29, 4 33. 21, 4 34. 30 ft
Practice Level A
Practice Level B
1. B 2. A 3. C 4. 2(x 2 3)(x 1 5)
1. (x 1 7)(x 1 1) 2. (b 2 5)(b 2 2)
5. 2(m 2 1)(m 2 2) 6. 2(p 1 2)(p 2 7)
3. (w 2 13)(w 1 1) 4. (p 1 5)2
7. (2w 1 1)(w 1 3) 8. (3y 1 2)(y 1 1)
5. (m 2 6)(m 2 4) 6. (y 2 8)(y 1 3)
9. (2b 2 1)(b 1 1) 10. 3(n 2 1)(n 1 1)
7. (a 1 9)(a 1 4) 8. (n 2 6)(n 1 8)
11. (5a 2 2)(a 1 3) 12. (2z 2 1)(z 1 5)
9. (z 2 10)(z 2 4) 10. 29, 28 11. 23, 12
12. 6, 7 13. 22, 7 14. 28, 23 15. 3, 9
16. 210, 5 17. 212, 24 18. 25, 6 19. 24, 9
20. 210, 2 21. 3, 8 22. 27, 24 23. 212, 1
24. 26, 3 25. 212, 25 26. 24, 8
27. 25, 23 28. 27, 1 29. 22, 5 30. 29
31. a. x 2 1 150x 1 5000 b. 20 ft
32. a. x 2 2 7x 1 12 b. 144 in.
Practice Level C
1. (x 2 8)(x 1 7) 2. (m 1 6)(m 1 8)
3. (y 2 9)(y 2 6) 4. (p 1 10)(p 1 2)
5. (w 2 9)(w 2 5) 6. (x 1 6)(x 2 4) 7. 24, 15
8. 211 9. 12 10. 225, 20 11. 212, 11
16. 215 17. 210, 15 18. 3, 10 19. 220, 30
20. 214, 22 21. 28, 25 22. 23, 7
23. 26, 4 24. 3, 8 25. 29, 24 26. 22, 12
27. 7, 8 28. 25, 4 29. 214, 23 30. 2, 9
31. a. x 2 1 600x 1 80,000 b. 25 ft c. $234,375
32. a. 50 ft by 40 ft b. 110 ft by 90 ft
Review for Mastery
1. (x 1 8)(x 1 2) 2. ( y 1 5)( y 1 1)
3. (z 2 3)(z 2 4) 4. (x 1 1)(x 2 11)
5. ( y 2 7)( y 1 9) 6. (z 1 4)(z 2 9) 7. 6, 5
Challenge Practice
( y1/3 1 4)( y1/3 1 2)
1
1
2. (y 2 2)(y 1 2)( y 2 1 3) 3. } 2 9 } 1 1
y
y
1
}
}
21
}
2
5
5
4
4
4. 1 Ï
y 1 12 2 1 Ï
y 1 4 2 5. 1 Ï
y 1 11 21 Ï
y 1 12
6. x 5 22, 2 7. x 5 22, 2, 23, 3
A24
Algebra 1
Chapter 9 Resource Book
13. (7d 2 1)(d 2 2) 14. 2(r 2 5)(r 2 1)
1
3
15. (6s 2 1)(s 2 2) 16. 25, } 17. 24, 2}
3
2
5
1
1
1
18. 21, } 19. 23, } 20. 24, } 21. 22, 2}
3
2
2
3
3
1
2 1
2
2
22. 2} , 2} 23. 2}, } 24. 2} , 2 25. 27, }
5
2
2
3 3
3
3 1
1
1
2
26. 2} , 2} 27. 2} , } 28. 25, 1 29. 2} , 5
2
4
2 3
3
1
3
1
30. 2} , 5 31. 2, 3 32. } , 2 33. 26, }
2
4
2
3
1
1
34. 24, } 35. 2} , 1 36. 22, 2} 37. 1 sec
2
4
2
2
38. a. 4x 2 39x 1 90 b. 18 in. by 72 in.
Practice Level B
1. 2(x 2 4)(x 1 7)
2. 2(p 2 2)(p 2 6)
3. 2(m 1 8)(m 1 5) 4. (2y 1 1)(y 1 7)
12. 29, 28 13. 26, 10 14. 26, 12 15. 8
}
}
Lesson 9.6
35. a. x(x 1 1) 5 6 b. 2 ft, 3 ft c. 3 ft 2
1.
}
10. x 5 22Ï 3 , 2Ï 3 , 22, 2
5. (3a 2 1)(a 2 4) 6. (5d 1 2)(d 2 4)
7. (3c 1 2)(2c 1 1) 8. 2(5n 2 3)(n 2 2)
9. (2w 1 3)(6w 2 5) 10. 2(b 1 4)(2b 2 3)
11. 2(r 1 5)(3r 1 2) 12. 22(s 2 2)(2s 1 1)
1
13. 24, 5 14. 28, 22 15. 6, 7 16. } , 5
2
5
5
1
1
}
}
}
}
17. 2 , 2 18. 2 , 2 19. 26, 2
2
8
2
3
2 4
1
5
1 2
20. }, } 21. 2} , } 22. 24, 2} 23. 22, }
3 5
2
4 3
3
1 5
1
4
}
}
}
}
24. 2 ,
25. 23, 9 26. 28,
27. 26,
2 4
2
3
5
5
3
28. 21, 2 29. } , 4 30. 27, } 31. 26, 2}
4
3
8
3
1
32. 210, } 33. 21, } 34. $90 35. 3 sec
2
2
36. a. 4x 2 1 24x 1 32 b. 8 in. by 16 in.
Practice Level C
1. 2(x 2 9)(x 1 20) 2. 2(2m 2 3)(m 2 8)
3. 2(3p 1 4)(p 2 10) 4. (2r 1 5)(4r 1 3)
5. 2(b 1 3)(7b 2 2) 6. 2(y 2 3)(5y 2 3)
Copyright © Holt McDougal. All rights reserved.
ANSWERS
27. B 28. 24, 3 29. 22, 5 30. 21, 6
9. x 5 9
Lesson 9.6, continued
5
1
2 2
1
5 3
3
7. 2} , } 8. 2} , 2} 9. 2} , } 10. 2} , 2}
5 3
2
2
3
4 8
2
3 1
2 5
4 2
3 3
11. 2} , } 12. } , } 13. 2}, } 14. 2} , }
5 3
9 7
8 2
10 4
5 5
7 3
1
11
15. }, 5 16. }, } 17. }, } 18. 22, }
3
8 2
10 2
10
3 5
7
1
1
1
}
}
}
}
}
19. 2 , 2 20. 2 ,
21. 21,
22. 2} , 5
2
3
2 6
2
3
3 1
3 4
5
2
2
23. 21, } 24. 2} , } 25. } , 2 26. } , } 27. }
2 4
5
5
4 3
6
28. 3.5 sec 29. 2 sec 30. a. h 5 216t 2 1 8t
b. 0.25 sec c. It takes the frog 0.25 second
18. (w 2 6)2 19. (m 2 4)2 20. (r 2 10)2
to reach a height of 12 inches and it reaches
the ground at 0.5 second, so it can’t go any
higher because it will take another 0.25 second
to reach the ground. d. h 5 216t 2 1 8t 1 4
e. No, because the frog is higher when it jumps, it
will take the frog longer to reach the ground.
39. a. 6 ft b. about 0.79 sec
3. (6c 2 7)(c 1 2) 4. 2(3r 1 4)(r 1 1)
5. 2(3s 1 4)(s 2 4) 6. 2(4t 2 1)(2t 2 1)
Problem Solving Workshop:
Using Alternative Methods
1. 2.75 seconds 2. The linear term should be
positive in the vertical motion equation. The diver
enters the water after 2.75 seconds.
3. 1.5 seconds 4. 1 second 5. 2 seconds
1 1)(2y
3
3
3. } 1 1 } 2 5
y
y
1
21
}
1 5) 2. (4y 1 1)(2y 2 3)
2
5. 1 2Ï y 2 3 21 2Ï y 1 1 2 6. x 5
Î Î}53
}
}
}
}
3
3
4. 1 7Ï
y 1 1 21 5Ï
y 1 12
}
5
7. x 5 2 } ,
3
2
}
2
1. (x 2 6)(x 1 6) 2. (5p 2 12)(5p 1 12)
3. 4(b 2 5)(b 1 5) 4. 9(2m 2 3)(2m 1 3)
5. 22(x 2 4)(x 1 4) 6. 24(r 2 5s)(r 1 5s)
10. (4n 2 7)2 11. 22(3a 1 1)2 12. 5(2z 2 7)2
5 5
1
13. 27 14. 2} , } 15. } 16. 23, 3 17. 25
2 2
8
5
1
18. 4 19. 25, 5 20. 10 21. } 22. 2}
3
2
3
3 3
23. 2} 24. 2} , } 25. 8 26. 3 27. 1 sec
5
8 8
28. a. 0; 3.75; 5; 3.75; 0 b. Any other values
between 0 and 20 because the ladder is on the
ground at x 5 0 and meets the ground again at
x 5 20.
Î Î12
}
3
2
3
2} ,
}
3
}
5
49
1
8. x 52}, 2} 9. x 5 }
5
3
9
}
2Ï5 2Ï5
10. x 5 2}, }
5
5
Lesson 9.7
Practice Level A
1. B 2. A 3. C 4. (x 2 1)(x 1 1)
5. (b 2 9)(b 1 9) 6. (m 2 10)(m 1 10)
7. (p 2 15)(p 1 15) 8. (2y 2 1)(2y 1 1)
9. (4n 2 5)(4n 1 5) 10. (3w 2 10)(3w 1 10)
11. (8z 2 6)(8z 1 6) 12. (7d 2 5)(7d 1 5)
13. (2r 2 11)(2r 1 11) 14. (3s 2 12)(3s 1 12)
15. (c 2 25)(c 1 25) 16. (x 1 3)2 17. (b 1 5)2
Height (feet)
Copyright © Holt McDougal. All rights reserved.
1. (2y
Practice Level B
c.
Challenge Practice
1/3
27. (3p 2 2)2 28. 23, 3 29. 27 30. 5
3 3
1 1
31. 2} , } 32. 1 33. 210 34. 2} , }
5 5
2 2
4 4
35. 2}, } 36. 22 37. a. π(x 2 y)(x 1 y)
3 3
3
b. 55π cm2 38. a. 8.36 ft b. } sec
4
7. (y 1 12)2 8. (3c 1 4)2 9. (5w 2 2)2
1. (7a 2 1)(a 2 7) 2. (2b 2 5)(2b 1 1)
1/3
24. (2c 1 1)2 25. (4d 1 1)2 26. (3y 2 1)2
ANSWERS
Review for Mastery
21. (z 1 8)2 22. (s 1 11)2 23. (x 2 8)2
y
5
4
3
2
1
0
0
5
10
15
20 x
Distance from left end (feet)
d. 10 ft
Practice Level C
1. (5x 2 9)(5x 1 9) 2. 25(3p 2 2)(3p 1 2)
3. (11w 2 25)(11w 1 25) 4. 4(3m 2 4)(3m 1 4)
1
5. } (3r 2 1)(3r 1 1) 6. (9x 2 7y)(9x 1 7y)
16
7. 23(y 1 8)2 8. 4(n 2 5)2 9. 3(2z 1 1)2
10. 6(2a 2 5b)2 11. 22(3s 1 4t)2
8 8
1
1 2
12. } (5z 1 1)2 or 5 z 1 }
13. 2} , } 14. 29
5 5
5
5
7 7
15. 2}, }
2 2
3
11
4
4
16. 10 17. } 18. } 19. 2} 20. }
5
6
15
7
1
8
3
}
}
}
21. 22, 2 22.
23. 2 24.
25. 90
5
3
4
1
2
Algebra 1
Chapter 9 Resource Book
A25
26. 140 27. 36 28. 16 29. 9 30. 28
31. 23, 1, 3 32. 22, 0, 2 33. 28, 0, 8
31. Only once, because the squirrel reaches the
34. 4(x 1 1)(x 1 2) 35. a. 8πr 2 2 72π 5 0
ground in 1 second and it takes the squirrel 0.5
second (half this time) to reach the height of 4 feet.
32. a. 0; 4.5; 6; 4.5; 0 b. Any other values between
0 and 80 because the bridge is on the ground at
x 5 0 and meets the ground again at x 5 80.
b. 3 in. 36. 2 sec
c.
Height (feet)
ANSWERS
Lesson 9.7, continued
y
6
5
4
3
2
1
0
Practice Level B
1. (4x 2 3)(x 1 5) 2. 22(a 2 6)(a 2 3)
3. (w 2 2 5)(w 1 8) 4. (2b2 1 3)(b 1 6)
5. (y 2 1)(x 1 15) 6. 3(x 2 2)(y 1 4)
7. (x 2 1 5)(x 1 1) 8. ( y 2 1 1)(y 2 14)
9. (m2 1 2)(m 2 6) 10. ( p2 1 4)(p 1 9)
11. (t 2 2 2)(t 1 12) 12. (3n2 1 1)(n 2 1)
13. 7x 2(x 1 4) 14. 4m(m 2 2)(m 1 2)
0 10 20 30 40 50 60 70 80 x
Distance from left end (feet)
15. 22p(8p2 1 1) 16. 6r 2(8r 2 5)
d. 40 ft
17. 15y(1 2 4y) 18. 6x(3y 2 4x)
Review for Mastery
19. 5(m2 1 4m 1 8) 20. 6(x 1 5)(x 2 4)
1. (m 1 11)(m 2 11) 2. (3n 2 8)(3n 1 8)
21. 4z(z 2 2)(z 1 1) 22. 9(x 3 1 4x 2 1 4)
1 2
3. 3( y 1 7z)( y 2 7z) 4. m 2 }
4
23. (x 2 1 5)(x 1 1) 24. (d 2 1 5)(d 1 4)
1
2
9 9
5. (4r 1 5s)2 6. 9(2x 2 1)2 7. 5 8. 2}, }
4 4
Challenge Practice
1. (x 2 3y)2 2. (2x 2 5y)2 3. (5xy 1 4)2
4. 4(x 2 10)2 5. (5x 1 13)2
}
}
1
6. 5 2 Ï 5 , 5 1 Ï 5 7. }
2
5
4
9. 2} 10. 2}
7
2
Î72 1 72
}
3
}
13
5
} 8. 2}
1
7
25. 24, 22 26. 25, 5 27. } 28. 2} , 21
2
2
5
4
29. } 30. 2} 31. 2(2x 1 3)(x 1 1)
3
3
32. a. 8πr 2 2 32π 5 0 b. 2 cm
33. a. h 5 216t 2 1 12t 1 4 b. 5.04 ft c. 6 ft
d. 1 sec
Practice Level C
1. 13a(1 2 2a) 2. 15x(2y 2 3x)
Lesson 9.8
5. r(r 1 5)2 6. 5b2(b 1 4)2 7. 4n3(n 1 6)(n 2 5)
Practice Level A
8. 7c(c 2 2)2 9. 25(2t 2 5)(t 1 3)
1. C 2. A 3. B 4. (x 1 1)(x 1 4)
10. (x 2 y)(x 1 9) 11. (x 2 2 8)(x 1 5)
5. (b 2 1)(b 1 3) 6. (2m 1 1)(m 1 1)
12. (3x 2 8y)(3x 1 8y) 13. 3x 3y(x 2 9)(x 1 9)
7. (5r 2 1)(r 1 2) 8. (w 1 3)(w 1 6)
14. 8rs4(r 2 3)(r 1 3) 15. 25x 2y(x 2 4)
5
1
16. 23, 21, 0 17. 22, 2 18. 2} , 0, }
3
2
15 15
1
}
}
}
19. 2 ,
20. 2 , 0, 5 21. 26, 0
7
4 4
5 5
3
5 9
22. 22, 0, } 23. }, } 24. 2}, }
6 6
5
6 2
9
9
25. 2} , 0, } 26. 25, 7 27. 14
4
4
3
10
10
28. 2} , 0 29. 23, 9 30. 2} , 0, }
5
9
9
31. a2 2 2ab 1 b2 5 a2 2 ab 2 ab 1 b2 5
9. (y 2 6)(y 1 4) 10. (n 2 7)(n 2 3)
11. (3z 1 8)(z 2 4) 12. (2p 2 3)(p 1 5)
13. (x 1 3)(x 1 1) 14. (x 1 2)(x 2 1)
15. (x 2 1)(x 1 8) 16. (x 2 1 2)(x 2 5)
17. (x 2 2 6)(x 2 4) 18. (x 2 1 5)(x 1 3)
19. (x 2 1 7)(x 2 1) 20. (x 2 2 3)(x 1 3)
21. (x 2 2 1)(x 1 3) 22. not completely factored
23. completely factored 24. not completely
factored 25. x 3(x 2 1)(x 1 1)
26. a2(2a 2 5)(2a 1 5) 27. 5y 4(y 2 5)(y 1 5)
28. 25, 21, 5 29. 24, 21, 4 30. 22, 1, 2
A26
Algebra 1
Chapter 9 Resource Book
a(a 2 b) 2 b(a 2 b) 5 (a 2 b)(a 2 b) 5 (a 2 b)2
1
32. 3(2x 2 1)(3x 1 2) 33. a. 2πr 2 2 } π 5 0
2
1
b. } ft 34. about 11 sec
2
Copyright © Holt McDougal. All rights reserved.
3. 22(m 1 1)(m 1 7) 4. 7(2p 2 3)(p 2 1)
Lesson 9.8, continued
Review for Mastery
Chapter Review Game
1. (11x 1 3)(x 2 8) 2. (9x2 2 7)(x 1 1)
3. (5x2 2 3y)(2x 2 7) 4. 0, 27, 3 5. 0, 26
9. Leading 11. Monomial 13. Degree
14. Projectile
Problem Solving Workshop:
Mixed Problem Solving
1. a.
Down 1. Grouping 3. Trinomial
5. FOIL pattern 6. Factoring 8. Binomial
10. Prime 12. Vertical
h in.
h 1 12 in.
1
h 2 3 in.
G
2
3
E
P
R
F
E
R
T
C
b. h3 1 9h2 2 36h c. length: 18 inches; width:
3 inches; height: 6 inches
2. Answers will vary. 3. a. 6x 2 2 8x 2 30
b. length: 12 inches; width: 4 inches; height:
9 inches 4. 14 feet; The zeros of the function,
0 and 14, are where the underpass touches the
ground. The difference between the zeros is
the width of the underpass at its base.
5. a. h 5 216t 2 1 60t 1 4 b. 1.25 seconds and
2.5 seconds c. Yes; The ball reaches a height of
54 feet on the way up and on the way down.
6. 0.75 second 7. a. 4x3 2 44x 2 1 117x
b. 77 cubic inches; 90 cubic inches; 63 cubic
inches; 20 cubic inches; 2 inches c. No; You
cannot cut two squares with a side length of 5
inches from a side of a piece of cardboard that is
9 inches.
ANSWERS
6. 0, 1, 3
Across 2. Perfect 4. Roots 7. Polynomial
4
R
O
O
T
S
A
L
U
I
5
P
N
F
6
7
O
P
F
L
Y
N
O
M
I
8
A
B
C
I
L
T
N
P
I
N
M
9
11
M
O
N
O
I
M
10
A
R
M
T
I
I
T
N
A
E
G
L
E
A
D
I
N
P
G
A
R
12
L
L
V
I
E
M
13
D
E
G
E
C
T
R
E
E
L
E
T
14
P
R
N
O
J
I
C
A
L
Copyright © Holt McDougal. All rights reserved.
Challenge Practice
1. (y 1 3)(2y 1 9)(4y 1 17)
2. (y 2 3)(y 1 1)( y 2 2 2y 1 5)
3. (3x 2 5)(3x 1 1) 4. (7x 1 5)(3x 1 2)
5. 2y(y 2 2)(y 1 2)( y 2 1 4) 6. x 5 28, 21
3
7
5
7. x 5 23, 0, 3 8. x 5 2}, 2} 9. x 5 }, 6
2
4
2
1
10. x 5 } 11. 150 mi/h
3
12. t 5 1 min and t 5 3 min
Algebra 1
Chapter 9 Resource Book
A27
Answers
Lesson 10.1
18.
Practice Level A
x
y
2.
3.
4.
22
21
20
5
6
0
1
0
2
2
5
23
20
19.
x
22
21
0
1
2
y
216
24
0
24
216
x
22
21
0
1
2
y
10
7
6
7
10
x
22
21
0
1
2
y
24
27
28
27
24
3
23
21
21
9.
1
1
23
3 x
21
23
10.
y
23
21
1
3 x
domain: all reals;
range: y ≤ 0;
vertical shrink by
1
a factor of }3 and
reflection in x-axis
domain: all reals;
range: y ≤ 0;
vertical stretch by
a factor of 6 and
reflection in x-axis
11. (0, 8); x 5 0 12. (0, 24); x 5 0
13. (0, 21.5); x 5 0 14. A 15. C 16. B
y
1
23
21
21
1
3 x
23
A28
Algebra 1
Chapter 10 Resource Book
domain: all reals;
range: y ≥ 23;
vertical stretch by a
factor of 2 and vertical
shift 3 units down
22. a. 210 ≤ x ≤ 10 b. 0 ≤ y ≤ 8
23. a. 26 ≤ x ≤ 6 b. 0 ≤ y ≤ 2
Practice Level B
1.
2.
3.
4.
5.
6.
17.
3 x
23
3 x
y
y
3
20. 5 units up 21. vertically stretching; 10
domain: all reals;
range: y ≥ 0;
vertical stretch by
a factor of 5
y
5
3 x
1
1
5. C 6. B 7. A
8.
21
22
domain: all reals;
range: y ≥ 25;
vertical shift 5 units down
x
22
21
0
1
2
y
36
9
0
9
36
x
22
21
0
1
2
y
220
25
0
25
220
x
24
22
0
2
4
y
41
11
1
11
41
x
216
28
0
8
16
y
234
210
22
210
234
x
22
21
0
1
2
y
213
21
3
21
213
x
22
21
0
1
2
y
19
1
25
1
19
7. F 8. A 9. D 10. B 11. C 12. E
13. shift the graph 8 units down 14. shift the
graph 4 units up and reflect over x-axis
15. stretch vertically by a factor of 2 and shift
3 units up 16. stretch vertically by a factor of 5,
reflect in x-axis, and shift 1 unit up
Copyright © Holt McDougal. All rights reserved.
ANSWERS
1.
domain: all reals;
range: y ≥ 7;
vertical shift 7 units up
y
10
Lesson 10.1, continued
1
17. shrink vertically by a factor of } and shift
2
2 units down 18. shrink vertically by a factor
3
of }4 , reflect over x-axis, and shift 5 units up
1.
2.
3
29
23
23
3
x
22
21
0
1
2
y
36
6
24
6
36
x
22
21
0
1
2
y
23
1.5
3
1.5
23
9 x
y
3.
y
20.
3
1
23
3 x
21
ANSWERS
domain: all reals;
range: y ≥ 9;
vertical shift 9 units up
y
19.
Practice Level C
domain: all reals;
range: y ≤ 0;
vertical shrink by a factor
1
of }5 and reflection in x-axis
3
1
23
21
21
1
3 x
23
domain: all reals;
range: y ≥ 2;
vertical shrink by a
1
factor of }6 and vertical
shift 2 units up
23
y
4.
y
21.
3
1
23
21
1
3 x
domain: all reals;
range: y ≤ 0;
vertical stretch by a factor
3
of }2 and reflection in x-axis
2
23
21
22
1
3 x
domain: all reals;
range: y ≤ 23;
vertical stretch by a factor
of 4, reflection in x-axis, and
shift 3 units down
23
y
5.
y
22.
1
23
21
21
1
3 x
domain: all reals;
range: y ≥ 23.5;
vertical shift 3.5 units down
6
2
23
21
1
3 x
25
y
y
23.
9
3
29
23
3
9 x
domain: all reals;
range: y ≥ 29;
vertical stretch by a factor
of 2 and shift 9 units down
5
3
1
23
21
21
1
3 x
29
y
7.
y
24.
3
23
21
1
3 x
domain: all reals;
range: y ≤ 2;
vertical stretch by a factor
of 5, reflection in x-axis, and
vertical shift 2 units up
25. a. 218 ≤ x ≤ 18 b. 0 ≤ y ≤ 20
26. a.
Height (feet)
Copyright © Holt McDougal. All rights reserved.
6.
y
100
80
60
40
20
0
3
1
21
21
0
t
1
2
Time (seconds)
x
y
8.
30
b. 0 ≤ t ≤ 2.5;
0 ≤ y ≤ 100
c. 84 ft
d. about 1.8 sec
e. 2.5 sec
1
18
6
23
21
26
1
3 x
domain: all reals;
7
range: y ≥ 2}2 ;
vertical stretch by a factor
7
of 9 and vertical shift }2
units down
domain: all reals;
1
range: y ≥ }5 ;
vertical shrink by a factor
3
1
of }5 and vertical shift }5
unit up
domain: all reals;
range: y ≤ 4;
vertical shrink by a factor
1
of }2 , reflection in x-axis,
and vertical shift 4 units up
domain: all reals;
3
range: y ≥ }4 ;
vertical stretch by a factor
3
of 6 and vertical shift }4
unit up
Algebra 1
Chapter 10 Resource Book
A29
Lesson 10.1, continued
15
9
ANSWERS
3
23
21
23
3 x
1
y
10.
2
21
23
3 x
1
26
210
5
23
21
25
3 x
1
c. The second graph is a transformation of the first
domain: all reals;
1
range: y ≤ 2}2 ;
vertical stretch by a factor
of 2, reflection in x-axis, and
1
vertical shift }2 unit down
Review for Mastery
graph. The first graph has been reflected in the
x-axis and shifted 20 units up to obtain the second
graph. For the first graph, find the value of t when
y 5 8. For the second graph, find the value of t
when y 5 12.
domain: all reals;
range: y ≤ 15;
vertical stretch by a factor
of 5, reflection in x-axis,
and vertical shift 15 units up
y
11.
domain: all reals;
2
range: y ≥ 2}3 ;
vertical stretch by a factor
2
of 4 and vertical shift }3
unit down
215
12. shift the graph of f 8 units down 13. shift
the graph of f 5 units down 14. shift the graph of
f 4 units down 15. shift the graph of f 16 units up
16. stretch the graph of f vertically by a factor of 3
1
17. shrink the graph of f vertically by a factor of }
2
y
18.
y
19.
y5
10
2x 2
23
y 5 x2 1 6
21
3
x
23
2
23
11
21
1
3
x
y
3
1
21
23
3 x
1
Both graphs have the same vertex, (0, 0), and the
same axis of symmetry, x 5 0. However, the graph
of y 5 28x 2 is narrower than the graph of y 5 x 2
and it opens down. This is because the graph of
y 5 28x 2 is a vertical stretch (by a factor of 8) of
the graph of y 5 x 2 and a reflection in the x-axis
of the graph of y 5 x 2.
Both graphs have the
same vertex, (0, 0),
and the same axis of
3
symmetry, x 5 0. Both
1
graphs open upward.
x
21
1
23
3
However, the graph of
1 2
y 5 }7 x is wider than the graph of y 5 x 2. This is
2.
y
5
1
because the graph of y 5 }7 x 2 is a vertical shrink
1 by a factor of }17 2 of the graph of y 5 x2.
y
3.
y
20.
1.
3
6
1
2
23
23
21
22
1
3
x
y 5 x2 2 4
21
3 x
23
Distance (feet)
22. a.
A30
w
100,000
80,000
60,000
40,000
20,000
0
y
20
16
12
8
4
0
b. about 1.5 in.
t
0.4
0.8
Time (seconds)
Algebra 1
Chapter 10 Resource Book
y
20
16
12
8
4
0
1
y 5 2}3 x 2 is a vertical shrink 1 by a factor of }3 2 of
0.5
1.0
1.5
2.0 d
Diameter (inches)
b.
0
Both graphs have the same vertex, (0, 0), and the
same axis of symmetry, x 5 0. However, the graph
1
of y 5 2}3 x 2 is wider than the graph of y 5x 2
and it opens down. This is because the graph of
1
0
Height (feet)
21. a.
Weight (pounds)
26
the graph of y 5 x 2 and a reflection in the x-axis
of the graph of y 5 x 2.
0
t
0.4
0.8
Time (seconds)
Copyright © Holt McDougal. All rights reserved.
y
9.
Lesson 10.1, continued
4.
Lesson 10.2
y
3
Practice Level A
1
1. a 5 7, b 5 2, c 5 11 2. a 5 3, b 5 25, c 5 1
3 x
1
Both graphs have the same axis of symmetry,
x 5 0, and both open up. However, the graph of
y 5 x 2 2 3 has a lower vertex than the graph of
y 5 x 2. This is because the graph of y 5 x 2 2 3 is
a vertical translation (3 units down) of the graph
of y 5 x 2.
11. upward; x 5 21 12. downward; x 5 4
3
7
13. upward; x 5 2} 14. downward; x 5 }
2
2
15. upward; x 5 21 16. (0, 5) 17. (0, 3)
18. (25, 222) 19. (2, 2) 20. (21, 22)
1 9
1 3
1 11
21. (2, 5) 22. }, } 23. 2}, } 24. }, }
2 2
2 4
2 4
25.
y
5.
3. a 5 4, b 5 2, c 5 22 4. a 5 23, b 5 9,
1
c 5 4 5. a 5 }2, b 5 21, c 5 25 6. a 5 21,
b 5 7, c 5 26 7. upward; x 5 0 8. downward;
x 5 0 9. upward; x 5 23 10. upward; x 5 2
5
3
1
21
23
1
3
x
Both graphs open up, and have the same axis
of symmetry, x 5 0. However, the graph of
26.
1
y 5 }4 x 2 1 2 is wider than the graph of y 5 x 2,
and has a higher vertex. This is because the
1
graph of y 5 }4 x 2 1 2 is a vertical shrink
1
by a factor of }4 and a vertical translation (2 units
1
27.
2
up) of the graph of y 5 x 2.
y
6.
28.
3
Copyright © Holt McDougal. All rights reserved.
1
21
23
1
ANSWERS
21
21
23
3 x
1
2
1
1
2
x
1
2
3
4
5
y
3
0
21
0
3
x
4
5
6
7
8
y
27
30
31
30
27
x
23
22
21
0
1
y
23
2
25
2
23
x
23
22
21
0
1
y
25
1
3
1
25
2
29. C 30. B 31. A
y
32.
Both graphs have the same axis of symmetry,
23
1
x 5 0. However, the graph of y 5 2}2 x 2 2 1 is
wider than the graph of y 5 x 2, opens down and
has a lower vertex. This is because the graph of
1
21
22
3
1
10
x
6 (0, 7)
(0, 26)
2 x 50
210
a reflection in the x-axis, and a vertical translation
of the graph of y 5 x 2.
y
34.
1
3
x
22
22
2
6
10
x
5
(21, 4)
1. y 5 3x 1 4 2. y 5 22x 1 1
2
1
4. y 5 2x 1 5 5. y 5 2} x 2 1 2
2
21
22
y
35.
7
Challenge Practice
3. y 5 4x 2 2 10
23
x 50
1
y 5 2}2 x 2 2 1 is a vertical shrink 1 by a factor of }2 2,
2
y
33.
2
x 5 21
25
23
26
3
x 54
210
1
1
3 x
214
(4, 215)
2
6. 1 kilogram 7. about 5.2 3 1027 kilograms
8. 1 3 1010 meters per second
9. about 1 3 1029 kilograms 10. 3.125 kilograms
Algebra 1
Chapter 10 Resource Book
A31
Lesson 10.2, continued
23
37.
21
21
x5
ANSWERS
23
y
23.
7
5
215
1
4
(
y
(22, 7)
x
3
1
25
25
x 5 22
1
,
4
2
23
8
)
15 x
5
(0, 3)
3
27
1
23
23
21
y
3
25.
(
23
21
3. a 5 7, b 5 23, c 5 21 4. a 5 22, b 5 9,
y
27.
23
(
b 5 3, c 5 27 7. upward; x 5 0; (0, 25)
2
)
1
31
,2 8
4
y
29.
26
6
10 x
(4, 211)
x
10
14. downward; x 5 0; (0, 224) 15. downward;
(26, 210)
57
1 }74, }
82
16
1
1 2}3, 2}
32
(2
4 10
,
3 3
210
y
)
1
25
21
x
4
x 5 2 3 23
19. vertex: (5, 222)
x
3
4
5
6
7
32. minimum; 240 33. maximum; 3
y
218
221
222
221
218
7
34. minimum; } 35. 12 ft 36. 24 in.
2
Practice Level C
20. vertex: (3, 7)
13
21. vertex: 1, }
2
1
22. vertex: (3, 0)
x
10
230
31.
y
2
22
210
x 5 28
33
1 8
x54
x 5 26
30.
(28, 35)
x 5 4}; 1 }4, 2}
13. upward; x 5 25; 1 25, 2} 2
8 2
2
x 5 }2; 1 }2, 2}4 2 16. upward; x 5 }3 ; 1 }3, }3 2
2
210
11. upward; x 5 21; (21, 25) 12. downward;
119
22
22
1
4
x5
23
3
8. downward; x 5 0; (0, 9) 9. downward; x 5 };
2
3 23
} , } 10. upward; x 5 2; (2, 211)
2 2
3 x
1
)
y
2
28.
21
21
3
A32
3 x
1
21
(
c 5 0 5. a 5 }4 , b 5 0, c 5 210 6. a 5 28,
7
17. downward; x 5 };
4
1
18. upward; x 5 2};
3
1
y
23
3 x
1
3
1. a 5 6, b 5 3, c 5 5 2. a 5 }, b 5 21, c 5 8
2
1
3 x
1
25
1
1
25, 25
5
1
3
x5
)
1 3
,
2 2
Practice Level B
3
21
21
26.
1
2
x5
1
41. 10 ft 42. 3.5 ft
3
x50
x
38. minimum value; 27 39. maximum value; 9
40. minimum value; 22
7 7
5
(0, 210)
1
1
y
24.
x50
2
x
1
2
3
4
5
y
3
6
7
6
3
1 1 23
2
1. downward; x 5 }; }, } 2. upward; x 5 };
5
2 2 4
2 3
1 1 23
}, } 3. upward; x 5 }; }, } 4. downward;
5 5
8 8 8
1 1 9
x 5 }2; }2, }4 5. upward; x 5 0; (0, 29)
14
1 1
6. downward; x 5 }; }, 2} 7. upward;
5
5 5
1
1
2
2
1
1
2
x
21
0
1
2
3
y
}
17
2
7
}
13
2
7
}
x
1
2
3
4
5
1 }32, 21 2
y
}
4
3
}
1
3
0
}
1
3
}
4
3
22
2 2
12. downward; x 5 }; }, 2}
3
3 3
Algebra 1
Chapter 10 Resource Book
17
2
2
1
2
x 5 8; (8, 28) 8. downward; x 5 0; (0, 7)
3
9. downward; x 5 1; (1, 11) 10. upward; x 5 };
2
11. upward; x 5 21; (21, 28)
1
2
Copyright © Holt McDougal. All rights reserved.
y
36.
Lesson 10.2, continued
13. vertex: (4, 1)
x
y
3
2
5
}
4
5
215
215
6
1
5
}
4
2
27. minimum; 236 28. maximum; 101
29. minimum; 217 30. a. lamp A: 25 mm;
lamp B: 20 mm b. 5 mm 31. 6 ft; Find the
maximum of the top part of the window and
subtract 1.5 from the result.
0
1
2
3
y
21
}
13
2
9
}
4
Review for Mastery
1. x 5 22: (22, 211) 2. x 5 6: (6, 231)
13
21
2
3. minimum value; 219
5
y
4.
16.
x50
25
25
5
x
y
15.
4
x52
6
x
15
(0, 215)
2
(0, 8)
6
(2, 3)
22
x50
ANSWERS
14. vertex: (2, 9)
2
6
x
2
26
y
17.
18.
( , 4)
1
2
22
2
x
6
y
(0, 20)
1. about 215 feet 2. 28 feet 3. 8 feet
4. about 2.54 feet
3
5
23
21
1
x5
215
x
3
1
2
Problem Solving Workshop:
Worked Out Example
x
15
x50
Challenge Practice
1. y 5 2x 2 2 3x 1 1 2. y 5 2x 2 2 x 1 4
3. y 5 x 2 2 2x 1 3 4. y 5 x 2 2 4x 1 4
y
19.
5
Copyright © Holt McDougal. All rights reserved.
23
3
12
1
4
21
21
y
21.
20
x51
3
212
x
x53
9. f (x) 5 26x 2 1 5x 2 1
4
x
12
x54
53
2
Practice
10
6
(4, 6)
6
26
26
18
26
(1, 7)
2
x51
22
2
215
25
x5
6
22
26
X 9
2
10 x
6
2
15
(
(225, 255)
9
,
2
2
127
4
)
X y
3.
50
2150
/
x
x 5 210
Y
2.
x
26.
5
10 x
6
230
6
X
domain: all real numbers;
range: y ≥ 28
/
218
y
30
25.
2
y
26
10
26
22
24.
6
2
x
y
14
23.
X
domain: all real numbers;
range: y ≥ 29
Y
1.
18
218
10. f (x) 5 3x 2 2 19x 1 6
Focus On 10.2
y
22.
(3, )
30
7. f (x) 5 22x 2 1 3x 1 5
8. f (x) 5 2x 2 1 5x 2 7
24
24
(1, 21)
5. y 5 23x 2 1 6x 1 9 6. f (x) 5 x 2 2 3x 1 2
x53
(3, 23)
y
20.
50
150
x
domain: all real numbers;
range: y ≤ 4
X Y
/
X
x 5 225
(210, 2179)
Algebra 1
Chapter 10 Resource Book
A33
Lesson 10.2, continued
Y
4.
2.
Y
/
X
domain: all real numbers;
range: y ≥ 236
X ANSWERS
domain: all real numbers;
range: y ≤ 4
X /
X
domain: all real numbers;
range: y ≥ 0
Y
5.
3.
y
(2, 0) O
1 (2, 0) x
2
domain: all real numbers;
range: y ≥ 212
x0
X / X
(0, 12)
6.
Y
/
X
domain: all real numbers;
range: y ≥ 245
4.
Y
domain: all real numbers;
range: y ≤ 12
X X Y
7.
X X
/
domain: all real numbers;
range: y ≤ 0
/
X
Lesson 10.3
Practice Level A
1. x 2 1 3x 1 12 5 0 2. x 2 2 8x 2 14 5 0
Y
/
X
domain: all real numbers;
25
range: y ≥ 2}
2
X domain: all real numbers;
range: y ≤ 45
Y X /
1
5. x 2 1 3x 2 14 5 0 6. }x 2 1 3x 1 7 5 0
2
7. not a solution 8. solution 9. solution
9.
3. x 2 2 9x 1 1 5 0 4. x 2 1 10x 2 6 5 0
10. solution 11. not a solution 12. not a
solution 13. no solution 14. 22, 2 15. 23, 21
16. 24, 4 17. no solution 18. 24, 2
5
9
3
X
1
10. a. 23 and 1; b. 21; c. y 5 2(x + 3)(x 2 1)
1
11. y 5 2} (x 2 0)(x 2 50)
25
Review for Mastery
23
21
/
A34
29
2
x
X
26, 22
Algebra 1
Chapter 10 Resource Book
3
9 x
3
215
3 x
y
22.
6
22
1
21, 2}2
y
21.
21
1
no solution
X 23
3 x
1
domain: all real numbers;
range: y ≤ 18
Y y
20.
15
1.
y
19.
23, 5
Copyright © Holt McDougal. All rights reserved.
8.
Lesson 10.3, continued
23.
24.
y
3
21
23
16.
y
10
y
5
3
3 x
1
15.
y
25
25
6
5
15
x
1
29
25.
23, 1
5
9 x
23
27.
28.
x
2
21
3 x
1
22, 0
20.
2
22
22
2
y
30
6 x
210
22, 6
30.
y
215
3, 6
220
50
6 x
212
23
29.
2
y
26
9 x
23
23
29
22
22
19.
4
1
23
29
3
y
y
3
2
26
23, 3
y
3
10
6
29
25, 5
18.
y
3
29
225
0, 10
17.
y
15 x
215
x
2, 5
26.
y
6
ANSWERS
22
22
215
no solution
215
2
x
21
21
no zeros
y
22
212
218
2
x
6
21.
6
18 x
2
6 x
2
6 x
26
y
218
26
26
212
6
18 x
22.
y
10
22
210
30
236
10
22
210
260
2
28, 5
Height (inches)
y
12
10
8
6
4
2
0
b. 0 ≤ x ≤ 16;
0 ≤ y ≤ 12
c. 16 in.
d. 12 in.
23.
24.
3
23
21
y
26
22
22
3 x
23
0 2 4 6 8 10 12 14 16 x
Width (inches)
21, 1
25. a.
Practice Level B
1. not a solution 2. not a solution
3. not a solution 4. solution 5. not a solution
6. solution 7. 24 8. 26, 6 9. 28, 3
10. 26, 25 11. 25, 5 12. no solution
13.
14.
y
y
26. a.
3
1
3 x
23
22
22
2
x
23
26, 0
27, 7
y
21, 1
Height (feet)
Copyright © Holt McDougal. All rights reserved.
31. a.
212, 0
0, 10
Height (feet)
26
230
x
24, 0
h
50
40
30
20
10
0
h 5 216t 2 1 10t 1 50
0
h
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
b. about 2.1 sec
0.5
1.0
1.5
2.0 t
Time (seconds)
h 5 216t 2 1 3t 1 1.3
0
0.1
0.2
0.3
0.4 t
Time (seconds)
Algebra 1
Chapter 10 Resource Book
A35
Lesson 10.3, continued
b. about 0.39 sec
y
13.
3
c. about 0.34 sec
ANSWERS
1
23
3 x
21
1
Practice Level C
y
1.
y
14.
23
y
2.
21
x
1
6
1
23
21
21
20.2, 2.7
2
3 x
1
26
22
22
6 x
2
21.3, 1.3
y
15.
1
22, 2
23
24, 1
y
3.
5
22
25
210
20
20.7, 1.4
16. 4.5 in. 17. 1.9 ft 18. 9.9 cm
19. a.
h 5 216t2 1 5t 1 2.5;
h
2.5
First, write 30 inches
2.0
1.5
in feet and then use
1.0
the vertical motion
0.5
model.
0
0
0.2
0.4
0.6 t
215
4
27
x
6
Height (feet)
2
22, 8
5.
3 x
1
y
4.
x
y
25
21
y
15
6.
9
Time (seconds)
3
25
15 x
5
25
23
21
23
b. 1 ft
x
c. about 0.6 sec
25, 23
y
7.
20. a.
y
8.
3
4
1
23
26
21
21
x
2
3 x
1
212
h
40
30
20
10
0
h 5 216t2 1 50t 1 6
0
23
y
9.
b. about 3.2 sec
22, 6
no solution
c. about 3.1 sec; Determine t when y 5 5.
y
10.
25
4
Review for Mastery
1
3
x
5
215
25, 5
Challenge Practice
21, 5
y
11.
1.
y
12.
30
1. 23, 5 2. 22 3. about 25.2, about 20.8
4. 22, 2 5. 27, 2
212
15 x
1
2
3 t
Time (seconds)
2.
y
23
6
1
10
3 x
2
6
23
21
26
1
22
22
x
24, 2
y5
21, 2
2x 2
23
15
21
Algebra 1
Chapter 10 Resource Book
1
y 5 22 x2 1 8
2
6
y 5 3x 2 1 1
1
3
(22, 13), (2, 13)
A36
1
y 5 2 x2 2 1
14
2
18
y
10
x
26
1 23, }72 2, 1 3, }72 2
10
x
Copyright © Holt McDougal. All rights reserved.
25
25
Height (feet)
215
Lesson 10.3, continued
y
3.
8. 23.32, 3.32 9. no solution 10. 24.12, 4.12
y
4.
9
7
7
5
7
y 5 x2 1 2
5
y 5 2x 2 2
1
21
1
3
5
1
2
y 5 x2 1 x 1 3
y 5 2x 2 1 4x 1 3
x
23
1
5.
15
15
22, }
, 2, }
2
2
21
2
21
1
3
5 x
(23, 9), (0, 3)
y
3
Review for Mastery
y5
2x 2
1 3x 1 1
1 y 5 22x 2 2 3x 1 1
21
21
1
3
5
x
23
1 2}32, 1 2, (0, 1)
6. The baseball hits the fence. 7. The baseball
hits the ground before reaching the fence.
8. The baseball goes over the fence.
Lesson 10.4
Practice Level A
1. 7 2. 15 3. 10 4. x 2 5 2 5. x 2 5 3
6. x 2 5 4 7. 26, 6 8. 23, 3 9. 22, 2
10. 23, 3 11. 23, 3 12. 0, 4 13. 2.24
14. 3.16 15. 3.46 16. 22.83, 2.83
Copyright © Holt McDougal. All rights reserved.
ANSWERS
3
11. 22.45, 2.45 12. 6.27, 9.73 13. 211.45,
26.55 14. 0.26, 7.74 15. 210.46, 23.54
16. 22.32, 10.32 17. 23.16, 3.16
18. 21, 1 19. 23, 3 20. 226, 34
21. 21.03, 1.03 22. 27, 1 23. 28, 12
24. 26, 16 25. 1, 13 26. 22, 4 27. 28, 0
28. about 12 cm 29. about 64 ft/sec
17. 21.73, 1.73 18. 21.41, 1.41 19. 5 m
20. 11 in. 21. about 9.59 cm
22. about 5.96 knots 23. 6 in., 7 in., 10 in.
Practice Level B
1. 22, 2 2. 24, 4 3. 26, 6 4. 27, 7
5. 25, 5 6. 29, 9 7. 25, 5 8. 23, 3
9. 21, 1 10. 22.83, 2.83 11. 21.73, 1.73
12. no solution 13. 22.24, 2.24 14. 0
15. 22.45, 2.45 16. 24.12, 4.12 17. 22.5, 2.5
18. no solution 19. 0.76, 5.24 20. 25.16, 1.16
11 11
1. 23, 3 2. 22, 2 3. no solution 4. 2}, }
6 6
15 15
5. 0 6. 2}, } 7. 21.73, 1.73
2 2
8. 22.24, 2.24 9. 21.12, 1.12
10. 21.83, 3.83 11. 21, 27 12. 2.17, 7.83
Problem Solving Workshop:
Mixed Problem Solving
1. a. 1994 b. $3,582,000 2. a. 4
b. 48 square inches 3. Yes; The vertex, which
is a maximum, of the parabola occurs at around 1
year after 1998, or 1999. 4. Answers will vary.
5. 45 6. a. h 5 216t2 1 25t 1 6
b. h 5 216t 2 1 30t 1 5.5 c. The second throw
is in the air longer. Find the x-intercept of the
graph of each equation. The second
equation has a larger x-intercept. 7. 4
8. a. R 5 25n 2 160n 1 800 b. 980
c. The T-shirts should be sold for $14 each. The
maximum occurs at an x-coordinate of 6, which
means that there should be six $1 increases on the
price of a T-shirt. Since the price was $8, you need
to add $6 to this.
Challenge Practice
9
1
1. x 5 212, x 5 6 2. x 5 2}, x 5 2}
2
2
1
1
3. x 5 24 4. x 5 2}, x 5 }
2
6
13
3
5. x 5 2}, x 5 2} 6. 20 min 7. 12 min
7
7
8. 32 min
21. 1.55, 6.45 22. 13 m 23. about 6.16 in.
24. about 13.42 cm 25. about 3 in.
Lesson 10.5
26. 5 ft, 8 ft, 10 ft
Practice Level A
Practice Level C
1. 23, 3 2. 27, 7 3. 24, 4 4. 22, 2
1. B 2. C 3. A 4. (x 1 1)2 5. (x 2 7)2
6. (x 1 9)2 7. (x 2 2)2 8. (x 1 11)2
5. 26, 6 6. 28, 8 7. 22.24, 2.24
Algebra 1
Chapter 10 Resource Book
A37
Lesson 10.5, continued
9. (x 2 12)2 10. 25; (x 2 5)2 11. 16; (x 2 4)2
2
12. 9; (x 2 3)
13. 121; (x 1 11)
ANSWERS
14. 36; (x 2 6)2 15. 100; (x 1 10)2
16. 225; (x 2 15)2 17. 169; (x 1 13)2
3 2
9
18. 400; (x 1 20)2 19. } ; x 1 }
2
4
7 2
11 2
49
121
20. } ; x 1 }
21. } ; x 2 }
2
2
4
4
1
1
2
2
1
2
22. 26.32, 0.32 23. 210.10, 0.10
24. 20.65, 4.65 25. a. 64 5 216t 2 1 64t 1 32
b. about 0.59 sec, about 3.41 sec
c. 32 5 216t 2 1 64t 1 32; 0 sec; 4 sec
26. a. 4 ft b. 152 ft2; Subtract the interior area,
28 square feet, from the total area, 12(15) 5 180
square feet.
Practice Level B
Review for Mastery
81
9 2
121
11
1. }; x 2 }
2. }; x 1 }
4
2
4
2
1
2
1
2
2
3. 64; (x 2 8)2 4. 20.80, 8.80
5. 20.26, 211.74 6. 13.71, 0.29
Challenge Practice
1. 14 and 16 2. 11 and 13 3. 23 and 24
4. 14 and 15 5. 17
}
}
2b 2 Ï b2 128
2b 1 Ïb2 128
6. x 5 }}, x 5 }}
2
2
}
}
5 2 Ï37 2 4c
5 1 Ï37 2 4c
7. x 5 }}, x 5 }}
2
2
}
}
2b 2 Ï b2 2 4c
2b 1 Ï b2 2 4c
8. x 5 }}, x 5 }}
2
2
}
}
2
2b 2 Ï b 2 4ac
2b 1 Ï b2 2 4ac
9. x 5 }}, x 5 }}
2a
2a
1. 36; (x 1 6)2 2. 625; (x 1 25)2
10. 17 feet wide by 35 feet long 11. 2.08 sec
3. 169; (x 2 13)2 4. 81; (x 2 9)2
Focus On 10.5
13
169
5. }; x 1 }
2
4
1
2
2
9
81
6. }; x 2 }
2
4
1
2
2
Practice
1
9
25
1
2
1
Y
1.
11 2
1 2
121
1
7. }; x 2 }
8. }; x 1 }
2
4
4
16
2
X 3 2
10. 26.16, 0.16
5
9. }; x 2 }
Y
2.
X 2
/
X
/
11. 26.12, 2.12 12. 21.32, 11.32
13. 29.10, 1.10 14. 21.83, 3.83
Y
3.
X 22. about 4.05 sec 23. a. l 1 2w 5 60;
lw 5 400 b. 20 ft by 20 ft, 40 ft by 10 ft
/
/
Practice Level C
1
1
Y
6.
X 1 2
1
1. 3.24; (x 1 1.8)2 2. }; x 2 }
8
64
1 2
1
3. }; x 1 }
4. 20.5, 3.5 5. 210.65, 20.35
3
9
2
/
X
/
2
6. 22.67, 3 7. 20.82, 9.82 8. 0.21, 4.79
9. 26.89, 20.11 10. 21.27, 6.27
11. 217.66, 20.34 12. 0.76, 13.24
X
X
Y
5.
17. 25.54, 0.54 18. 20.62, 1.62 19. 6
X
Y
4.
X 15. 21.55, 13.55 16. 23.56, 0.56
20. 5 21. about 272 mi by about 383 mi
X
X 7. y 5 2(x 2 3)2 2 16
Y
/
X
13. 28.89, 20.11 14. 21.05, 6.05
15. 22.08, 1.08 16. about 4.71 ft 17. 6
18. 216, 215 19. about 39 mi/h
20. a. l 1 2w 5 100; lw 5 1000
b. about 27.6 ft by 36.2 ft, about 72.4 ft by 13.8 ft
A38
Algebra 1
Chapter 10 Resource Book
X Copyright © Holt McDougal. All rights reserved.
2
Lesson 10.5, continued
8. y 5 24(x + 1)2 1 20;
21. no solution
y
( 1, 20)
22. a. 300 5 1.55x 2 2 5.1x 1 197; 2000
b. 237 5 1.55x2 2 5.1x 1 197; 1997
O
1
9. y 5 }(x 2 2)2 2 3;
2
x
1
b.
Y
X /
X
2
1. 213.10, 6.10
2.
58
X=7.0212766 Y=72.9984
0 1 2 3 4 5 6 7 8
Years since 1995
2. 22.15, 2.48 3. 21.82, 2.07
X
3 1
9. 22.11, 2.36 10. 21.45, 1.25 11. 2}, }
2 3
12. no solution 13. Sample answer: Use finding
X
3. y 5 22(x + 2)2 1 1;
X Y
X
Copyright © Holt McDougal. All rights reserved.
62
7. 24.61, 21.39 8. 213.44, 7.44
Y X 4. y 5 2(x + 1)2 2 3;
66
4. 23.73, 20.27 5. 23, 4 6. 22.61, 1.28
Y
X=5.3191489 Y=73.0418
58
x 1 y
1
1
x
( 1, 3)
Lesson 10.6
Practice Level A
1. a 5 5, b 5 7, c 5 1 2. a 5 2, b 5 26,
c 5 11 3. a 5 21, b 5 17, c 5 223 4. a 5 10,
b 5 28, c 5 213 5. a 5 23, b 5 1, c 5 22
6. a 5 5, b 5 218, c 5 23 7. B 8. C 9. A
3
10. 27.36, 1.36 11. 21.61, 5.61 12. 21, }
5
13. 27.74, 20.26 14. 29.90, 20.10
1
15. no solution 16. 22, } 17. no solution
3
1
18. 0.42, 3.58 19. no solution 20. }, 2
3
square roots because the equation can be written in
the form x 2 5 d. 14. Sample answer: Use finding
square roots because the equation can be written in
the form x 2 5 d. 15. Sample answer: Use
factoring because the equation is easily factored.
16. Sample answer: Use factoring because the
equation is easily factored.
17. Sample answer: Use the quadratic formula
because the equation cannot be factored easily.
18. Sample answer: Use the quadratic formula
because the equation cannot be factored easily.
19. 22.24, 2.24 20. 8 21. 24.70, 1.70
22. 7 23. 27.80, 1.80 24. no solution
25. a. 500 5 1.36x 2 1 27.8x 1 304; 1995
b. 575 5 1.36x 2 1 27.8x 1 304; 1997
26. a. 80 5 20.27x 2 1 3.3x 1 77; 1998
b.
Number of eggs
(billions)
X 62
Practice Level B
Review for Mastery
1.
66
74
70
0 1 2 3 4 5 6 7 8
Years since 1995
3
1 2
10. y 5 } x 2 } 2 1
2
2
3
11. y 5 2}(x 2 16)2 1 12
64
1
74
70
ANSWERS
23. a. 73 5 20.31x 2 1 3.8x 1 61.6; 2000, 2002
4
Millions of acres
1
Millions of acres
x 95
90
85
80
75
70
65
X=1.0425532 Y=80.146958
0
1
2 3 4 5 6
Years since 1997
7
Practice Level C
1
1
1
1. 2}, 2} 2. }, 1 3. 21.10, 0.10
5
3
2
4. 21.90, 7.90 5. 20.27, 2.77 6. 23.30, 0.30
7. 20.30, 3.30 8. 4.35, 9.65 9. 20.87, 3.67
Algebra 1
Chapter 10 Resource Book
A39
Lesson 10.6, continued
10. no solution 11. 20.34, 0.80
factoring because the equation is easily factored.
14. Sample answer: Use the quadratic formula
because the equation cannot be factored easily.
15. Sample answer: Use the quadratic formula
because the equation cannot be factored easily.
16. 22.45, 2.45 17. no solution 18. 23.30, 0.30
19. no solution 20. 27.36, 1.36 21. no solution
3 3
22. 2} , } 23. 23 24. 217, 21
2 2
25. a. 1992
c.
Billions of dollars
b. 1999
40
30
20
10
0
X=2.6808511 Y=20.149205
0
2 4 6 8 10 12
Years since 1990
Practice Level A
1. a 5 2, b 5 1, c 5 210 2. a 5 4, b 5 25,
c 5 2 3. a 5 1, b 5 28, c 5 11 4. a 5 21,
b 5 6, c 5 23 5. a 5 21, b 5 23, c 5 12
6. a 5 3, b 5 24, c 5 15 7. 215 8. 223
9. 44 10. 4 11. 279 12. 52 13. 84
14. 2271 15. 105 16. no solution
17. two solutions 18. two solutions
19. two solutions 20. two solutions
21. two solutions 22. two solutions
23. no solution 24. one solution 25. two
26. two 27. two 28. none 29. none
30. one 31. a. 155 5 2x 2 1 5x 1 150
b. discriminant: 5 > 0 c. about 1.4 ft; about 3.6 ft
32. 15 5 216t 2 1 20t 1 5.5; no
Practice Level B
1. no solution 2. two solutions 3. two solutions
b.
4. no solution 5. two solutions 6. two solutions
Billions of dollars
26. a. 1994
14
12
10
8
6
4
2
7. two solutions 8. no solution 9. two solutions
10. no solution 11. two solutions
X=4.0851064 Y=7.0589632
0
2 4 6 8 10 12
Years since 1990
Review for Mastery
12. one solution 13. two 14. two 15. two
16. none 17. two 18. two 19. one 20. none
21. none 22. two 23. one 24. two
25. Answers will vary. 26. Answers will vary.
1. 21.07, 13.07 2. 2.72, 20.52 3. 20.93, 0.60
27. Answers will vary. 28. Answers will vary.
4. 2005 5. factor or complete the square
29. Answers will vary. 30. Answers will vary.
6. quadratic formula 7. complete the square
31. a. 150 5 2x 2 1 x 1 156
b. discriminant: 25 > 0 c. 3 ft 32. no
Problem Solving Workshop:
Using Alternative Methods
1. 2001 2. The steps taken are to find the zero
of the function and not when 50 million cassettes
were shipped. There were 50 million cassettes
shipped in 2001. 3. 1.1 seconds 4. 2000
Challenge Practice
41
3
7
1. } x 2 1 4x 1 1 5 0 2. } x 2 1 6x 1 } 5 0
14
2
2
134
1
3 2
15 2
3. } x 2 x 1 } 5 0 4. } x 117x 1 } 5 0
6
15
2
2
5
11 2
5. } x 2 11x 1 5 5 0 6. x 5 2}
6
2
1
7. x 5 1 8. x 5 2}
12
A40
Lesson 10.7
Algebra 1
Chapter 10 Resource Book
Practice Level C
1. no solution 2. two solutions 3. no solution
4. two solutions 5. no solution
6. two solutions 7. one solution 8. no solution
9. two solutions 10. two 11. two 12. none
13. two 14. one 15. two 16. two 17. two
18. two 19. Answers will vary.
20. Answers will vary. 21. Answers will vary.
22. Answers will vary. 23. Answers will vary.
24. Answers will vary. 25. below; the graph
opens upward and the discriminant is positive
Copyright © Holt McDougal. All rights reserved.
ANSWERS
12. 21.78, 0.28 13. Sample answer: Use
Lesson 10.7, continued
18. quadratic 19. quadratic 20. a. quadratic
b. no; The salaries should not continue to fall; at
some point they would rise. 21. linear
Practice Level B
ANSWERS
26. on the x-axis; the graph opens upward and
the discriminant is 0 27. above; the graph opens
upward and the discriminant is negative
28. a. h 5 216t 2 1 42t b. yes; at about 0.9 sec;
at about 1.7 sec 29. a. yes b. yes; First write
5 square feet as 720 square inches, substitute 720
for y in the equation and solve.
1. B 2. C 3. A
y
4.
y
5.
20
3
Review for Mastery
12
1
1. two solutions 2. no solution 3. one solution
4
23
4. 2 5. 0 6. 1
Challenge Practice
21
24
22
24
y
10
Copyright © Holt McDougal. All rights reserved.
2
26
6
22
22
6 x
2
2
23
26
linear
y
1
3 x
13.
15
1
5
1
3
1
21
22
3 x
1
1
3
b.
x
0
1
2
3
4
y
1
4
16
64
256
c. y 5 4 x 19. Answers will vary.
21
25
x
y
5
exponential
quadratic
10. exponential 11. linear 12. quadratic
13. linear 14. exponential 15. quadratic
16. linear 17. exponential
18. a. exponential; The graph rises quickly.
y
3
21
21
23
21
22
exponential
12.
9.
10
2
11.
x
linear
y
23
6. linear 7. linear
6
5
25
1. C 2. A 3. B 4. quadratic
10.
x
6 x
6
9. exponential
23
21
29
4
26
8.
y
3
1
12
Practice Level A
8. exponential
x
y
7.
linear
Lesson 10.8
1
quadratic
y
6.
10. No solution 11. 10 < x < 20
5. quadratic
3 x
1
exponential
40
1
1. k 5 22, k 5 2 2. k 5 } 3. k 5 }, k 5 0
4
3
9
1
1
4. k 5 0, k 5 } 5. k 5 0, k 5 } 6. k > }
4
8
3
147
3
1
7. k < 2} 8. k > } 9. k < }
4
4
21
21
21
23
1
3
5 x
20. linear; V 5 275t 1 800
Practice Level C
quadratic
14.
1. C 2. A 3. B
linear
y
15.
20
y
4.
y
3
y
5.
18
6
12
1
4
21
21
21
24
1
3
5 x
6
1
3
2
5 x
25
23
21
22
1 x
22
26
2
6 x
218
23
exponential
quadratic
16. linear 17. exponential
26
quadratic
linear
Algebra 1
Chapter 10 Resource Book
A41
Lesson 10.8, continued
y
6.
22
21
2
6
10 x
6
Problem Solving Workshop:
Mixed Problem Solving
y
7.
1
1. a. quadratic function b. y 5 20x 2, where y is
2
ANSWERS
23
23
21
22
3 x
1
25
linear
exponential
y
8.
9.
10
6
192
2
210
26
y
320
64
22
22
21
264
2 x
1
5 x
3
quadratic
exponential
10. linear 11. quadratic 12. exponential
13. linear 14. quadratic 15. exponential
16. linear 17. quadratic
18. a. exponential; The graph falls quickly.
b.
x
23
22
21
0
1
y
64
16
4
1
0.25
c. y 5 (0.25)
x
Challenge Practice
1. linear model 2. 13 3. y 5 2x 1 3
4. exponential model 5. 34.171875
6. y 5 2(1.5) x
7. quadratic model
8. 361 9. y 5 3x2 2 2 10. about 935 pounds
Chapter Review Game
19. linear; V 5 280t 1 2000
20. exponential; B 5 1020.20(1.02)
the power and x is the current
2. a. h 5 216t 2 1 80t 1 6.5 b. 5.0 seconds
3. a. h 5 216t 2 2 30t 1 80 b. 1.5 seconds
4. Answers will vary. 5. a. The discriminant is
positive, so there are two x-values that correspond
to y 5 29. b. The average monthly basic rate for
cable television reached $29 in 1999. The other
value can be disregarded since it is negative.
6. 8.9 7. a. A 5 4x 2 1 136x b. 3 feet
c. You can ignore the negative value because a
negative width does not make sense.
t
Review for Mastery
1. (21, 2) 2. (21, 6) 3. (3, 2) 4. (3, 6)
5. (1, 4) 6. (3, 4) 7. (6, 2) 8. (6, 6) 9. (4, 4)
10. (6, 4)
y
1. quadratic function 2. linear function
7
3. quadratic function: y 5 x 2 2 5x 1 6
3
1
21
1
144 feet
A42
Algebra 1
Chapter 10 Resource Book
3
5
7 x
Copyright © Holt McDougal. All rights reserved.
5
4. exponential function: y 5 (0.25)(2) x
Answers
Lesson 11.1
16.
y
6
Practice Level A
2
1. C 2. A 3. B
domain: x ≥ 0;
range: y ≥ 0;
vertical stretch by a
factor of 6
y
10
6
22
22
2
6
10
17.
y
x
1
domain: x ≥ 0;
range: y ≥ 0;
vertical shrink by a
factor of 0.4
y
1.0
0.6
23
18.
10
x
3
1
5
x
23
domain: x ≥ 0;
range: y ≤ 0;
vertical stretch by a
factor of 2 and reflection
in x-axis
19. a.
7. B 8. F 9. D 10. E 11. A 12. C
y
6
2
Copyright © Holt McDougal. All rights reserved.
22
22
2
6
x
domain: x ≥ 0;
range: y ≥ 25;
vertical translation
5 units down
y
3
1
21
21
1
3
x
domain: x ≥ 0;
range: y ≥ 3;
vertical translation
3 units up
20. a.
domain: p ≥ 0;
range: f ≥ 0
0 5 10 15 20 25 30 35 p
Nozzle pressure (lb/in.2 )
d
20
15
10
5
0
domain: h ≥ 0;
range: d ≥ 0
0
100 200 h
Eye level (feet)
b. about 292 nautical miles
1.
y
6
2
22
22
f
700
600
500
400
300
200
100
0
Practice Level B
23
26
10 x
6
b. 36 lb/in.2
26
14.
2
domain: x ≥ 5;
range: y ≥ 0;
horizontal translation
5 units right
26
25
15.
x
2
22
22
Flow rate (gal/min)
6
1
23
3
y
Distance
(nautical miles)
2
y
26
1
domain: x ≥ 23;
range: y ≥ 0;
horizontal translation
3 units left
23
0.2
13.
21
21
6
22
20.2
21
21
x
26
2
6.
6
3
22
22
5.
2
ANSWERS
4.
26
domain: x ≥ 2;
range: y ≥ 0;
horizontal translation
2 units right
2
6
x
domain: x ≥ 0;
range: y ≥ 0;
vertical stretch by
a factor of 7
y
14
domain: x ≥ 0;
range: y ≥ 26;
vertical translation
6 units down
10
6
2
26
26
22
2
2.
6
x
y
3
1
23
21
21
1
3
x
domain: x ≥ 0;
range: y ≥ 0;
vertical shrink by a
1
factor of }5
23
Algebra 1
Chapter 11 Resource Book
A43
Lesson 11.1, continued
2
22
22
x
6
2
26
y
21.
3
1
25
21
21
3 x
1
23
210
}
y
3
1
23
21
21
3
1
x
23
y
17.
1
25
23
21
21
3 x
1
domain: x ≥ 24;
range: y ≥ 24;
vertical translation
4 units down and
horizontal translation
4 units left
domain: x ≥ 25;
range: y > 1;
vertical translation
1 unit up and horizontal
translation 5 units left
22. a.
Side length (inches)
4. translate graph of y 5 Ï x horizontally 8 units
}
right 5. translate graph of y 5 Ï x vertically
}
3 units up 6. translate graph of y 5 Ïx
horizontally
7 units left 7. translate graph of
}
5 units down 8. translate graph
y 5 Ïx vertically
}
of y 5 Ïx vertically 3.5 units up 9. translate
}
1
graph of y 5 Ïx horizontally }2 unit right 10. E
11. C 12. A 13. F 14. B 15. D
16.
6
2
26
22
22
10 x
6
2
26
y
19.
6
2
26
22
22
10 x
6
2
26
y
20.
3
1
23
21
21
domain: V ≥ 0;
range: x ≥ 0
0
200 400 V
Volume
(cubic inches)
23. a.
domain: w ≥ 0;
range: d ≥ 0
d
15
12
9
6
3
0
0 10 20 30 40 50 60 70 80 90 w
Weight (pounds)
b. about 99 lb c. about 4 lb
Practice Level C
y
1.
3
1
y
18.
x
7
6
5
4
3
2
1
0
1
3
23
Algebra 1
Chapter 11 Resource Book
x
domain: x ≥ 6;
range: y ≥ 4;
vertical translation
4 units up and
horizontal translation
6 units right
domain: x ≥ 5;
range: y ≥ 27;
vertical translation
7 units down and
horizontal translation
5 units right
domain: x ≥ 1;
range: y ≥ 2;
vertical translation
2 units up and
horizontal translation
1 unit right
domain: x ≥ 25;
range: y ≥ 24;
vertical translation
4 units down and
horizontal translation
5 units left
b. 225 in.3 c. 576 in.3
23
A44
23
Diameter (inches)
ANSWERS
26
domain: x ≥ 0;
range: y ≤ 0;
vertical stretch by a
factor of 4 and reflection
in x-axis
23
21
21
1
3
x
domain: x ≥ 0;
range: y ≥ 0;
vertical stretch by a
factor of 2.5
23
y
2.
1.8
0.6
23
21
20.6
1
3
x
domain: x ≥ 0;
range: y ≤ 0;
vertical shrink by a
3
factor of }5 and reflection
in x-axis
21.8
y
3.
0.75
0.25
23
21
20.25
20.75
1
3
x
domain: x ≥ 0;
range: y ≤ 0;
vertical shrink by a factor
of 0.25 and reflection in
x-axis
}
4. translate graph of y 5 Ï x horizontally
}
2.5 units left 5. translate graph of y 5 Ï x
3
vertically }2 units down 6. translate graph of
Copyright © Holt McDougal. All rights reserved.
y
3.
Lesson 11.1, continued
}
22. a.
1
}
of y 5 Ïx horizontally }4 unit right 8. translate
}
graph of y 5 Ïx horizontally 5.5 units left
}
3
9. translate graph of y 5 Ï x vertically } unit up
4
y
16.
6
2
26
22
x
domain: x ≥ 26;
range: y ≥ 24;
vertical translation 4 units
down and horizontal
translation 6 units left
domain: h > 0;
range: t > 0
0 5 10 15 20 25 h
Height (meters)
b. 11.025 m
23. a.
Radius of gyration
(inches)
10. D 11. C 12. E 13. A 14. F 15. B
t
2.5
2.0
1.5
1.0
0.5
0
r
2.5
2.0
1.5
1.0
0.5
0
26
y
17.
6
2
26
22
22
2
6
x
26
domain: x ≥ 1;
range: y ≤ 5;
vertical translation 5 units
up, horizontal translation
1 unit right, and reflection
in x-axis
domain: d > 0;
range: r > 1
0 1 2 3 4 5 6 7 d
Inside diameter (inches)
b. about 3.3 in.
Review for Mastery
1.
domain: x ≥ 0;
range y ≥ 0; The graph is a
vertical stretch (by a factor}
of 4) of the graph of y 5 Ï x .
y
14
y54 x
10
6
y
18.
3
1
21
21
1
3
5
x
domain: x ≥ 3;
range: y ≥ 23;
vertical translation 3 units
down and horizontal
translation 3 units right
2
y
Copyright © Holt McDougal. All rights reserved.
2
26
22
22
x
26
y
10
6
2
2
2.
6
14 x
10
domain: x ≥ 26;
range: y ≤ 2;
vertical translation 2 units
up, horizontal translation
6 units left, and reflection
in x-axis
domain: x ≥ 7;
range: y ≥ 8;
vertical translation 8 units
up and horizontal
translation 7 units right
y
6
2
22
22
26
x
2
6
10
domain: x ≥ 4.5;
range: y ≤ 2.5;
vertical translation
2.5 units up, horizontal
translation 4.5 units right;
and reflection in x-axis
10
x
y
y5 x
2
1
3
5 x
y 5 26 x
26
210
3.
5
y5 x11
3
y5 x
1
1
4.
domain: x ≥ 0;
range y ≤ 0; The graph is a
vertical stretch (by a factor
of 6) and a reflection in the
x-axis
of the graph of y 5
}
Ïx .
domain: x ≥ 0;
range y ≥ 1; The graph is a
vertical translation (of 1 unit
}
up) of the graph of y 5 Ï x .
y
7
3
5
x
y
3
y5 x
1
21
21.
6
22
6
20.
y5 x
2
23
19.
ANSWERS
Time (seconds)
y 5 Ïx vertically 12 units up 7. translate graph
23
1
3
y5 x23
x
domain: x ≥ 0;
range y ≥ 23; The graph
is a vertical translation
(of 3 units down)
of the
}
graph of y 5 Ï x .
5. domain: x ≥ 22; range: y ≥ 24
Algebra 1
Chapter 11 Resource Book
A45
Lesson 11.1, continued
Challenge Practice
y
5.
y
1.
5
3
1
1
21
3
1
x
5
25
The domain is all real numbers less than or equal
to 6. The range is all real numbers
greater than or
}
equal to 0. The graph of y 5 Ï6 2 x is a reflection
in the y-axis and a horizontal translation
of
}
6 units right of the graph of y 5 Ï x .
2.
y
21
21
3
1
7 x
5
23
The domain is all real numbers greater than or
equal to 0. The range is all real numbers
less than
}
or equal to 0. The graph of y 5 2Ï 2 x is a
reflection in the x-axis and a vertical stretch
}
(by a factor of 2) of the graph of y 5 Ï x .
3.
y
27
25
23
21
x
The domain is all real numbers less than or equal
to 2. The range is all real numbers
greater than or
}
equal to 3. The graph of y 5 Ï 2x 1 3 is a
reflection in the y-axis, a horizontal translation of
2 units right, and a vertical
translation of 3 units
}
up of the graph of y 5 Ï x .
}
}
6. y 5 Ï x 2 2 1 1 7. y 5 Ï 4 2 x
}
}
8. y 5 2Ï x 1 1 9. y 5 2Ï 5 2 x 1 3
}
10. y 5 Ï x 1 1
Lesson 11.2
Practice Level A
}
}
}
1. C 2. A 3. B 4. 3Ï 11 5. 2Ï 7 6. 3Ï 6
}
}
}
7. 5Ï 2 8. 3Ï 3a 9. 4 x 10. 10 nÏ n
}
y
21
1
}
}
}
Ï23
Ï10
11. 5p Ï 5p 12. 3Ï 5 13. }
} 14. }
}
Ï23
Ï10
The domain is all real numbers less than or equal
to 1. The range is all real numbers
less than or
}
equal to 0. The graph of y 5 2Ï1 2 x is a
reflection in the x-axis, a reflection in the y-axis,
and a horizontal} translation of 1 unit right of the
graph of y 5 Ï x .
23
21
}
x
23
4.
23
}
}
}
}
Ï17
Ï 5x
Ï5
7Ï3
15. }
} 16. } 17. } 18. } 19. 7Ï 5
5
17
3
Ï 5x
}
}
}
}
20. 7Ï 2 21. 23Ï 7 22. 15Ï 2 23. 2Ï 2
}
}
}
24. 5Ï 3 25. 2 1 Ï 2 26. 3 2 2Ï 3
}
27. 6 1 Ï 3 28. about 81.2 volts 29. a. 8 in.
b. 6 in.
Practice Level B
}
}
}
}
}
}
3
1. 10Ï 2 2. 3Ï 5 3. 4Ï 7 4. 20Ï d 5. 3y
1
6. 5nÏ n 7. 3Ï 7 8. 10Ï 3 9. 2 x Ï 5
1
3
}
}
x
The domain is all real numbers greater than or
equal to 24. The range is all real numbers greater
}
Î1
than or equal to 0. The graph of y 5 }2 x 1 2 is a
1
vertical shrink 1 by a factor of }2 2 and a horizontal
}
translation of 4 units left of the graph of y 5 Ï x .
}
}
}
Ï6
Ï5
4Ï 5
x
4
10. } 11. } 12. } 13. } 14. }
7
5
12
10
}
}
}
3y
2
p
Ï
Ï
Ï3
9Ï 2x
15. } 16. }
17. } 18. }
p
5
3y
2x
9
}
}
}
}
19. 13Ï 7 20. 23Ï 5 21. 27 1 4Ï 7
}
}
}
22. 40Ï 2 1 Ï 5 23. 37 1 20Ï 3 24. 33
25. a. about 3.87 mi/h b. about 4.61 mi/h
26. a. about 4.90 ft/sec b. about 9.80 ft/sec
Practice Level C
}
}
1. 3sÏ 5s 2. 14r 2 3. 15c 2Ï 2c
}
}
4. 2m2n5 Ï 31 5. 11x3y 4 Ï x 6. a 2b
A46
Algebra 1
Chapter 11 Resource Book
Copyright © Holt McDougal. All rights reserved.
ANSWERS
3
Lesson 11.2, continued
}
}
Ï10
d
11
7. 3y 2Ï 15x 8. } 9. } 10. }
5
4m
4
}
Lesson 11.3
}
Practice Level A
}
}
m2Ï77m
5Ï 5x
11. } 12. }
13. Ï 15 2 Ï 3
2
11
2x
}
}
}
}
}
1. solution
14. 214 1 3Ï 7 15. 6Ï 7 2 Ï 14
4. not a solution 5. solution 6. not a solution
}
}
}
}
}
}
16. 9 17. 36 18. 72 19. 25 20. 3 21. 31
}
22. 7x 1 3 5 7x 2 1 23. 5x 2 8 5 1 2 6x
24. 9 2 2x 5 25x 2 25. 4x 2 5 3x 1 1
2
b. about 100 watts 23. a. about 1.5 m
b. about 1.6 m2
26. x 2 1 2x 1 1 5 1 2 3x
27. 4x 2 3 5 x 2 2 4x 1 4 28. 1 29. 6 30. 2
Review for Mastery
}
}
}
}
1. 6Ï 2 2. xÏ 3 3. 3y 2 Ï 5y 4. 6xÏ 3 5. 5Ï 2
}
}
}
Ï5
3Ï 2x
xÏ2
6. 3x Ï 5y 7. } 8. } 9. }
9
3y
2x
}
}
}
7. Add 5 to each side, then square each side,
}
}
4. 4xÏ 6 1 12xÏ 2 2 xÏ 3 2 3x 5. y (10 1 Ï 2 )
6. 91.5 mi/h 7. 129.4 mi/h 8. 24,500 ft
Focus On 11.2
Copyright © Holt McDougal. All rights reserved.
Practice
}
3}
3
3}
3}
Ïz
Ïy
1. 9 2. 2 } 3. 7Ï x 4. 24 5. } 6. Ï 3
3
2
3}
3}
3}
}
3
7. 2Ï 4 8. 24Ï 3 9. 2Ï x 10. 0 11. 6Ï
p
3}
3}
3}
12. 0 13. 22Ï z 14. 4Ï 3 1 3Ï x
3}
3}
3}
15. 22Ï 2 2 2 16. 4Ï 2 2 2 Ï 4
3}
Ï25
Ïx
3}
1. 8 2. } 3. 2Ï
y 4. Ï4 5. }
6. Ï 9
5
2
}
3}
3}
3}
3
7. 2Ï
p 8. 6Ï 5 9. 2Ï 4 2 4Ïx
3}
3}
10. 5Ï 5 2 Ï 25
7
32
3
1
18. } 19. } 20. 4 21. } 22. } 23. 1
5
3
2
4
27. no solution 28. 2, 3 29. 6 30. no solution
Review for Mastery
3}
subtract 3 from each side, and divide each side
by 7. 8. Add 3 to each side, divide each side by
6, square each side and solve the linear equation
for x. 9. Square each side and solve the
resulting linear equation for x. 10. Divide each
side by 2, square each side, and solve the resulting linear equation for x. 11. Add the second
radical expression to each side, square each side,
and solve the resulting linear equation for x.
12. Add 2 to each side, square each side, and then
solve the resulting quadratic equation for x.
13. 16 14. 80 15. 46 16. 42 17. 40
24. no solution 25. 6 26. no solution
3}
17. 24Ï 5 2 5 18. 216 19. x 5 1
3}
Practice Level B
4. not a solution 5. solution 6. solution
11. 23Ï 5
Challenge Practice
}
}
}
}
Ï2
1. } 2. 24Ï 3 3. 10Ï 2 2 15Ï 3
4
}
31. a. 16 ft b. 9 ft 32. a. 2 wk b. 5 wk
1. not a solution 2. not a solution 3. solution
}
10. 3Ï 7 1 8Ï 10
}
12. Ï 2x 1 3 5 13 13. 225 14. 64 15. 9
13Ïx
21. } 22. a. about 50 watts
2x
}
}
10. Ï x 1 5 5 9 11. Ï x 2 4 5 4
}
5mÏ5mn
5Ï 7 1 Ï14
2Ï 3
18. }
19. } 20. }
n
7
3
}
}
7. Ï x 5 3 8. Ï x 5 4 9. Ï x 5 3
16. 133 1 60Ï 3 17. 8Ï 3 1 Ï 2 2 Ï 6 2 24
}
2. not a solution 3. not a solution
ANSWERS
}
3}
31. a. about 560,000 subscriptions
b. 312,500 subscriptions 32. about 94.25 ft2
Practice Level C
1. Subtract 1 from each side, square each side,
and then solve the resulting linear equation for x.
2. Subtract 15 from each side, square each side,
and then solve the resulting equation for x.
Algebra 1
Chapter 11 Resource Book
A47
3. Subtract 4 from each side, divide each side by
22, square each side, and then solve the resulting
equation for x. 4. Add 5 to each side, divide each
side by 6, square each side, and then solve the
resulting linear equation for x. 5. Square each
side and solve the resulting linear equation for x.
6. Add the second radical expression to each side,
square each side, and solve the resulting linear
equation for x. 7. Divide each side by 3, square
each side, and then solve the resulting linear
equation for x. 8. Square each side and solve the
resulting quadratic equation for x. 9. Subtract x
from each side, square each side, and solve the
resulting quadratic equation for x.
20
10. no solution 11. } 12. no solution 13. 11
3
14. 1 15. no solution 16. 4 17. no solution
2
1
1
18. } 19. 3 20. 10 21. 2 22. } 23. 4 24. }
3
4
4
25. Answers will vary. 26. a. about 4.8°C
b. 0 m/sec 27. a. about 38.9 in. b. about 155.6 in.
Review for Mastery
1. 9 2. 39 3. 3 4. 7
Revenue (millions of dollars)
Final velocity
2. a.
y
260
250
240
230
220
210
200
190
180
170
0
v
30
25
20
15
10
5
0
}
}}
1. Ï x 2 2 3x 1 2 5 0 2.
Ïx 3 2 7x 2 1 36 5 0
}}
}
3. Ï x 3 2 x 5 0 4. Ï x3 2 2x 2 1 x 5 0
}}
5. Ï 8 x 3 2 12 x 2 2 2 x 1 3 5 0
}
}
6. x 5 Ï 3 2 2x 7. x 5 2Ï 3 2 2x
}
}
8. x 5 2Ï 20 1 x 9. x 5 Ï 20 1 x
}
}
10x 2 x 3
}
3
10x 2 x 3
11. x 5 2 }
3
Ï
12. x 5 }
Ï15 2 x7 2 7x
13. x 5 2 }
Ï15 2 x7 2 7x
10. x 5
Ï
}}
3
}}
3
}}
b. 2004
}}
15. x 5 Ï x 3 2 44x 1 84
Lesson 11.4
Practice Level A
0 1 2 3 4 5 x
Years since 1999
b. about 24.49 meters
per second
c. about 32 meters per
second
1. legs: x, y; hypotenuse: z 2. legs: m, n;
hypotenuse: p 3. legs: c, t; hypotenuse: r
}
}
}
}
}
4. 2Ï 5 5. Ï 10 6. Ï 34 7. 2Ï 13 8. Ï 58
}
}
}
9. 5Ï 2 10. 3Ï 7 11. 10 12. 4Ï 6 13. 4
14. 3 15. 2, 4 16. not a right triangle
17. right triangle 18. not a right triangle
19. about 28.3 in. 20. about 73.2 in.
Practice Level B
}
}
}
}
}
1. Ï 26 2. Ï 65 3. 6Ï 2 4. Ï 95 5. 2Ï 17
}
}
}
}
6. 6Ï 26 7. Ï 241 8. 6Ï 2 9. 10Ï 5 10. 50
}
0 5 10 15 20 25 v0
Initial velocity
3. Answers will vary. 4. 49 5. a. 625 joules
b. The kinetic energy increases. Since the velocity
increases, that means the right-hand side of the
equation must increase. Since the mass stays
constant, that means the kinetic energy must
increase. 6. Answers will vary. 7. a. 4 times
A48
Challenge Practice
14. x 5 2Ï x 3 2 44x 1 84
Problem Solving Workshop:
Mixed Problem Solving
1. a.
b. about 1.07 times c. When the mass increases,
r decreases. That means that the rate of effusion
for the gas as compared to oxygen will get lower
and eventually when the mass is larger than 32,
the rate of effusion for the gas will be less than
1 time greater than the rate of effusion for oxygen.
Algebra 1
Chapter 11 Resource Book
11. 20 12. 11Ï 5 13. 2, 8 14. 12, 16, 20
15. 9, 12, 15 or 15, 36, 39 16. 4 in., 7 in.
17. not a right triangle 18. right triangle
19. not a right triangle 20. about 155 in.
21. about 33 ft 22. about 9.2 ft
Practice Level C
}
}
}
1. 15 2. 5Ï 11 3. Ï 18.25 4. Ï 42.75
Copyright © Holt McDougal. All rights reserved.
ANSWERS
Lesson 11.3, continued
Lesson 11.4, continued
}
}
}
}
5. Ï 19.24 6. Î 19.01 7. Ï 273.44 8. Ï 4.29
}
}
}
9. Ï 0.4 10. Ï 151.21 11. Ï 221.44
24. not a right triangle 25. right triangle
}
26. right triangle 27. not a right triangle
15. 6 in., 10 in. 16. right triangle
28. a. book and basket; about 447 ft
b. book and backpack; about 894 ft
29. a. about 12 mi b. (4, 7) c. about 3 mi;
19. about 25 ft 20. about 10 mi
ANSWERS
12. Ï 0.58 13. 5 in., 9 in. 14. 4 in., 8 in.
17. not a right triangle 18. not a right triangle
21. No, because 152 1 182 5 549 Þ 529 5 232
1 }52, }92 2; Find the midpoint between (1, 2) and
Review for Mastery
(4, 7) and then find the distance between these
points.
1. 15 2. 7 inches 3. no 4. no 5. yes
Review for Mastery
Challenge Practice
}
1. 3 2. 12 3. 10 4. 9 5. 10 6. 5.7
}
}
}
1. C 2. A 3. B 4. Ï 61 5. Ï 26 6. Ï 41
}
}
7. Ï 13 8. 2Ï 10 9. Ï 17 10. 0, 24 11. 1, 9
12. 26, 10 13. 28, 16 14. 2 15. 3 16. (6, 7)
17. (3, 6) 18. (3, 0) 19. a. about 2.06 mi
b. about 2.24 mi c. the distance between stop 1
and home; 0.18 mi 20. $860,000
Practice Level B
}
}
}
}
1. Ï 5 2. Ï 10 3. 5Ï 2 4. 10 5. Ï 37
}
}
}
}
6. Ï 109 7. 4Ï 2 8. Ï 305 9. 2Ï 13 10. 2
11. 2 12. 3 13. 23, 5 14. 24, 10 15. 27, 3
17
16. 3, }
2
1
2
3
11
19. }, 2}
2
2
1
17
17. 2}, 8
2
19
18. 21, 2}
2
3
20. (25, 0) 21. 2}, 2
2
1
2
2
1
1
2
2
24. not a right triangle 25. not a right triangle
26. right triangle 27. not a right triangle
28. 15 mi 29. a. (1750, 2000) b. 1953 ft
30. 8.5 books
Practice Level C
}
}
}
1. 13 2. Ï 122 3. Ï 458 4. 2Ï 101 5. Ï 661
}
}
}
Ï37
6. 8 7. Ï 122 8. Ï 10 9. } 10. 1, 5
3
11. 2, 6 12. 23, 21 13. 24, 22 14. 26, 0
1
5
15. 24, 24 16. 22, 2} 17. }, 8
2
2
169
}
18. 101, 2
19. (3.75, 9.25)
2
1
1
2
1
2
20. (4.75, 21.75) 21. (25.5, 4)
2
Problem Solving Workshop:
Mixed Problem Solving
}
}
3
1
1. a. Ï 17 , 4Ï 2 , 5 b. 2}, 0 , (1, 0), }, 22
2
2
}
} Ï 17
5
c. }, 2Ï 2 , } d. The perimeter of the original
2
2
1
2
1
2
triangle is twice the perimeter of the triangle
using the midpoints. 2. a. 9 miles b. about
2.6 miles 3. 125 4. Answers will vary.
}
5. a. Ï 34 ø 5.83 miles
1
1
b. You should meet } mile north and } mile east
2
2
of your original starting point. You have to
}
Ï34
hike }
miles, or about 2.92 miles.
2
22. right triangle 23. right triangle
}
}
5. (10, 6) 6. (24.5, 24)
Practice Level A
}
}
1. 2Ï 10 2. Ï 29 3. 4Ï 10 4. 28, 22
Lesson 11.5
Copyright © Holt McDougal. All rights reserved.
22. right triangle 23. not a right triangle
6. You should attach the guy wires about
4.8 feet up the tree. Each guy wire is 6.25 feet
long. The guy wire is the hypotenuse of the
triangle and one of the legs is the distance from
the trunk to the stake, which is 4 feet.
7. a. Molly: r miles, Julie: r 1 3 miles
b. Molly: 9 miles per hour, Julie: 12 miles per
hour c. They are 30 miles apart after 2 hours.
After two hours Molly has biked 18 miles and
Julie has biked 24 miles. These are two sides
of a right triangle. The hypotenuse is how far
apart they are.
Challenge Practice
5
1. a 5 61 2. a 5 }, b 5 1
2
}
Ï2
3. a 5 6}, b 5 27 4. a 5 21, b 5 1
2
}
21
5. a 5 61, b 5 6Ï 2 6. a 5 1, }
17
Algebra 1
Chapter 11 Resource Book
A49
Lesson 11.5, continued
108
65
7. a 5 22, } 8. a 5 5, }
61
187
9. a 5 0, a 5 24 10. b 5 0, a > 0 11. 25 mi
ANSWERS
12. 10 mi 13. 38 min
Chapter Review Game
1. rationalizing 2. hypotenuse 3. midpoint
4. triple 5. square 6. radical 7. conjugates
8. distance 9. legs 10. Pythagorean
11. extraneous 12. theorem 13. simplest form
O
F
S
F
C
D
R
I
S
T
A
C
N
E
U
N
S
T
A
M
A
P
L
E
F
S
T
E
U
O
A
H
A
H
J
H
L
Q
U
Q
N
C
T
F
E
L
Q
N
F
C
G
K
B
U
R
P
Y
O
I
N
I
N
C
I
M
U
Y
A
K
Q
O
A
P
Z
A
Z
K
E
N
A
E
R
O
G
A
H
T
Y
P
R
I N G
P M S
F C I
R O M
U N P
E J L
K U E
M G S
V A T
R T F
S E O
A S R
H I M
L X N
T X E
Copyright © Holt McDougal. All rights reserved.
F R A T I
L L A S E
X E X D I
P A G T I
O U F S M
N Y M D I
E R I W D
M T D A P
E C P E O
R K O X I
O Z T D N
E S Z R T
H P E S U
T R I P L
J W Q P S
A50
Algebra 1
Chapter 11 Resource Book
Answers
33.
1. direct variation 2. inverse variation
3. neither 4. inverse variation
9. direct variation 10. C 11. B 12. A
13. Domain and range:
y
all real numbers
except 0
1
23
x
21
21
1
Walking speed (mi/hour)
7. direct variation 8. neither
Practice Level A
s
8
7
6
5
4
3
2
1
0
23
14. Domain and range:
6 x
3
9 x
3
23
23
y
are all real numbers
except 0.
Copyright © Holt McDougal. All rights reserved.
2
y
9
16. Domain and range
23
23
3
2
26
29
y
12. Domain and range:
3
29
9 x
x
22
22
2
26
17. Domain and range:
all real numbers
except 0
3
y
all real numbers
except 0
x
23
23
3. neither 4. inverse variation 5. inverse
variation 6. direct variation 7. neither
8. inverse variation 9. direct variation
y
10. Domain and range:
9
all real numbers
3
except 0
11. Domain and range:
3
29
0 1 2 3 4 5 6 7 8 t
1. direct variation 2. inverse variation
22
22
all real numbers
except 0
3
t
Practice Level B
2
15. Domain and range:
s5
Time (hours)
y
6
all real numbers
except 0
yes; Answers will
vary.
ANSWERS
5. inverse variation 6. direct variation
25 25
214
26. y 5 } ; 27 27. y 5 } ; } 28. no
x
x
2
22
29. yes; y 5 } 30. direct variation
x
31. inverse variation
4000
32. a. t 5 } b. 125 h
p
Lesson 12.1
23
23
x
3
y
6
all real numbers
except 0
2
22
22
2
6 x
29
18. Domain and range:
all real numbers
except 0
y
6
13. Domain and range:
2
22
22
2
6 x
y
6
all real numbers
except 0
2
26
22
22
x
2
26
3 3
19. C 20. B 21. A 22. y 5 } ; }
x 2
8
18
216
23. y 5 } ; 4 24. y 5 } ; 9 25. y 5 } ; 28
x
x
x
Algebra 1
Chapter 12 Resource Book
A51
Lesson 12.1, continued
y
12
all real numbers
except 0
37.
4
ANSWERS
24
15. Domain and range:
4
12 x
y
12
all real numbers
except 0
Average speed (mi/hour)
14. Domain and range:
s
8
7
6
5
4
3
2
1
0
yes; Answers will
vary.
s5
5
t
0 1 2 3 4 5 6 7 8 t
Time (hours)
4
4
12 x
Practice Level C
1. direct variation 2. inverse variation
16. Domain and range:
3. neither 4. inverse variation
y
5. inverse variation 6. direct variation
3
29
x
23
23
3
9. inverse variation
10. Domain and range:
29
17. Domain and range:
1
21
3
29
23
23
11. Domain and range:
y
3
all real numbers
except 0
1
23
21
21
1
3 x
2
2
23
6 x
12. Domain and range:
27 27
23
3
14
19. y 5 } ; 7 20. y 5 } ; } 21. y 5 } ; 2}
x
x 2
x
2
211
11
144
22. y 5 } ; 2} 23. y 5 } ; 72
x
2
x
72
50
24. y 5 }; 36 25. y 5 } ; 25
x
x
228
36
26. y 5 } ; 214 27. y 5 } ; 18
x
x
236
2200
28. y 5 } ; 218 29. y 5 } ; 2100
x
x
55 55
20
30. y 5 } ; } 31. no 32. yes; y 5 }
x 2
x
18
33. no 34. yes; y 5 }
x
2400
35. a. t 5 }
p b. 120 minutes c. 300 minutes
2500
36. a. V 5 } b. 10 lb/in.3
P
Algebra 1
Chapter 12 Resource Book
x
21
x
y
6
22
22
A52
1
3
29
all real numbers
except 0
y
all real numbers
except 0
y
all real numbers
except 0
18. Domain and range:
7. neither 8. direct variation
y
6
all real numbers
except 0
2
22
22
13. Domain and range:
2
6 x
y
all real numbers
except 0
3
29
23
23
x
3
29
14. Domain and range:
all real numbers
except 0
y
15
5
5
15 x
Copyright © Holt McDougal. All rights reserved.
all real numbers
except 0
Lesson 12.1, continued
15. Domain and range:
y
b.
all real numbers
except 0
1
23
21
21
1
3 x
23
16. Domain and range:
y
3
all real numbers
except 0
0
1
0
100
200
300
400
a
Amount saved each month
(dollars)
21
21
17. Domain and range:
inverse variation;
Answers will vary.
m
9
8
7
6
5
4
3
2
1
1
3 x
400
c. m 5 }
a
Review for Mastery
y
1. neither 2. direct 3. inverse 4. direct
all real numbers
except 0
5. Domain and range:
x
21.2 20.4
20.4
0.4
y
all real numbers
except 0
21.2
6
2
26
18. Domain and range:
y
12
all real numbers
except 0
6. Domain and range:
12 x
2
6
x
2
6
x
y
all real numbers
except 0
Copyright © Holt McDougal. All rights reserved.
22
22
4
4
6
2
26
251
51
144
19. y 5 } ; 2} 20. y 5 } ; 72
x
2
x
36
242
21. y 5 }; 221 22. y 5 } ; 18
x
x
230
49 49
23. y 5 } ; 215 24. y 5 } ; }
x
x 2
2150
120
25. y 5 } ; 275 26. y 5 }; 60
x
x
244
2114
27. y 5 } ; 222 28. y 5 } ; 257
x
x
105 105
70
29. y 5 } ; } 30. y 5 }; 35
x
x
2
16
210
31. yes; y 5 } 32. yes; y 5 }
x
x
299,008
33. a. f 5 } b. 1.024 3 105 hertz
w
34. a.
ANSWERS
Number of months
3
22
22
26
7. Domain and range:
y
all real numbers
except 0
3
1
23
x
21
21
1
23
8. Domain and range:
y
all real numbers
except 0
6
2
26
22
22
x
2
6
26
a
40
50
80
100
200
400
m
10
8
5
4
2
1
As the amount of money you save each month
increases, the number of months you need to
save decreases.
16
9. y 5 }; y 5 24
x
Challenge Practice
a
1. x 5 }; inverse variation
dv
a
2. v 5 } y; direct variation
bd
dkc
3. u 5 }; inverse variation
w
Algebra 1
Chapter 12 Resource Book
A53
Lesson 12.1, continued
ANSWERS
4. u 5 dkz; direct variation
a
5. v 5 }; inverse variation
dkz
bd
6. y 5 }; inverse variation
u
ck
7. w 5 }; inverse variation
x
a
8. v 5 } w; direct variation
dkc
ck
9. w 5 } y; direct variation
b
b
10. y 5 }; inverse variation
kz
1
11. a 5 21 12. c 5 22 13. x 5 2}
3
1
1
14. y 5 2} 15. x 5 2}
4
500
23
21
21
3 x
domain: all reals except 26;
range: all reals except 0;
horizontal translation 6 units
to the left
y
12.
6
2
210
22
22
x
26
13. C 14. B 15. A 16. x 5 8, y 5 0
17. x 5 0, y 5 214 18. x 5 6, y 5 5
19. x 5 213, y 5 1 20. x 5 210, y 5 22
y
6
22.
Practice Level A
1. C 2. A 3. B 4. domain: all reals except
6; range: all reals except 1 5. domain: all reals
except 22; range: all reals except 21 6. domain:
all reals except 1; range: all reals except 21
y
7.
domain: all reals except 0;
6
range: all reals except 0;
2
vertical stretch
2
1
21. x 5 25, y 5 27
Lesson 12.2
22
domain: all reals except 0;
range: all reals except 22;
vertical translation 2 units
down
y
1
11.
22
y
5
23.
2
3
x
22
1
23
1
3 x
y
24.
2
210
26
22
2 x
6 x
26
1
1
3
2
1
3
1
3
y
9.
2
26
x
22
22
2
domain: all reals except 0;
range: all reals except 0;
vertical stretch and
reflection in x-axis
domain: all reals except 0;
range: all reals except 4;
vertical translation 4 units
up
y
10
6
26
A54
22
22
C
175
150
125
100
75
50
25
0
0 10 20 30 40 50 60 70 p
Number of people
26.
26
10.
b.
1 x
2
6 x
Algebra 1
Chapter 12 Resource Book
Average number of
flowers per person
21
500
25. a. C 5 } 1 13
p
Cost (dollars/person)
domain: all reals except 0;
range: all reals except 0;
vertical shrink
y
8.
f
175
150
125
100
75
50
25
0
f5
400
31p
0 2 4 6 8 10 12 14 p
Number of extra workers
Copyright © Holt McDougal. All rights reserved.
210
Lesson 12.2, continued
Practice Level B
17. x 5 8, y 5 12 18. x 5 25, y 5 10
1. domain: all reals except 3; range: all reals
except 1 2. domain: all reals except 4; range: all
reals except 3 3. domain: all reals except 26;
range: all reals except 24 4. domain: all reals
except 26; range: all reals except 28
5. domain: all reals except 23; range: all reals
except 3 6. domain: all reals except 3; range: all
reals except 22
y
7.
domain: all reals except 0;
6
range: all reals except 0;
2
vertical stretch
19. x 5 14, y 5 1 20. x 5 212, y 5 23
22.
23.
y
y
5
10
3
6
1
23
21
22
21
3 x
1
24.
2
6
10 x
ANSWERS
22
22
21. x 5 5, y 5 214
y
6
22
2
x
6 x
2
26
1
1
9.
x
y
1
23
x
21
21
1
domain: all reals except 0;
range: all reals except 0;
vertical stretch and
reflection in x-axis
23
10.
y
Copyright © Holt McDougal. All rights reserved.
23
21
22
1
3 x
26
11.
515
25. a. C 5 } 1 14.5
p
b.
C
Cost (dollars/person)
domain: all reals except 0;
range: all reals except 0;
vertical shrink
y
domain: all reals except 0;
range: all reals except 27;
vertical translation 7 units
down
domain: all reals except 0;
range: all reals except 10;
vertical translation 10 units
up
y
12.
21
0 5 10 15 20 25 30 35 p
Number of people
140
120
100
80
60
40
20
0
1.
3 x
y
1
6
10 x
c. 75 pizzas
0 1 2 3 4 5 6 7 p
Number of extra workers
y
x
21
1
3
21
120
100
80
60
40
20
0
c. $40.25
Practice Level C
2
23
140
450
26. a. n 5 }
41p
b.
n
Average number of pizzas
8.
domain: all reals except 0;
range: all reals except 0;
vertical shrink and reflection
in x-axis
21
domain: all reals except 4;
range: all reals except 0;
horizontal translation 4 units
right
2.
domain: all reals except 0;
range: all reals except 0;
vertical shrink
y
3
1
1
3 x
23
13. x 5 6, y 5 4 14. x 5 25, y 5 26
15. x 5 3, y 5 28 16. x 5 27, y 5 7
Algebra 1
Chapter 12 Resource Book
A55
Lesson 12.2, continued
1
23
ANSWERS
domain: all reals except 0;
range: all reals except 0;
vertical stretch and
reflection in x-axis
y
3.
x
21
21
1
13.
1
23
domain: all reals except 0;
range: all reals except 0;
vertical shrink and reflection
in x-axis
y
21
21
x
6
y
15.
6
10 x
6
210
y
16.
2
22
22
22
22
2
26
x
210
domain: all reals except 0;
range: all reals except 0;
vertical stretch
y
3
1
23
1 x
10 x
23
5.
21
2
22
22
4.
y
14.
6
23
23
y
10
21
21
1
y
17.
2
y
18.
2
26
x
210
3 x
22
2
x
26
26
22
22
2
6 x
26
domain: all reals except 0;
range: all reals except 5;
vertical translation 5 units
up
y
7.
10
6
2
26
8.
22
22
2
y
2
22
22
2
10 x
9 people
n
35
30
25
20
15
10
5
0
n5
225
61p
0 2 4 6 8 10 12 14 p
Number of extra parents
17.25
20. a. C 5 }
41r
b.
6 x
6
19.
domain: all reals except 6;
range: all reals except 0;
horizontal translation 6 units
to the right
about 4 rentals
C
4
3
2
1
0
0 1 2 3 4 5 6 7 r
Number of additional
rentals
Review for Mastery
26
y
9.
6
2
26
22
22
x
domain: all reals except 28;
range: all reals except 0;
horizontal translation 8 units
to the left
1.
8
y
8
y 5x
3
1
21
1
y 5x
1
3
x
26
10. x 5 213, y 5 210 11. x 5 2, y 5 2
12. x 5 21, y 5 23
2.
y5
6
26
Algebra 1
Chapter 12 Resource Book
1
x 15
1
y 5x
2
26
A56
1
y
2
The graph of y 5 }x is a
vertical stretch of the
1
graph of y 5 }x . Domain:
all real numbers except 0;
Range: all real numbers
except 0.
6
x
The graph of y 5 }x 1 5 is
a vertical translation (of
5 units up) of the graph of
1
y 5 }x . Domain: all real
numbers except 0; Range:
all real numbers except 5.
Copyright © Holt McDougal. All rights reserved.
domain: all reals except 0;
range: all reals except 29;
vertical translation 9 units
down
y
26
Average cost per
rental (dollars)
6.
Average number of
box lunches per person
23
Lesson 12.2, continued
3.
1
y
1
y 5 x 1 10 6
2
2 x
y5
1
x
26
4.
3
4. vertical asymptote: x 5 };
4
horizontal asymptote: y 5 2
y
6
26
22
22
ANSWERS
26
The graph of y 5 }
x 1 10
is a horizontal translation
(of 10 units to the left) of
1
the graph of y 5 }x .
Domain: all real numbers
except 10; Range: all real
numbers except 0.
6 x
2
y
4
5. vertical asymptote: x 5 2};
3
6
2
22
22
26
2
6
x
1
horizontal asymptote: y 5 2}4
y
26
6
2
26
Challenge Practice
22
2
x
1. vertical asymptote: x 5 2;
horizontal asymptote: y 5 4
y
6
6
8. f (x) 5 } 2 1 9. f (x) 5 }
2x 2 2
2x 1 1
6
2
22
6
x
1
2. vertical asymptote: x 5 2};
2
horizontal asymptote: y 5 23
y
24
10. f (x) 5 } 1 2
22x 1 3
Lesson 12.3
Practice Level A
1. 3x 2 2. 23x 3. 21 4. 3x 2 2 2x 1 6
5. 2x 2 1 3x 2 4 6. 2x 3 2 2x 2 2 3x
7. 4x 3 2 x 1 2 8. x 2 2 3x 2 2
2
22
2
6 x
9. 2x 2 1 4x 2 5 10. A 11. C 12. B
13. x 1 4 14. x 2 5
3. vertical asymptote: x 5 21;
horizontal asymptote: y 5 1
y
15. x 2 6 16. x2y 1 3x 1 2
20h 1 5
17. a. C 5 }
h
b.
2
26
22
2
6 x
Average cost per
hour (dollars)
Copyright © Holt McDougal. All rights reserved.
6
26
23
7
6. f (x) 5 } 1 2 7. f (x) 5 } 1 1
2x 2 6
7x 2 1
C
50
40
30
20
0
0 1 2 3 4 5 6 7 h
Number of hours rented
Algebra 1
Chapter 12 Resource Book
A57
Lesson 12.3, continued
ANSWERS
b.
Ratio of sports cars
sold to total cars sold
1
1
18. a. R 5 } 1 }
6
24(t 1 5)
Practice Level C
1. 3x 3 2 4x 1 2 2. 212x 2 1 8x 1 3
R
0.178
0.176
0.174
0.172
0.170
0.168
0
27
5
3. 7x 1 16 1 } 4. x 1 4 2 }
x22
x21
54
4
5. 3x 1 8 1 } 6. x 1 3 2 }
x24
x13
17
3
7. 2x 2 10 1 } 8. 3x 2 1 2 }
2x 1 2
3x 1 1
0 1 2 3 4 5 6 7 8 t
Years since 1995
9
2
9. 5x6y4 2 x4y3 2 } y 10. 8a2b2 1 2b 2 } a
7
3
Practice Level B
y
11.
1. 3x 2 2 4x 1 2 2. x 2 2 3x 1 6
15
3. 211x 3 1 9x 2 3 4. x 1 1 5. 5x 2 3
3
29
10
6. 4x 1 5 7. 6x 2 2 8. 4x 1 9 1 }
x22
y
15
x
x
215
4
5
y
6
4
212
25
25
29
13.
y
12.
12
x
23
9. 5x2y3 1 2x 2 1 10. 3a2 1 2ab 2 1
11.
y
12.
9
25
2
12 x
26
22
22
2
215
6 x
212
24 1 0.06m
14. a. C 5 }
m
y
6
b.
6
Average cost per
mile (dollars)
2
x
40h 1 4.5
14. a. C 5 }
h
C
28
24
20
16
12
8
4
0
0 1 2 3 4 5 6 7 8 m
Number of miles
C
80
70
60
50
40
30
0
1049
1
15. a. R 5 } 1 }}
6
546t 1 12,552
b.
0 1 2 3 4 5 6 7 8 h
Time (hours)
b.
Ratio of fruit drinks
sold to total drinks sold
23
1
15. a. R 5 } 1 }
2
2(2t 1 9)
R
1.8
1.5
1.2
0.9
0.6
0.3
0
Ratio of walking shoes
sold to total shoes sold
Average cost per
hour (dollars)
b.
R
0.255
0.250
0.245
0.240
0.235
0.230
0
0
1
2
4 t
Review for Mastery
1. 2p2 2 5p 1 6 2. 6r2 1 4r 2 11
3. 25t 2 2 2t 1 6 4. 4x 1 3
0 1 2 3 4 5 6 7 8 t
Years since 1995
4
5. 3x 2 2 1 } 6. 2x 1 5
8x 2 1
3
7. 4x 2 7 1 }
4x 2 7
A58
3
Years since 1999
Algebra 1
Chapter 12 Resource Book
Copyright © Holt McDougal. All rights reserved.
13.
Lesson 12.3, continued
Challenge Practice
11. not in simplest form 12. in simpest form
2
1. x 1 3 2. x 2 1 5x 1 6 3. 1 1 }
x2 2 1
6. 18x 3 1 3x 2 1 30x 1 5
19. in simplest form; x 5 5
7. x 4 1 9x 3 1 14x 2 2 27x 2 5
1
20. } ; x 5 26, 6
x16
8. 6x 3 1 2x2 1 3x 1 6
10. x 4 1 x 3 1 x 2 1 x 1 2 11. x 1 3
21. in simplest form; x 5 210, 10
2(2x) 1 2x
3
22. a. } b. }
x
2x(x)
2x 1 1
12. x 2 4 1 }
13. x 2 2 5x 1 3
x 2 1 2x 1 1
245t 1 32,800
49t 1 6560
23. a. R 5 }} b. R 5 }
465t 1 56,780
93t 1 11,356
9. x 5 1 6x 3 1 x 2 1 7x 1 4
14. 2x 3 1 5x 2 1 15. x 2 2 5
ANSWERS
96x 1 31
5. 5x 2 2 15x 1 37 2 }
x 2 1 3x 1 1
2
15. in simplest form 16. } ; x 5 0
3x
7
17. } ; x 5 0 18. 2; x 5 22
2x
4. x 2 1 2x 1 1
Practice Level B
1. x 5 0 2. x 5 5 3. x 5 210 4. x 5 2
Focus On 12.3
7
5. x 5 23 6. x 5 2} 7. x 5 1 8. x 5 26, 2
3
Practice
2
1. x 1 3 2. x 1 1 3. x2 2 2x 2 }
x22
5
4. 2x3 2 2x2 1 x 1 1 2 } 5. x2 2 x 2 1
x11
1
6. 2x2 2 x 2 2 7. x2 1 x 1 1 2 }
x15
8
5
8. x2 2 3 2 } 9. x3 1 x2 2 2x 2 2 2 }
x23
x21
1
10. x2 1 x 1 6 11. x2 1 1 2 } 12. x2 1 2
1
x2}
2
Copyright © Holt McDougal. All rights reserved.
13. in simplest form 14. not in simplest form
13. No. To use synthetic division, the divisor must
be of the form x 2 k.
14. 50
Review for Mastery
3
1. x2 2 2x 2 }
x21
3
2. 2x3 1 4x2 1 4x 1 2 2 }
x22
5
3. x2 2 3x 1 7 2 }
x11
9. x 5 25, 5 10. 22x; x 5 0 11. 6; x 5 4
1
12. 24; x 5 3 13. } ; x 5 211, 11
x 2 11
1
14. } ; x 5 23, 27
x17
1
15. in simplest form; x 5 23, 28 16. }
2x
3x 1 5
3x 1 2
17. } 18. }
x(x 1 5)
x(x 1 1)
2(4x 1 3) 1 2(4x 2 2)
8x 1 1
19. a. }} b. }}
(4x 1 3)(4x 2 2)
(4x 1 3)(2x 2 1)
2(4t 1 5)
20. }}
; about 11 thousand pounds
0.1t 2 2 0.2t 1 3
Practice Level C
5
1
1. x 5 24, } 2. x 5 2}, 1 3. x 5 7
8
3
4. x2 1 2x 1 2
5
22x 2
1
4. }, x 5 21, 7 5. }, x 5 0, }
x11
3x 2 5
3
3
3(x 2 4)
4
3x
1
6. } ; x 5 0, } 7. } ; x 5 2}, 0
3
2
3x 1 4
4(1 2 2x)
Lesson 12.4
1
4x
2
8. }; x 5 210, 2 9. }; x 5 2}, 3
2
x 1 10
2x 1 1
Practice Level A
1. none 2. x 5 0 3. x 5 6 4. x 5 23
1
5. x 5 1 6. x 5 2 7. x 5 22 8. x 5 }
2
2
9. x 5 2} 10. not in simplest form
3
3
x26
1
10. }; x 5 210, 2} 11. }; x 5 0, 4
2
2x 1 3
x(x 2 4)
x15
12. } ; x 5 22, 0, 2
2x(x 2 2)
13. (5x 1 1)2; Answers will vary.
14. Answers will vary.
Algebra 1
Chapter 12 Resource Book
A59
Lesson 12.4, continued
2[x 1 x 1 (x 1 3)]
15. a. }}}
1
2 } (x 1 3 1 2x 1 4)(x 2 1)
b.
6(t 1 4)
16. a. }}
2
0.01t 2 0.5t 1 18
b. about 4 hundred thousand
0 1 2 3 4 5 6 7 8 x
c. You cannot use the model to conclude that the
revenue of the new trucks sold had increased
because the prices may have decreased which led
to more trucks being sold.
Challenge Practice
0 1 2 3 4 5 6 7 8 t
1. a 5 7 2. b 5 9 3. c 5 1
Years since 1995
The number of flyers increased as time went by.
Review for Mastery
1. 3 2. 24, 4 3. no excluded values 4. 26, 2
4x
1
5. }2 , x Þ 0 6. } x Þ 3
x
23
5x
4. d 5 2 5. e 5 2
5
5
6. p(x) 5 2x 2 2 6x 1 }, q(x) 5 4x 2 2 11x 1 }
2
2
7. p(x) 5 2x 2 1 7x 1 5, q(x) 5 x 2 2 2x 2 3
8. p(x) 5 2x 2 1 x 2 1, q(x) 5 2x 2 1 9x 2 5
9. p(x) 5 x 4 2 1, q(x) 5 2x 4 1 x 2 2 1
5
7. simplest form, x Þ 23 8. }, x Þ 0, x Þ 22
3x
x16
9. }, x Þ 27, x Þ 9
x29
2x 1 5
10. }, x Þ 22.5, x Þ 2.5
2x 2 5
10. p(x) 5 6x 3 1 13x 2 1 8x 1 3,
q(x) 5 3x 3 2 x 2 2 x 2 1
Lesson 12.5
Practice Level A
Problem Solving Workshop:
Mixed Problem Solving
397,000 2 15,500x
1. a. N 5 }} b. 3,550,000 people
125 2 7x
3x 1 64
2. a. p 5 }
11x 1 291
b.
The number of
trucks sold from
1993 to 2002
increased.
Years since 1993
A
7
6
5
4
3
2
1
0
N
9
8
7
6
5
4
0
x
0
1
2
3
p
0.2199
0.2219
0.2236
0.2253
x
4
5
6
7
p
0.2269
0.2283
0.2297
0.2310
7
1
1. C 2. B 3. A 4. 21x 5. }2 6. }
x
6x
5
x
1
7. } 8. 6x 2 9. }2 10. }
8
x(x 1 1)
4x
3x
3x
x21
11. 5(x 1 2) 12. } 13. } 14. }
x14
2
4
9x
8
2x
4x 2
4
15. } 16. }2 17. } 18. } 19. }
5
x
3
2(x 1 2)
3x
2(3t 1 10)
20. a. T 5 }
11 2 t
b. about 8 hundred thousand dollars
c. increasing 3. 5 4. Answers will vary.
5. a. The density and the volume are inversely
related because as the density increases, the
14
volume decreases; V 5 }
b. Because the height
D
is increasing, the volume is increasing which
means that the density in decreasing. 6. 11
A60
Algebra 1
Chapter 12 Resource Book
Practice Level B
6x 4
1
14
1
1
1. }3 2. } 3. } 4. } 5. }
5
5
4
2(x 1 5)
6x
x(x 1 3)
x16
6. } 7. } 8. x 9. 3(x 1 5)
2(x 1 2)
3(2x 2 1)
20
x6
2
1
1
10. } 11. } 12. } 13. } 14. }
x
4
9
9
3(x 1 9)
1
1
2
15. } 16. } 17. }2
4
x
(x 1 3)
18. a. 4x 2 b. 100 tiles
Copyright © Holt McDougal. All rights reserved.
c.
Number of new
trucks sold (millions)
G
6(x 1 1)
b. }}
(3x 1 7)(x 2 1)
Number of flyers
(hundreds of thousands)
ANSWERS
F2
2842 1 337x
7. a. N 5 }; about 9 million
500 1 14x
Lesson 12.5, continued
1250
19. } ; about $8.74 per unit
150 2 t
Practice Level C
2(x 1 3)(x 1 6)
x14
6. 4(2x 2 5) 7. }}
8. }
x(5x 1 16)
x2 1 6
x 2(x 2 2 5)(x 2 7)
2
9. 21 10. } 11. }}
x(x 1 4)
4(x 2 2 7)
6(x 2 4)
3
x2 1 4
12. } 13. } 14. }
2x 2 1
x
2x 2
15. (x 1 6)(x 2 6) 16. (2x 1 1)(x 2 7)
b.
Average number of cars
admitted per month
500(t 1 20)
17. A 5 } ; about $13,529
1 2 0.05t
5(3122t 1 13,124)
18. a. x 5 }}
7(t 1 42)
Answers will vary.
800
600
400
200
0
0
2 4 6 8 10 12
Years since 1988
Review for Mastery
2(x 1 4)
2(x 1 1)
3x
x15
1. } 2. } 3. } 4. }
x25
x22
3x(x 2 6)
3(x 1 6)
Copyright © Holt McDougal. All rights reserved.
Challenge Practice
1. p(x) 5 4x 2 1 2. p(x) 5 x 1 2
3. p(x) 5 22x 2 1 3x 2 1 4. p(x) 5 2x 2 1 3
5. p(x) 5 16x 2 3 6. p(x) 5 1
7. p(x) 5 x 2 2 6x 1 9 8. p(x) 5 2x 1 3
Focus On 12.5
Practice
3x2
3x4
2x3
1
1. } 2. } 3. } 4. 22x 5. } 6. 6x2
6
4
4
12
2x
x
7. } 8. } 9. (x 1 3)2 10. (x2 2 4)2
25
23
x
x
11. }} 12. }2
3(x 2 3)(x 2 4)
(x 2 3)
4x 2 1
1
13. }} 14. }
4(x 2 3)(x 2 1)(x 1 1)
2(x 1 2)
x(x 2 7)
15. }}
(x 1 1)(x 2 1)
the fraction is in the numerator, the value of the
1 1
1
expression is }2 ? }3 5 }6.
a2 2 b2
a2b
}
}
17. Yes.
a2 2 b2
}
a1b
}
a2b
a1b
a2 2 b2
1
a2b a1b
a2 2 b2
a 2b
5}?}5}
and
2
2
a2 2 b2
1
a1b a2b
ANSWERS
7(x 1 1)
1
x24
x5
1. x 2. } 3. 2} 4. } 5. }
2
3x
9
9x(x 1 3)
16. No, when the fraction is in the denominator,
3
3
the value of the expression is 1 ? }2 5 }2. When
a2 2 b2
a 2b
5}?}5}
, so the
2
2
complex fractions are equivalent.
Review for Mastery
5
7x
x
x
1
1. } 2. } 3. } 4. } 5. }}
18
2
10
2
3(x 2 5)(x 2 4)
2
6. }}
(x 1 6)(x 2 6)
Lesson 12.6
Practice Level A
3
2
1
1. } 2. } 3. }
4x
x
3x 2
2x 1 7
2
7. } 8. } 9.
5x
x2
x 2 27
x14
2
4. }3 5. } 6. }
2x
9x
x
2x 1 4
} 10. 10x
x21
11. 12x 3 12. x(x 1 1) 13. x(x 2 4)
14. (x 1 2)(x 1 4) 15. (x 2 3)(x 1 7)
40x 2 1 3
7x 2 2 1
73
16. } 17. } 18. }
15x
2x
36x
9(x 1 2)
2(5 2 12x)
7x 1 16
19. } 20. } 21. }}
2
x(x
1
4)
(x
2
2)(x 1 7)
15x
48
96r
48
22. a. }; } b. t 5 }} c. 9 h
(r 2 4)(r 1 4)
r24 r14
80r 2 160
23. t 5 }; about 1.6 h
r(r 2 4)
Practice Level B
3(2x 1 1)
x18
211
4x
1. } 2. } 3. } 4. }
x15
x29
x12
x24
10x 2 7
x15
5. }
6. }
7. 15x 3 8. x(x 1 7)
2
x 29
3x 2
9. (x 2 4)(x 1 6) 10. 2(x 2 2)
11. x(x 2 5)(x 1 2) 12. (x 1 1)2(x 1 4)
x(5x 1 31)
85
32 2 5x 2
13. } 14. } 15. }}
3
14x
(x
2 5)(x 1 2)
12x
2
5x 1 3x 2 5
25 2 3x
16. }} 17. }
(6x 2 5)(5x 2 3)
x(x 2 7)
x 3 1 3x 2 1 5x 2 5
2x 2 1 5x 1 1
18. }} 19. }}
2
(x 2 1)(x 1 1)
x (x 2 1)
Algebra 1
Chapter 12 Resource Book
A61
Lesson 12.6, continued
ANSWERS
8
8
22. a. t 5 } 1 } b. 6 h c. about 1.0 h
r21
r11
50
50
23. a. t 5 }
b. about 6.1 h
r 1}
r13
c. about 1.4 h
Practice Level C
3(x 2 2)
x28
23x 1 2
1. } 2. } 3. }
x25
x13
2x 2 5
11x 2 1 19x 2 20
22x 2 2 22x 2 9
4. }} 5. }}
(x 2 5)(x 1 2)
(x 1 10)(x 2 1)
22x 2 1 31x 2 16
9x 3 2 16x 2 1 3x 2 2
6. }} 7. }}
(2x 2 3)(x 1 5)
3x 2(x 2 2)
5x 3 2 19x 2 1 8x 1 12
215x 2 1 x 2 30
8. }}
9. }}
2
8x(x 1 6)
5x (x 1 2)
3x 2 1 28x 1 1
5x 2 7
10. }} 11. }}
(x 2 1)(x 1 1)(x 1 7)
(x 2 3)2(x 1 5)
2(x 2 1 6)
2x 2 1 19x 1 50
12. }} 13. }}
(x 2 6)(x 1 2)(x 1 1)
(x 1 2)(x 1 1)
6(4x 2 1 32x 1 7)
21x 2 1 79x 2 2
14. }} 15. }}
(x 1 1)(x 1 4)(x 1 6)
(x 1 8)(x 2 3)
22x 3 1 8x 2 1 51x 1 27
16. }}
(x 2 7)(x 1 2)
11c 2 2 34c 2 105
17. a 5 }}
(3c 1 4)2
10
10
18. a. t 5 }
b. about 2.4 h
r 1}
r 2 2.75
1
c. about 0.2 h 19. a. y 5 }
t
b. no; Answers will vary.
Review for Mastery
3x 1 16
2x 1 1
1. } 2. } 3. 30x5
7x
3x 2 4
Challenge Practice
12x 2 2 72x 1 60
2x 2 1 1
1. w 5 }}
2. w 5 }
2
4
9x 1 6x 1 1
x 1 2x 2 1 6
2
2x 2 2x 1 13
2x
3. w 5 } 4. w 5 }}
3x 2 1
210x 1 5
2
2325x 2 115x 2 10
5. w 5 }}
x 2 2 2x 1 1
1
1
6. y(t) 5 } t 1 }
xt
60
7. Approximately 25 hours and 43 minutes
8. 60 hours 9. 30 hours 10. 60 hours
Lesson 12.7
Practice Level A
1. x 5 6 2. x 5 210, 24 3. x 5 23, 3
4. 26, 6 5. 28, 8 6. 15 7. 224 8. 216
9. 25, 3 10. x(x 1 4) 11. x(x 2 3)
12. (x 1 2)(x 2 5) 13. 29 14. 11
23
9
10
15. no solution 16. } 17. 2} 18. }
3
4
3
19. 12 days 20. 7 field goals 21. a. 2 1 p; 4 1 p
21p
b. } 5 0.6 c. 1 pt d. 5 pt
41p
Practice Level B
1. 29, 9 2. 212 3. 27 4. 18 5. 25
1
6. 24, 4 7. 21 8. no solution 9. 2}
3
10. x 2 3 11. 2(x 2 1) 12. (x 2 2)(x 2 3)
9
13. no solution 14. 2} 15. no solution
5
17
16. 24, 2} 17. no solution 18. 3, 6
7
19. 1 pt 20. a.
Person
Fraction
of room
papered
each hour
Time
(hours)
Fraction
of room
papered
Assistant
}
1
x
3
}
Expert
}
3
2x
3
}
4. (3x 2 1)(x 1 6) 5. (x 1 5)2(x 1 3)
7r 1 24
x2 2 15
}}
6. }
7.
(x 2 3)(x 1 3)(x 2 5)
18r3
13t 2 11
8. }}
(t 2 7)(t 1 3)
Problem Solving Workshop:
Worked Out Example
270r 2 2025
1. t 5 }; 7.5 hours
r(r 2 15)
50r
2. t 5 }}; 3.75 hours
(r 1 5)(r 2 15)
A62
Algebra 1
Chapter 12 Resource Book
3
x
9
2x
9
3
b. Answers will vary. c. } 1 } 5 1; 7.5 h
2x
x
Practice Level C
}
1
1
4
1. } 2. 62Ï 2 3. 6 4. 3 5. } 6. }
4
2
3
Copyright © Holt McDougal. All rights reserved.
x16
25
20. } 21. }}
x23
(x 1 1)(x 1 4)(x 2 4)
Lesson 12.7, continued
}
in the r% account and $3000 in the (r 1 1)%
account
19. a.
3.50
3.40
3.30
3.20
3.10
3.00
0.00
ANSWERS
3
11. 22, 21 12. 21 13. no solution 14. 2}
2
1
15. 5 16. a 5 }; b 5 22 17. 4 pt 18. $1000
3
1175.8 1 213.3x
6. a. C 5 }}}
(1 1 0.107x)(356.2 1 28.1x 2 3.2x 2)
b. $3.34
c.
C
Average cost
(dollars per passenger)
}
5
3 6 Ï89
5 6 Ï35
19
7. } 8. } 9. 2} 10. 2}, 21
3
10
2
3
0 1 2 3 4 x
Years since 1997
Person
Fraction
of roof
completed
each hour
Time
(hours)
Fraction
of roof
completed
Expert
}
1
x
24
}
Assistant
}
3
5x
24
}
24
x
72
5x
72
24
b. Answers will vary. c. } 1 } 5 1; 38.4 h
5x
x
Challenge Practice
1
1. a 5 }, b 5 4 2. a 5 1, b 5 9
2
d. 64 h
3. a 5 24, b 5 4 4. a 5 49, b 5 5
Review for Mastery
5. a 5 119, b 5 47 6. Yes, 20,000 barrels of
1. 8, 6 2. 29, 12 3. 27.5 4. 23, 9
Copyright © Holt McDougal. All rights reserved.
The average cost decreased from 1997 to 1998
and then increased from 1998 to 2002. You cannot
use the graph to describe how the amount of
passenger fares changed during the period because
this graph just shows the average cost. You do not
know what happened to the number of passengers
during this period.
Problem Solving Workshop:
Mixed Problem Solving
6
8
1. a. t 5 } 1 } b. 5 hours
x22
x
(67 1 168x)(1 1 0.001x)
2. a. A 5 }} b. The best
(1 2 0.003x)(355 1 555x)
approximation of the model for the years shown
is 1964. In 1964, Clemente had a career batting
average of 0.307 and the estimate of the model is
0.306. 3. Answers will vary.
6003 1 1216.23x 1 58.05x 2
4. a. S 5 }}}
1 1 0.17x 1 0.007x 2
b. about 6,684,000 students 5. 5
87 octane and 40,000 barrels of 93 octane.
7. Yes, 60,000 barrels of 87 octane and 30,000
barrels of 93 octane. 8. The order cannot be
filled. 9. Yes, 16,667 barrels of 87 octane and
3333 barrels of 93 octane for the 88 octane order,
and 13,333 barrels of 87 octane and 26,667 barrels
of 93 octane for the 91 octane order.
Chapter Review Game
x(x 1 4)
x13
1. y 5 2 2. 4x 1 5 3. } 4. }
2x 1 1
2(x 2 2)
3x
213x 2 35
5. }} 6. }} 7. x 5 9
(x 1 5)(x 1 6)
x(x 1 3)(x 1 7)
8. x 5 0 or x 5 2
VINCULUM
Algebra 1
Chapter 12 Resource Book
A63