Name Class 9-1
Date Additional Vocabulary Support
Quadratic Graphs and Their Properties
Complete the vocabulary chart by filling in the missing information.
Word or
Word Phrase
Definition
Picture or Example
axis of symmetry
The fold or line that divides the
parabola into two matching halves
y
6
4
2
x
−4
parabola
−2
1. The graph of a quadratic
function is a U-shaped curve.
O
6
2
4
y
4
HSM11ALTR_0902_T00101
2
x
−4
−2
O
2
4
−2
quadratic function
A function that can be written in the
form y 5 ax2 1 bx 1 c, where a 2 0
2. y 5 x2 1 16
y 5 3x2 2 5x 1 1
HSM11ALTR_0902_T00102
quadratic parent
function
vertex
The simplest quadratic function
f (x) 5 x2 or y 5 x2
3. y 5 x2 out of
y 5 x2, y 5 2x2 , and
y 5 3x2
4. The highest or lowest point on a
parabola, which is on the axis of
symmetry
4
y
2
x
−4
−2
O
2
4
−2
−4
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HSM11ALTR_0902_T00103
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1
Name
Class
9-1
Date
Think About a Plan
Quadratic Graphs and Their Properties
Physics In a physics class demonstration, a ball is dropped from the roof of a
building, 72 feet above the ground. The height h (in feet) of the ball above the
ground is given by the function h 5 216t2 1 72 , where t is the time in seconds.
a. Graph the function.
b. How far has the ball fallen from time t 5 0 to t 5 1?
c. Reasoning Does the ball fall the same distance from time t 5 1 to time
t 5 2 as it does from t 5 0 to t 5 1? Explain?
1. Complete the following table of values.
t
h âŹ16t2 à72
(t, h)
0
72
(0, 72)
1
56
(1, 56)
2
8
(2, 8)
3
Ź72
(3, Ź72)
2. Use the completed table to graph the function h 5 216t2 1 72 .
72
h
48
24
3. What was the height of the ball at t 5 0? 72 ft
What was the height of the ball at t 5 1? 56 ft
t
O
Ź24
Ź48
Ź72
How far has the ball fallen from time t 5 0 to t 5 1? 16 ft
4. What is the height of the ball at t 5 2? 8 ft
How far has the ball fallen from time t 5 1 to t 5 2? 48 ft
5. Does the ball fall the same distance from time t 5 1 to t 5 2 as it does from
t 5 0 to t 5 1? Explain.
no, it falls three times farther
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2
2
4
6
8
Name
Class
Date
Practice
9-1
Form G
Quadratic Graphs and Their Properties
Identify the vertex of each graph. Tell whether it is a minimum or a maximum.
1.
2.
y
2
x
4
3.
y
2
Ź2
4
x
4
2
Ź2
(1, 23); minimum
y
x
2
Ź2
(3, 21); minimum
4
(1, 5); maximum
Graph each function.
4. f (x) 5 3x2
10
Ź4 Ź2
5. f (x) 5 22.5x2
y
y
2
Ź4 Ź2 O
x
1
6. f (x) 5 25 x2
y
4
8
Ź2
Ź2
6
Ź4
Ź4
4
Ź6
Ź6
2
Ź8
Ź8
Ź10
Ź10
2
O
4
x
2
O
Ź4 Ź2
x
4
Order each group of quadratic functions from widest to narrowest graph.
7. y 5 23x2, y 5 25x2, y 5 21x2
8. y 5 4x2, y 5 22x2, y 5 26x2
2x2; 23x2; 25x2
22x2; 4x2; 26x2
1
9. y 5 x2, y 5 3 x2, y 5 2x2
1 2
3x ;
1
1
1
10. y 5 6 x2, y 5 4 x2, y 5 2 x2
1 2 1 2 1 2
6x ; 4x ; 2x
x2; 2x2
Graph each function.
11. f (x) 5 x2 1 1
12. f (x) 5 x2 2 2
y
8
8
6
4
2
O
4
x
1
14. f (x) 5 22 x2 1 5
6
18
4
12
Ź4 Ź2
2
4
x
Ź4 Ź2
x
2
4
2
4
x
16. f (x) 5 5x2 2 10
y
2
O
4
40
x
y
30
Ź4
2
O
Ź6
15. f (x) 5 23x2 2 4
Ź4 Ź2
y
6
O
y
4
O
24
Ź2
Ź2
Ź4 Ź2
y
2
2
Ź4 Ź2
13. f (x) 5 2x2 1 1
Ź8
20
Ź12
10
Ź2
Ź16
Ź4 Ź2
Ź4
O
Ź10
Ź20
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2
4
x
Name
9-1
Class
Date
Practice (continued)
Form G
Quadratic Graphs and Their Properties
17. For a physics experiment, the class drops a golf ball off a bridge
y
toward the pavement below. The bridge is 75 feet high. The function
h 5 216t2 1 75 gives the golf ball’s height h above the pavement
(in feet) after t seconds. Graph the function. How many seconds
does it take for the golf ball to hit the pavement?
100
80
60
40
20
about 2.2 s
x
O
18. A relief organization flew over a village and dropped a package of
food and medicine. The plane is flying at 1000 feet. The function
h 5 216t2 1 1000 gives the package’s height h above the ground
(in feet) after t seconds. Graph the function. How many seconds
does it take for the package to hit the ground?
1
2
3
4
y
1000
800
600
400
200
about 8 s
x
O
1 2 3 4 5 6 7 8 9 10
Identify the domain and range of each function.
1
20. y 5 22 x2 1 3
D: all real numbers; R: y K 3
19. y 5 5x2 2 5
D: all real numbers; R: y L 25
3
21. y 5 5x2 2 2
22. f (x) 5 29x2 1 1
D: all real numbers; R: y L 22
D: all real numbers; R: f (x) K 1
Use a graphing calculator to graph each function. Identify the vertex and axis of
symmetry.
23. y 5 2.75x2 1 3
(0, 3); x 5 0;
5
1
24. y 5 23 x2 2 8
(0, 28); x 5 0;
25. y 5 22x2 1 7
(0, 7); x 5 0;
26. Writing Discuss how the function y 5 x2 1 4 differs from the graph y 5 x2 .
The parent function of y 5 x2 1 4 is y 5 x2 . Both graphs open the same width and
are parabolas that open up. The graph of y 5 x2 has a vertex of (0, 0). The graph of
y 5 x2 1 4 has a vertex of (0, 4) — it is 4 units above the graph of y 5 x2 .
27. Writing Explain how you can determine if the parabola opens up or down by
simply examining the equation.
The coefficient of the x2 term determines if the parabola opens up or down. A
positive coefficient, the graph opens up; a negative coefficient, the graph opens
down.
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4
Name
Class
Date
Practice
9-1
Form K
Quadratic Graphs and Their Properties
Identify the vertex of each graph. Tell whether it is a maximum or a minimum.
1.
2.
y
y
2
2
x
2
2
x
(22, 1); minimum
(3, 2); maximum
2
2
2
Graph each function.
3. f (x) 5 5x2
4. f (x) 5 23x2
y
y
4
4
2
2
x
x
Ź4 Ź2
2
O
4
Ź4 Ź2
2
O
Ź2
Ź2
Ź4
Ź4
4
3
6. f (x) 5 25x2
2
5. f (x) 5 23x2
y
y
4
4
2
2
x
Ź4 Ź2
2
O
x
4
Ź4 Ź2
O
Ź2
Ź2
Ź4
Ź4
2
4
Order each group of quadratic functions from widest to narrowest graph.
1
1
8. y 5 3 x2, y 5 3x2, y 5 6 x2
y 5 16 x2, y 5 13 x2, y 5 3x2
7. y 5 22x2, y 5 24x2, y 5 23x2
y 5 22x2, y 5 23x2, y 5 24x2
Graph each function.
9. f (x) 5 x2 1 3
6
4
4
2
x
O
4
2
x
x
2
Ź2
y
y
y
Ź4 Ź2
1
11. f (x) 5 23 x2 2 1
10. f (x) 5 x2 2 5
2
4
Ź4 Ź2
O
2
4
O
Ź4 Ź2
Ź2
Ź2
Ź4
Ź4
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2
4
Name
Class
Date
Practice (continued)
9-1
Form K
Quadratic Graphs and Their Properties
12. Jared is casting his fishing line with a lead sinker attached over the edges
of a pier. The pier is 15 feet above the water. The function h 5 216t2 1 15
gives the sinker’s height h above the water (in feet) after t seconds. Graph the
function. How many seconds does it take for the sinker to hit the water?
20
about 0.97 s
h
16
12
8
4
t
0.25 0.5 0.75
O
1.25 1.5 1.75
1
2
13. A roofer is going to drop his hammer to the ground from the roof after making
sure the area is clear. The roof is 25 feet high. The function h 5 216t2 1 25
gives the hammer’s height h above the ground (in feet) after t seconds. Graph
the function. How many seconds does it take for the hammer to hit the
ground?
25
1.25 s
h
20
15
10
5
t
O
0.25 0.5 0.75
1
1.25 1.5 1.75
2
Identify the domain and range of each function.
1
15. y 5 24 x2 2 2
The domain is all real numbers.
The range is y K 22.
14. y 5 4x2 2 3
The domain is all real numbers.
The range is y L 23.
2
16. y 5 3 x2 1 1
The domain is all real numbers.
The range is y L 1.
17. f (x) 5 22x2 1 6
The domain is all real numbers.
The range is y K 6.
18. Writing Discuss how the graph of y 5 x2 2 7 differs from the graph of y 5 x2 .
The graph of y 5 x2 2 7 is shifted 7 units down.
19. Writing Explain how you can determine if the parabola has been shifted up
or down by examining the equation.
If the equation is in the form y 5 ax2 1 c, the sign of c determines
whether the parabola is shifted up or down. If c is positive, the parabola is
shifted up c units. If c is negative, the parabola is shifted down c units.
20. Open-Ended Write the equation of a quadratic function for which the graph
opens in the same direction as the graph of y 5 x2 , is wider than the graph of
y 5 x2 , and is shifted up compared to the graph of y 5 x2 .
Sample answer: y 5 0.25x2 1 3
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Name
Class
Date
Standardized Test Prep
9-1
Quadratic Graphs and Their Properties
Multiple Choice
For Exercises 1–4, choose the correct letter.
Ź2
1. What is the vertex of the parabola shown at the right? C
A. (21, 0)
C. (1, 24)
B. (0, 23)
D. (3, 0)
y
x
2
Ź2
4
Ź2
2. Which of the following has a graph that is wider than
Ź4
the graph of y 5 3x2 1 2? G
F. y 5 3x2 1 3
H. y 5 24x2 2 1
G. y 5 0.5x2 1 1
I. y 5 4x2 1 1
3. Which graph represents the function y 5 22x2 2 5? D
A.
B.
y
C.
y
4
4
Ź4
4
2
x
D.
Ź2
2
x
y
Ź2
y
Ź4
x
Ź2
2
4. What is the order, from narrowest to widest graph, of the quadratic functions
f (x) 5 210x2, f (x) 5 2x2, and f (x) 5 0.5x2 ? F
F. f (x) 5 210x2, f (x) 5 2x2, and f (x) 5 0.5x2
G. f (x) 5 2x2, f (x) 5 210x2 , and f (x) 5 0.5x2
H. f (x) 5 0.5x2, f (x) 5 2x2, and f (x) 5 210x2
I. f (x) 5 0.5x2, f (x) 5 210x2, and f (x) 5 2x2
Short Response
5. A ball fell off a cliff into the river from a height of 25 feet. The function
h 5 230t2 1 25 gives the ball’s height h above the water after t seconds.
Graph the function. How much time does it take for the ball to hit the water?
Check students’ graphs; about 0.9 s:
[2] Both parts answered correctly.
[1] One part answered correctly.
[0] Neither part answered correctly.
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x
4
Ź4
Name
9-1
Class
Date
Enrichment
Quadratic Graphs and Their Properties
When you shift and stretch parent functions, the functions you get are part of the
parent function’s family. The family of quadratic functions has the parent function
y 5 x2 .
1. Graph y 5 22x2, y 5 2x2, y 5 x2 , and y 5 2x2 on the
y
24
same coordinate grid.
16
8
Ź4 Ź2
2
O
4
x
Ź8
Ź16
Ź24
2. Compare and contrast the four equations you graphed in Exercise 1.
The graphs of y 5 2x2 and y 5 22x2 are open the same width with the same
vertex and axis of symmetry. y 5 2x2 and y 5 x2 open up; y 5 22x2 and
y 5 2x2 open down. The graphs of y 5 x2 and y 5 2x2 are open more than
y 5 2x2 and y 5 22x2 .
3. How does changing the coefficient of x2 affect the graph?
The coefficient determines the width of the parabola and whether it opens up or
down.
4. Graph y 5 22x3, y 5 2x3, y 5 x3 and y 5 2x3 on the same
y
8
coordinate grid.
4
Ź4 Ź2
O
Ź4
Ź8
5. Compare and contrast the four equations you graphed in Exercise 1 with the
four equations you graphed in Exercise 4.
Like parabolas, the graphs with smaller coefficients are wider. A negative
coefficient changes the direction of the graph for both types of functions. For
y 5 x3 graphs, a negative coefficient models the negative slope of a line.
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2
4
x
Name
Class
Date
Reteaching
9-1
Quadratic Graphs and Their Properties
A U-shaped graph such as the one at the right is called a parabola.
y
• A parabola can open upward or downward.
8
• A parabola that opens upward has a minimum or lowest point.
6
• A parabola that opens downward has a maximum or highest point.
• The vertex of a parabola is its minimum or maximum point.
All parabolas have a line or axis of symmetry.
4
2
Ź2
O
2
Problem
What is the vertex of the graph below? Is it a minimum or maximum?
y
2
O
x
Ź4 Ź2
Ź2
The graph opens downward, so you are looking for the highest point. The vertex is
(23, 2) and it is a maximum.
Exercises
Identify the vertex of each graph. Tell whether it is a minimum or a maximum.
y
1.
2.
4
2
O
x
2
4
(3, 1); minimum
O y
x
2
3.
4
Ź6 Ź4 Ź2
y O
x
Ź2
Ź2
Ź4
Ź4
Ź6
Ź6
(1, 26); minimum
(23, 21); maximum
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x
Name
Class
Date
Reteaching (continued)
9-1
Quadratic Graphs and Their Properties
Any function in the form y 5 ax2 1 bx 1 c where a 2 0 is called a quadratic
function. The graph of a quadratic function is a parabola.
Problem
What is the graph of y 5 12x2 2 4?
1
This is a quadratic function where a 5 2, b 5 0 and c 5 24. The graph will be a
parabola. Use a table to find some points on the graph. Then use what you know
about parabolas to complete the graph.
x
1
y â x2 Ź4
2
4
(x, y)
Ź4
yâ
1
(Ź4)2 Ź4 â4
2
Ź2
yâ
1
(Ź2)2 Ź4 âŹ2
2
0
yâ
1
(0)2 Ź4 âŹ4
2
(0, Ź4)
2
yâ
1
(2)2 Ź4 âŹ2
2
(2, Ź2)
4
yâ
1
(4)2 Ź4 â4
2
y
2
x
(Ź4, 4)
2
Ź4
4
Ź2
(Ź2, Ź2)
(4, 4)
Exercises
Graph each function.
4. y 5 2x2 1 5
5. y 5 x2 2 4
4
4
2
2
O
2 y
y
y
Ź4 Ź2
6. y 5 2x2 2 1
2
4
x
Ź4 Ź2
Ź4 Ź2
O
Ź2
O
2
4
x
Ź4
Ź2
Ź2
Ź6
Ź4
Ź4
Ź8
Ź10
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4
x
Name Class 9-2
Date Additional Vocabulary Support
Quadratic Functions
A ball is thrown into the air with an upward velocity of 12 meters per second. Its
height h in meters after t seconds is given by the function h 5 216t2 1 12t 1 5.
How long will it take the ball to reach its maximum height? What is the ball’s
maximum height? What is the range of the function?
Derrick wrote these steps to solve the problem on note cards, but they got
mixed up.
Find the h-coordinate of
Find the t-coordinate of the
2b
the vertex by plugging the
.
vertex with the formula t 5
2a
t-coordinate of the vertex into
the function. So, the vertex is
(0.375, 7.25).
The range of the function is
5 # h # 7.25.
Substitute 12 for b and 216 for a.
So, the t-coordinate of the vertex
is 0.375.
The ball will reach its maximum
height of 7.25 meters at 0.375
seconds.
Use the note cards to complete the steps below.
1. First,
find the t-coordinate of the vertex with the formula t 5 2b
2a .
2. Second,
substitute 12 for b and 216 for a. So, the t-coordinate of the vertex is 0.375.
3. Next, find the h-coordinate of the vertex by plugging the t-coordinate of the
vertex into the function. So, the vertex is (0.375, 7.25).
4. Then, the ball will reach its maximum height of 7.25 meters at 0.375 seconds.
5. Finally, the range of the function is 5 K h K 7.25.
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11
Name
Class
9-2
Date
Think About a Plan
Quadratic Functions
Business A cell phone company sells about 500 phones each week when it
charges $75 per phone. It sells about 20 more phones per week for each $1
decrease in price. The company’s revenue is the product of the number of phones
sold and the price of each phone. What price should the company charge to
maximize its revenue?
1. Let d 5 the total amount of dollar decrease to the price. Let r 5 the
company’s revenue. Write a quadratic function that reflects the company’s
revenue.
Revenue equals 500 phones plus d times 20 phones times $75 less d.
r5 a
r5
500
1 ad 3
20
bb 3 a
75
2 db
(20d 1 500)(75 2 d), or 220d2 1 1000d 1 37,500
r5
220(d2 2 50d 2 1875)
r5
220(d 1 25)(d 2 75)
2. Find the vertex of the quadratic function above. How will finding the vertex
help you determine at what price the company should charge to maximize its
revenue?
(25, 50,000); The vertex tells the amount of decrease to the price that results
in the maximum revenue.
3. What price should the company charge?
$50
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12
Name
Class
Date
Practice
9-2
Form G
Quadratic Functions
Find the equation of the axis of symmetry and the coordinates of the vertex of
the graph of each function.
1. y 5 4x2 2 2
(0, 22); x 5 0
2. y 5 2x2 1 4x 2 6
(2, 22); x 5 2
3. y 5 x2 1 4x 1 5
(22, 1); x 5 22
4. y 5 x2 2 8x 1 12
(4, 24); x 5 4
5. y 5 26x2 1 3
(0, 3); x 5 0
6. y 5 23x2 1 12x 2 7
(2, 5); x 5 2
7. y 5 2x2 1 x 2 14
8. y 5 26x2 2 8x 1 10
9. y 5 22x2 1 3x 1 6
1
1
2
1
2
3
2
Q 2 3 , 123 R ; x 5 2 3
Q 2 4 , 2148 R ; x 5 2 4
3
1
Q 4 , 78 R ; x 5 4
Graph each function. Label the axis of symmetry and the vertex.
10. f (x) 5 x2 2 2x 2 1
8
11. f (x) 5 22x2 1 8x 2 10
y
2
Ź4 Ź2
Ź2
2 x=1
Ź4
2
O
Ź2
4
x
x
2
2
Ź4 Ź2
4
2
Ź8
1
Ź16
y
y
x
4
(3, 16)
16
8
O
Ź2
(Ź0.5, 0.5)
Ź2 Ź1 O
x = Ź0.5
Ź1
Ź12
(3, 1)
2
15. f (x) 5 22x2 1 12x 2 2
3
(Ź1, Ź5)Ź4
O
Ź2
14. f (x) 5 2x2 1 2x 1 1
4
x=3
4
Ź8
y
y
6
4
(2, Ź2)
x=2
Ź6
(1, Ź2)
13. f (x) 5 23x2 2 6x 2 8
x = Ź1
Ź4 Ź2 O
2
O
4
4
8
x
6
Ź4 Ź2
12. f (x) 5 2x2 2 12x 1 19
y
1
2
x
x=3
2
4
6
Ź8
Ź16
16. A punter kicked the football into the air with an upward velocity of 62 ft/s. Its
height h in feet after t seconds is given by the function h 5 216t2 1 62t 1 2.
What is the maximum height the ball reaches? How long will it take the
football to reach the maximum height? How long does it take for the ball to hit
the ground?
62.06 ft; 1.94 s; about 3.91 s
17. A disc is thrown into the air with an upward velocity of 20 ft/s. Its height h in
feet after t seconds is given by the function h 5 216t2 1 20t 1 6. What is the
maximum height the disc reaches? How long will it take the disc to reach the
maximum height? How long does it take for the disc to be caught 3 feet off the
ground?
12.25 ft; 0.625 s; 1.385 s
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13
8
x
Name
Class
Date
Practice (continued)
9-2
Form G
Quadratic Functions
Graph each function. Label the axis of symmetry and the vertex.
3
18. f (x) 5 2 x2 1 6x 1 2
y
2
19. f (x) 5 3 x2 1 8x 1 5
8
Ź4 Ź2
(Ź2, Ź4)
x = Ź2
O
2
4
4
x
Ź12
Ź18
Ź18
(Ź6, Ź19)
Ź8
Ź24
Ź24
y
x = 12
12 18
(Ź8, Ź26)
3
22. f (x) 5 24 x2 1 2x 1 3
y
O
Ź4 Ź2
Ź12
Ź4
Ź24
Ź8
Ź36
Ź12
2
x = 4/3
4
y
16
12
x
x
24
5
23. f (x) 5 4 x2 2 4x 1 1
(4/3, 13/3)
4
6
6
Ź24 Ź18 Ź12 Ź6 O
x = Ź8
Ź6
Ź12
Ź4
12
O
O
y
Ź6
x
1
21. f (x) 5 2 x2 2 12x 1 11
Ź6
x
Ź16 Ź12 Ź8 Ź4
4
y
6
x = Ź6
1
20. f (x) 5 4 x2 1 4x 2 10
6
6
8
4 x = 8/5
x
O
Ź4 Ź2
2
4
(8/5, Ź11/5)
Ź4
Ź48
Ź60
(12, Ź61)
Open-Ended For Exercises 24–26, give an example of a quadratic function with
the given characteristic(s).
24. Its graph opens up and has its vertex at (0, 23).
Answers may vary. Sample: y 5 x2 2 3
25. Its graph lies entirely below the x-axis.
Answers may vary. Sample: y 5 2x2 2 2
26. Its vertex lies on the x-axis and the graph opens down.
Answers may vary. Sample: y 5 212 x2
27. A fountain that is 5 feet tall sprays water into the air with an upward velocity of
22 ft/s. What function gives the height h of the water in feet t seconds after it is
sprayed upward? What is the maximum height of the water?
h 5 216t2 1 22t 1 5; 12.6 ft
28. The parabola shown at the right is of the form
y
y 5 ax2 1 bx 1 c.
4
a. What is the y-intercept? 22
b. What is the axis of symmetry? x 5 21
2b
c. Use the formula x 5 2a to find b. b 5 4
d. What is the equation of the parabola? y 5 2 x2 1 4x 2 2
2
x
Ź4 Ź2
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14
O
Ź2
Ź4
2
4
Name
Class
Date
Practice
9-2
Form K
Quadratic Functions
Find the equation of the axis of symmetry and the coordinates of the vertex of
the graph of each function.
1. y 5 3x2 1 1 x 5 0; (0, 1)
2. y 5 x2 2 6x 1 2 x 5 3; (3, 27)
3. y 5 x2 2 8x 1 12 x 5 4; (4, 24)
4. y 5 22x2 2 5 x 5 0; (0, 25)
5. y 5 3x2 1 6x 2 8 x 5 21; (21, 211)
6. y 5 26x2 1 12x 2 3 x 5 1; (1, 3)
Graph each function. Label the axis of symmetry and the vertex.
7. f (x) 5 x2 1 6x 2 2
8. f (x) 5 23x2 1 9x 2 8
y
y
8
(X 5 1.5)
O
2
x
4
Ź8 Ź4
2
Ź4 Ź2
(X 5 23) x
4
O
4
Ź2
8
(1.5, 21.25)
Ź4
Ź4
Ź8
Ź6
(23, 211)
9. f (x) 5 4x2 2 8x 1 1
10. f (x) 5 5x2 1 10x 2 4
y
y
4
8
2
4
(X 5 1)
(X 5 21)
x
Ź4 Ź2
O
2
x
4
Ź8 Ź4
Ź2
Ź4
O
4
8
Ź4
(1, 23)
(21, 29)Ź8
11. A baseball player hit a ball with an upward velocity of 46 ft/s. Its height h in feet after
t seconds is given by the function h 5 216t2 1 46t 1 6. What is the maximum height
the ball reaches? How long will it take the baseball to reach the maximum height?
How long does it take for the ball to hit the ground? 39 ft; 1.44 s; 3 s
12. A golf ball is chipped into the air from a small hill with an upward velocity of 50 ft/s. Its
height h in feet after t seconds is given by the function h 5 216t2 1 50t 1 10. What
is the maximum height the ball reaches? How long will it take the ball to reach the
maximum height? How long does it take for the ball to hit the ground?
49ft; 1.56 s; 3.31 s
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15
Name
Class
Date
Practice (continued)
9-2
Form K
Quadratic Functions
Graph each function. Label the axis of symmetry and the vertex.
3
13. f (x) 5 4 x2 1 12x 2 2
1
14. f (x) 5 3 x2 1 4x 1 2
y
y
16
40
x
x
20
Ź40 Ź20 O
40
Ź16 Ź8
Ź8
Ź40
(26, 210) Ź16
2
15. f (x) 5 5 x2 2 8x 1 1
8
O
Ź20
(28, 250)
16
1
16. f(x) 5 22 x2 1 8x 2 6
y
y
(X 5 10)
(X 5 26)
8
20
(X 5 28)
40
(X 5 8)
20
(8, 26)
20
10
x
x
Ź40 Ź20 O
20
40
Ź20 Ź10 O
20
Ź10
Ź20
Ź40
10
(10, 239)
Ź20
For Exercises 17 and 18, give an example of a quadratic function with the given
characteristic(s). Justify your answer by graphing the function.
17. Its graph opens down and has its vertex at (0, 4).
y
Answers may vary. Sample:
4
2
y 5 2x 1 4;
2
x
Ź4 Ź2
O
2
4
Ź2
Ź4
18. Its graph opens upward and has its vertex at (0, 22).
y
Answers may vary. Sample:
4
2
y 5 x 2 2;
2
x
Ź4 Ź2
O
2
4
Ź2
Ź4
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16
Name
Class
Date
Standardized Test Prep
9-2
Quadratic Functions
Multiple Choice
For Exercises 1–5, choose the correct letter.
1. Which equation represents the axis of symmetry of the function
y 5 22x2 1 4x 2 6? B
A. y 5 1
B. x 5 1
C. x 5 3
D. x 5 23
2. What are the coordinates of the vertex of the graph of the function
y 5 2x2 1 6x 2 11? F
F. (3, 22)
H. (23, 229)
G. (3, 16)
I. (23, 220)
3. What are the coordinates of the vertex of the graph of the function
y 5 3x2 2 12x 1 3? C
B. (2, 215)
A. (22, 29)
C. (2, 29)
D. (3, 26)
4. Which graph represents the function y 5 3x2 1 12x 2 6? G
F.
8
G.
y
16
4
2
Ź2
16
I.
y
Ź6
2
Ź4 Ź2
y
4
x
x
4
8
8
8
x
Ź4
H.
y
2
Ź2
4
x
6
Ź4
2
Ź2
Ź4
Ź8
Ź8
Ź4
Ź8
Ź16
Ź16
Ź8
5. Which equation matches the graph shown at the right? D
A. y 5 8x2 1 2x 2 5
16
4
y
8
B. y 5 8x2 1 2x 1 5
x
C. y 5 2x2 1 8x 1 5
Ź6 Ź4 Ź2
D. y 5 2x2 1 8x 2 5
2
Ź8
Ź16
Short Response
6. A golf ball is driven in the air toward the hole from an elevated tee with an
upward velocity of 160 ft/s. Its height h in feet after t seconds is given by the
function h 5 216t2 1 160t 1 18. How long will it take for the golf ball to
reach its maximum height? What is the ball’s maximum height?
5 s; 418 ft
[2] Both parts answered correctly.
[1] One part answered correctly.
[0] Neither part answered correctly.
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17
Name
Class
Date
Enrichment
9-2
Quadratic Functions
A cubic function is one in the form f (x) 5 ax3 1 bx2 1 cx 1 d.
The most basic cubic function, y 5 x3 , is shown.
y
8
4
Ź8
Ź4
O
4
In Exercises 1–2, graph each function by making a table of values.
1. y 5 x3 1 4
8
2. y 5 x3 2 5
y
2
x
6
Ź4 Ź2
y
Ź4 Ź2
O
4
Ź2
2
Ź4
O
2
4
x
2
4
Ź6
Ź2
Ź8
3. What happens when you add or subtract a number from y 5 x3 ?
It shifts the graph up or down that many units; the y –intercept changes.
1
4. Graph y 5 2x3, y 5 x3, y 5 3 x3 and y 5 3x3 on the same coordinate grid.
y
8
4
x
Ź4 Ź2
O
2
4
Ź4
Ź8
5. Compare and contrast the graphs of the four equations in Exercise 4.
The graphs all go through the origin. Functions with a negative coefficient of x3
opens up on the left and down on the right. With a positive coefficient of x3
the graph opens down on the left and up on the right. The equation with a
coefficient of 3 has the most narrow graph, followed by the coefficients of 1
and 21. The function with a coefficient of 13 has the widest graph.
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18
8
x
Name
Class
9-2
Date
Reteaching
Quadratic Functions
Recall that the general equation for a quadratic function is y 5 ax2 1 bx 1 c.
Using this general equation, the equation for the axis of symmetry is x 5 2b
2a .
Since the vertex lies on the axis of symmetry, the x-coordinate of the vertex is 2b
2a .
Problem
What are the equation of the axis of symmetry and the coordinates of the vertex
of the graph of y 5 3x2 1 6x 2 4?
x 5 2b
2a
x5
Equation for axis of symmetry
26
2(3)
a 5 3 and b 5 6
x 5 21
Simplify.
Now, find the value of y when x 5 21.
y 5 3x2 1 6x 2 4
y 5 3(21)2 1 6(21) 2 4
y 5 27
The equation of the axis of symmetry is x 5 21 and the coordinates of the vertex
of the graph are (21, 27).
Exercises
Find the equation of the axis of symmetry and the coordinates of the vertex of
the graph of each function.
1. y 5 x2 1 8x
2. y 5 2x2 1 12x 1 10
(24, 216); x 5 24
4. y 5 2x2 2 4x 2 5
(1, 27); x 5 1
5. y 5 23x2 1 18x 2 25
(3, 2); x 5 3
7. f (x) 5 6x2 2 7
(0, 27), x 5 0
(23, 28); x 5 23
8. f (x) 5 25x2 2 10x 1 1
(21, 6); x 5 21
3. y 5 2x2 1 4x 2 8
(2, 24); x = 2
6. y 5 22x2 1 2x 2 6
1
11
9. f (x) 5 4x2 2 16x 2 2
(2, 218); x 5 2
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19
1
Q 2, 2 2 R ; x 5 2
Name
Class
Date
Reteaching (continued)
9-2
Quadratic Functions
You can use the axis of symmetry and the vertex to help graph a quadratic
b
equation. Use the equation x 5 22a
to find the equation of the axis of symmetry.
Because the vertex lies on the axis of symmetry, this value is also the x-coordinate
of the vertex.
Problem
What is the graph of y 5 2x2 2 4x 1 1?
1. Find the equation of the axis
2. Find the vertex.
y 5 2x2 2 4x 1 1
of symmetry.
x 5 2b
2a
2(24)
x5
2(2)
y 5 2(12) 2 4(1) 1 1
y 5 21
a 5 2 and b 5 24
x51
x51
Simplify.
The vertex is (1, 21)
Simplify.
3. Graph the axis of symmetry x 5 1 and the vertex (1, 21) .
(Ź1, 7)
4. Find a couple points on the graph.
y
6
For x 5 0, y 5 2(02) 2 4(0) 1 1 or 1.
Plot (0, 1).
(0, 1)
For x 5 21, y 5 2(21)2 2 4(21) 1 1 or 7.
O
Ź2
Plot (21, 7) .
x
2
(1, Ź1)
x â1
Ź2
5. Use the axis of symmetry to complete the graph.
Exercises
Graph each function. Label the axis of symmetry and the vertex.
10. y 5 x2 2 3
11. y 5 2x2 2 4x 1 1
y
O
4
4
2
x=0
Ź4 Ź2
y
y
(Ź2, 5)
4
12. y 5 2x2 1 8x 1 6
2
2
2
4
x
Ź4 Ź2
Ź2
Ź4 (0, Ź3)
2
O
4
x
O
Ź2
(Ź2, Ź2)
x = Ź2 Ź4
Ź2
x = Ź2
Ź4 Ź2
Ź4
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20
2
4
x
Name Class 9-3
Date Additional Vocabulary Support
Solving Quadratic Equations
Use the list below to complete the Venn diagram.
A quadratic equation can
have two, one, or no
real-number solutions.
The equation has an
x2 -term and a constant
term, but no x-term.
The x-intercepts show the
solutions of the equation.
x2 1 2x 5 0
x2 2 36 5 0
You can write the equation
in the form x2 5 k.
Solve Using
Square Roots
Solve Using
a Graph
The x-intercepts show the
A quadratic
The equation has an x2 -term,
solutions of the equation.
equation can
and a constant term, but no
x2 1 2x 5 0
have two,
x-term.
one, or no
You can write the equation
real-number
solutions.
in the form x2 5 k.
x2 2 36 5 0
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21
Name
9-3
Class
Date
Think About a Plan
Solving Quadratic Equations
Quilting You are making a square quilt with the design shown at the
right. Find the side length of the inner square that would make the area of
the inner square equal to 50% of the total area of the quilt. Round to the
nearest tenth of a foot.
x
6 ft
1. What is an expression for the area of the inner
x2
square?
36 ft 2
2. What is the area of the entire quilt?
3. What is 50% of the area of the entire quilt?
18 ft 2
4. Write an equation for the area of the inner square using the expressions from
x2 5 18
Steps 1 and 3.
5. Solve the quadratic equation.
x 5 w 3 !2
6. Which solution to the quadratic equation best describes the side length of the
inner square? Explain.
the positive solution, because length cannot be negative
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22
Name
9-3
Class
Date
Practice
Form G
Solving Quadratic Equations
Solve each equation by graphing the related function. If the equation has no
real-number solution, write no solution.
1. x2 2 16 5 0 4; 24
2. x2 1 12 5 0 no solution
3. 2x2 2 18 5 0 3; 23
4. 7x2 5 0 0
1
5. 2 x2 2 2 5 0 2; 22
6. x2 1 49 5 0 no solution
7. x2 2 15 5 215 0
8. 4x2 2 36 5 0 3; 23
9. x2 1 36 5 0 no solution
Solve each equation by finding square roots. If the equation has no real-number
solution, write no solution.
10. t2 5 25 5; 25
11. k2 5 484 22; 222
12. z2 2 256 5 0 16; 216
13. d2 2 14 5 250
no solution
14. 9y2 2 16 5 0
15. 2g2 2 32 5 232
0
16. 4a2 5 36 3; 23
17. 7x2 1 28 5 0 no solution 18. 6n2 2 54 5 0 3; 23
19. 81 2 c2 5 0 9; 29
20. 16x2 2 49 5 0 4; 2 4
4
3;
2 43
7
7
21. 64 1 j2 5 0 no solution
Model each problem with a quadratic equation. Then solve. If necessary, round
to the nearest tenth.
22. Find the side length of a square with an area of 196 ft2 .
x2 5 196; 14 ft
23. Find the radius of a circle with an area of 100 in2.
πr2 5 100; 5.6 in.
24. Find the side length of a square with an area of 50 cm2 .
x2 5 50; 5 !2 cm or 7.1 cm
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23
Name
Class
Date
Practice (continued)
9-3
Form G
Solving Quadratic Equations
25. The square tarp you are raking leaves onto has an area of 150 ft2 . What is the
side length of the tarp? Round your answer to the nearest tenth of a foot if
necessary.
12.2 ft
26. There is enough mulch to spread over a flower bed with an area of 85 m2 .
What is the radius of the largest circular bed that can be covered by the
mulch? Round your answer to the nearest tenth of a meter if necessary.
5.2 m
Mental Math Tell how many solutions each equation has.
27. q2 2 22 5 222
one
28. m2 1 15 5 0
none
29. b2 2 12 5 12
two
Solve each equation by finding square roots. If the equation has no real-number
solution, write no solution. If a solution is irrational, round to the nearest tenth.
30. 3.35z2 1 2.75 5 214
no solution
31. 100t2 1 36 5 100
0.8; 20.8
1
32. 5a2 2 125 5 0
0.04; 20.04
1
33. 3h2 2 12 5 0
6; 26
1
34. 22 m2 1 5 5 210
5.5; 25.5
35. 11x2 2 0.75 5 3.21
0.6; 20.6
36. Find the value of n such that the equation x2 2 n 5 0 has 24 and 224 as
solutions.
576
Find the value of x for the square and triangle. If necessary, round to the nearest tenth.
37.
38.
2.9 in.
4.6 m
3x
34 in.2
95 m2
2x
3x
39. Writing Explain how the number of solutions for a quadratic equation relates
to the graph of the function.
When there is no solution, the graph does not cross the x –axis. When there is only
one solution, the vertex of the graph is on the x-axis. When the graph has two
x-intercepts, the equation has two solutions.
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24
Name
Class
Date
Practice
9-3
Form K
Solving Quadratic Equations
Solve each equation by graphing the related function. If the equation has no
real-number solution, write no solution.
1. x2 1 9 5 0
no solution
2. x2 2 36 5 0
w6
y
16
8
20
x
Ź16 Ź8
O
8
y
40
x
16
O
Ź8 Ź4
Ź8
4
8
2
4
4
8
Ź20
Ź16
Ź40
3. 4x2 5 0
0
1
4. 9 x2 2 1 5 0
w3
y
4
y
4
2
2
x
x
Ź4 Ź2
O
2
4
Ź4 Ź2
O
Ź2
Ź2
Ź4
Ź4
5. x2 2 21 5 221
y
0
6. 2x2 2 32 5 0
w4
y
40
4
20
2
x
x
Ź4 Ź2
O
2
4
Ź8 Ź4
O
Ź2
Ź20
Ź4
Ź40
Solve each equation by finding square roots. If the equation has no real-number
solution, write no solution.
7. z2 5 49 w7
8. f 2 5 256 w16
3
10. 16n2 2 36 5 0 w2
9. h2 2 25 5 2125 no solution
11. 6c2 5 24 w2
12. 5p2 1 45 5 0 no solution
13. 64 2 a2 5 0 w8
9
14. 49t2 2 81 5 0 w7
Model each problem with a quadratic equation. Then solve. If necessary, round
to the nearest tenth.
15. Find the length of a side of a square with an area of 225 m 2 . s2 5 225; 15 m
16. Find the radius of a circle with an area of 121 yd 2 . πr2 5 121; 6.2 yd
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25
Name
Class
Date
Practice (continued)
9-3
Form K
Solving Quadratic Equations
17. The square yard you are mowing has an area of 9600 ft 2 . What is the side
length of the yard? Round your answer to the nearest tenth of a foot if
necessary. 98 ft
18. What is the radius of the largest circular quilt that can be made with an area
less than or equal to 70 ft 2 ? Round your answer to the nearest tenth of a foot if
necessary. 4.7 ft
Mental Math Tell how many solutions each equation has.
19. m2 1 46 5 46 1
20. w2 2 72 5 0 2
Solve each equation by finding square roots. If the equation has no real-number
solution, write no solution. If a solution is irrational, round to the nearest tenth.
21. 25n2 1 44 5 144 w2
3
22. 24 y2 1 5 5 222 w6
1
23. 2 a2 2 8 5 0 w4
24. 2.68b2 1 4.75 5 22.25 no solution
Find the value of x for the square and triangle. If necessary, round to the nearest
tenth.
25.
26.
1.3 ft
28 ft2
4.8 in.
2x
4x
46 in.2
2x
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26
Name
9-3
Class
Date
Standardized Test Prep
Solving Quadratic Equations
Multiple Choice
For Exercises 1–7, choose the correct letter.
1. What is the solution of n2 2 49 5 0? C
A. 27
B. 7
C. 47
D. no solution
2. What is the solution of x2 1 64 5 0? I
F. 25
G. 8
H. 48
I. no solution
3. What is the solution of a2 1 17 5 42? C
A. 25
B. 5
C. 45
D. no solution
4. What is the side length of a square with an area of 144x2 ? G
F. 12
G. 12x
H. 412x
5. What is the value of b in the triangle shown at the right? B
A. 24 in.
B. 4 in.
C. 44 in.
D. no solution
I. no solution
3b
24 in.2
b
6. What is the radius of a sphere whose surface area is 100 square centimeters?
Use the formula for determining the surface area of a sphere, S 5 4πr2 , and
3.14 for π. Round your answer to the nearest hundredth. F
F. 2.82 cm
G. 5 cm
H. 5.64 cm
I. 125,600 cm
7. What is the value of z so that 29 and 9 are both solutions of x2 1 z 5 103? C
A. 222
B. 3
C. 22
D. 184
Extended Response
8. A ball is dropped from the top of a building that is 250 feet tall. The height h of
the ball in feet after t seconds is modeled by the function h 5 216t2 1 250.
Round to the nearest tenth if necessary.
a. How long will it take for the ball to reach the ground? Show your work. 4 s
b. How long will it take for the ball to reach a height of 75 feet? Show your
work. 3.3 s
[2] Both parts answered correctly.
[1] One part answered correctly.
[0] Neither part answered correctly.
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27
Name
9-3
Class
Date
Enrichment
Solving Quadratic Equations
To simplify a square root, there must be no perfect square factors other than 1 in
the radicand. You can use the rule below to help you simplify square roots.
The square root of a product equals the product of the square roots of the factors.
For example, !9x 5 !9 ? !x 5 3 !x.
Problem
Simplify "49a2b4 ? "36a6b8 .
"49a2b4 ? "36a6b8 5 Q !49 ? "a2 ? "b4 R ? Q !36 ? "a6 ? "b8 R
5 (7 ? a ? b2) ? (6 ? a3 ? b4)
5 42a4b6
Exercises
Simplify each expression.
1. "64a6b10 ? "81a8b4
2. "25m12n8 ? "16m6n2
72a7b7
20m9n5
3. "100x20y14 ? "121x4y12
4. "144r22s14 ? "49r6s8
84r14s11
110x12 y13
5. "169j14k8 ? "100j 8k16
6. "225p10q16 ? "196p2q4
130j11k12
210p6q10
7. "36m18n10 ? "81m4n6
8. "256x24y4 ? "49x6y8
112x15y6
54m11n8
9. "4a8b6c4 ? "16a10b12c6
10. "49r6s10t12 ? "25r12s8t4
35r9s9t8
8a9b9c5
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28
Name
Class
Date
Reteaching
9-3
Solving Quadratic Equations
An equation in the form ax2 1 bx 1 c 5 0 where a 2 0 is called a
quadratic equation. Its related quadratic function is y 5 ax2 1 bx 1 c. If you
graph the related quadratic function, the solutions of the quadratic equation are
x-values where the graph crosses the x-axis.
A linear equation can have only one solution. However, a quadratic equation can
have 2, 1, or 0 real-number solutions.
The related function of
2x2 1 4 5 0 is
y 5 2x2 1 4. The graph of
y 5 2x2 1 4 is shown
below.
The related function of
x2 2 2x 1 1 5 0 is
y 5 x2 2 2x 1 1.
The graph of
y 5 x2 2 2x 1 1 is
shown below.
y
4
The related function of
x2 2 x 1 2 5 0 is
y 5 x2 2 x 1 2. The
graph of y 5 x2 2 x 1 2
is shown below.
y
4
2
2
x
Ź3
O
y
x
3
Ź2
O
x
2
Ź2
O
2
Ź2
The graph crosses the x-axis
where x 5 22 and x 5 2.
The equation 2x2 1 4 5 0
has two solutions, 22 and 2.
The graph touches the
x-axis where x 5 1.
The equation
x2 2 2x 1 1 5 0 has
one solution, 1.
The graph does not
touch the x-axis.
The equation
x2 2 x 1 2 5 0 has no
real-number solutions.
Exercises
Solve each equation by graphing the related function. If the equation has no
real-number solution, write no solution.
1. x2 1 3 5 0
2. x2 1 4x 1 4 5 0
22; 1
22
no solution
3. x2 1 x 2 2 5 0
4. How many times does the graph of y 5 x2 2 4 cross the x-axis? Explain.
twice; at x 5 2 and x 5 22
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29
Name
Class
Date
Reteaching (continued)
9-3
Solving Quadratic Equations
You can solve a quadratic equation by taking the square root of each side of the equation.
Problem
What are the solutions of 81x2 5 49?
81x2 5 49
81x2
49
81 5 81
Divide each side by 81.
49
x2 5 81
Simplify.
"x2 5 4Å49
81
Take the square root of each side.
7
x 5 49
Simplify.
Problem
What are the solutions of x2 1 9 5 0?
x2 1 9 5 0
x2 1 9 2 9 5 0 2 9
x2 5 29
Subtract 9 from each side.
Simplify.
Since x2 cannot equal 29 in the real numbers, x2 1 9 5 0 has no real-number
solutions.
Exercises
Solve each equation by finding square roots. If the equation has no real-number
solution, write no solution. If a solution is irrational, round to the nearest tenth.
5. x2 5 100
210; 10
8. 9x2 2 16 5 0
4
3;
2 43
11. 64x2 2 25 5 0
5
8;
2 58
6. x2 2 144 5 0
7. 5x2 2 125 5 0
5; 25
12; 212
9. 3x2 1 27 5 0
10. 7x2 2 49 5 0
no solution
2.6; 22.6
12. 3x2 2 30 5 0
3.2; 23.2
13. x2 1 7 5 0
no solution
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30
Name Class 9-4
Date Additional Vocabulary Support
Factoring to Solve Quadratic Equations
P roblem
What are the solutions of the equation x2 2 6x 5 28? Justify and explain your work.
Explain
Work
Justify
First, write the equation.
Second, add 8 to each
side to write in standard
form.
Then, factor x2 2 6x 1 8.
(x 2 4) (x 2 2) 5 0
Factor.
Next, use the Zero
Product Property.
x 2 4 5 0 or x 2 2 5 0
Use the Zero-Product Property.
Finally, solve for x to get
the solutions x 5 4 and
x 5 2.
x 5 4 or x 5 2
Solve for x.
x2 2 6x 5 28
x2 2 6x 1 8 5 0
Original equation
Write the equation in
standard form.
Solutions
x 5 4 or x 5 2
What are the solutions of the equation x2 2 3x 5 18? Justify and explain your work.
Explain
Work
x2 2 3x 5 18
Justify
First, write the equation.
__________________
Then, subtract 18 from
each side to write in
standard form.
__________________
x2 2 3x 2 18 5 0
Write the equation in
standard form.
________________________
Then, factor x 2 3x 2 18.
__________________
(x 1 3) (x 2 6) 5 0
Factor.
________________________
Next, use the Zero-Product
Property.
__________________
x 1 3 5 0 or x 2 6 5 0
Use the Zero-Product
Property.
________________________
2
Finally, solve for x to get
the solutions x 5 23 and
x 5 6.
__________________
Original equation
________________________
x 5 23 or x 5 6
Solutions
Solve for x.
________________________
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31
Name
9-4
Class
Date
Think About a Plan
Factoring to Solve Quadratic Equations
Sports You throw a softball into the air with an initial upward velocity of 38 ft/s
and an initial height of 5 ft.
a. Use the vertical motion model to write an equation that gives the ball’s height h
(in feet) at time t (in seconds).
b. The ball’s height is 0 ft when it is on the ground. Solve the equation you wrote
in part (a) for h 5 0 to find when the ball lands.
What do you know?
1. Write a vertical motion model that best describes the equation for the ball’s
height h at time t. What are the values of v and c?
h 5 216t2 1 v ? t 1 c
z
z
h 5 216t2 1 ?t1
38
5
z z
2. How would graphing the quadratic equation help you understand the
problem?
The graph would show the initial height, the maximum height (the vertex) and
when the softball would reach the ground (the x-intercept).
How do you solve the problem?
3. The ball’s height is 0 ft when it is on the ground. Solve the equation you wrote
in part (a) for h 5 0 to find when the ball lands.
(8t 1 1)(22t 1 5) 5 0; 52 s
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32
Name
Class
9-4
Date
Practice
Form G
Factoring to Solve Quadratic Equations
Use the Zero-Product Property to solve each equation.
1. (y 1 6)(y 2 4) 5 0 26 ; 4
2. (3f 1 2)( f 2 5) 5 0 5; 223
3. (2x 2 7)(4x 1 10) 5 0 72 ; 252
4. (8t 2 7)(3t 1 5) 5 0 87 ; 253
5. d(d 2 8) 5 0 0; 8
6. 3m(2m 1 9) 5 0 0; 292
Solve by factoring.
7. n2 1 2n 2 15 5 0
25 ; 3
10. 8x2 1 10x 1 3 5 0
8. a2 2 15a 1 56 5 0
7; 8
11. 3b2 1 7b 2 6 5 0
2
3;
234; 212
23
9. z2 2 10z 1 24 5 0
6; 4
12. 5p2 2 9p 2 2 5 0
2; 215
13. w2 1 w 5 12
3; 24
14. s2 1 12s 5 232
24; 28
15. d2 5 5d
0; 5
16. 3j 2 2 20j 5 212
17. 12y2 1 40y 5 7
18. 27r2 1 69r 5 8
2
3;
1
6;
6
2 72
1
9;
2 83
Use the Zero-Product Property to solve each equation. Write your solutions as a
set in roster form.
19. k2 2 11k 1 30 5 0
{6, 5}
20. x2 2 6x 2 7 5 0
{21 , 7}
21. n2 1 17n 1 72 5 0
{ 28 , 29 }
22. The volume of a sandbox shaped like a rectangular prism is 48 ft3 . The height
of the sandbox is 2 feet. The width is w feet and the length is w 1 2 feet. Use
the formula V 5 lwh to find the value of w.
4
23. The area of the rubber coating for a flat roof was 96 ft2 . The rectangular frame
the carpenter built for the flat roof has dimensions such that the length is
4 feet longer than the width. What are the dimensions of the frame?
8 ft by 12 ft
24. Ling is cutting carpet for a rectangular room. The area of the room is 324 ft2 .
The length of the room is 3 feet longer than twice the width. What should the
dimensions of the carpet be?
12 ft by 27 ft
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33
Name
Class
9-4
Date
Practice (continued)
Form G
Factoring to Solve Quadratic Equations
Write each equation in standard form. Then solve.
25. 21x2 1 5x 2 35 5 3x2 2 4x
26. 3n2 2 2n 1 1 5 23n2 1 9n 1 11
18x2 1 9x 2 35; 253; 76
6n2 2 11n 2 10; 52; 223
Find the value of x as it relates to each rectangle or triangle.
27. Area 5 60 cm2 6 cm
28. Area 5 234 yd2 13 yd
x
x
x à4
2x Ź8
29. Area 5 20 in.2 5 in.
30. Area 5 150 m2 12 m
x
x à3
2x à1
x
Reasoning For each equation, find k and the value of any missing solutions.
31. x2 2 kx 2 16 5 0 where 22 is one solution of the equation.
6; 8
32. x2 2 6x 5 k where 10 is one solution of the equation.
40; 24
1
33. kx2 2 13x 5 5 where 2 3 is one solution of the equation.
6; 52
34. Writing Explain how you solve a quadratic equation by factoring.
Write the equation in standard form equal to zero. Write two sets of
parentheses. Find factors of the x2 term. Find factors of the constant term.
Find the combination of factors whose sum equals the x-term.
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34
Name
9-4
Class
Date
Practice
Form K
Factoring to Solve Quadratic Equations
Use the Zero-Product Property to solve each equation.
1. (n 1 3)(n 2 2) 5 0 23, 2
2. (4a 1 2)(a 2 6) 5 0 6, 212
3. (5y 2 3)(2y 1 1) 5 0 35, 212
4. (3k 2 2)(6k 1 8) 5 0 32, 243
5. x(x 2 3) 5 0 0, 3
6. 2v(3v 1 4) 5 0 0, 243
Solve by factoring.
7. t2 1 3t 2 18 5 0 26, 3
8. j2 2 17j 1 72 5 0 8, 9
10. 8k2 2 2k 2 3 5 0 212, 34
9. 2c2 1 9c 1 4 5 0 212, 24
11. m2 1 6m 5 25 25, 21
12. y2 1 3y 5 28 27, 4
13. 2z2 1 z 5 6 22, 32
14. 15a2 2 a 5 6 235, 23
Use the Zero-Product Property to solve each equation. Write your solution in
roster form.
15. x2 2 10x 1 24 5 0 {6, 4}
16. d2 1 3d 2 10 5 0 {25, 2}
17. The volume of a storage tub shaped like a rectangular prism is 24 ft 3 . The height
of the tub is 3 feet. The width is w feet and the length is w + 2 feet. Use the formula
V 5 lwh to find the value of w. 2 ft
18. The area of a parking lot is 2475 ft 2 . The rectangular parking lot has
dimensions such that the length is 10 feet longer than the width. What are the
dimensions of the parking lot? 45 ft by 55 ft
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35
Name
Class
Date
Practice (continued)
9-4
Form K
Factoring to Solve Quadratic Equations
Write each equation in standard form. Then solve.
19. 3x2 2 x 2 7 5 2x2 1 5
20. x2 2 4x 2 2 5 29x 1 4
x2 1 5x 2 6 5 0; 26, 1
x2 2 x 2 12 5 0; 23, 4
Find the value of x as it relates to each rectangle or triangle.
21. Area = 15 m 2
22. Area = 408 in 2
3
x
12
x
x à2
3x Ľ2
23. Area = 36 ft 2
24. Area = 600 cm 2
8
24
x
x à1
2x à 2
x
25. Reasoning For each equation, find k and the value of any missing solutions.
a. x2 2 kx 2 15 5 0 where 23 is one solution of the equation. k 5 2; 5
b. x2 2 10x 5 k where 12 is one solution of the equation. k 5 24; 22
26. Writing Explain how you solve an equation by using the Zero-Product
Property.
When the product of two factors is zero, then one or both of the factors equal
zero. Set each factor equal to zero and find each solution.
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36
Name
Class
Date
Standardized Test Prep
9-4
Factoring to Solve Quadratic Equations
Gridded Response
Solve each exercise and enter your answer on the grid provided.
1. What is the positive solution of 3x2 2 10x 2 8 5 0? 4
2. A triangular-shaped wall has a base of 2x 1 4 and a height of x 1 3. The area
of the triangle is 56 in.2 . What is the value of x? 5
3. The product of two consecutive integers, n and n 1 1, is 42. What is the
positive integer that satisfies the situation? 6
4. One more rectangular-shaped piece of metal siding needs to be cut to cover
the exterior of a pole barn. The area of the piece is 30 ft2 . The length is 1 less
than 3 times the width. How wide should the metal piece be? Round to the
nearest hundredth of a foot. 3.33
5. What solution do 2x2 2 13x 1 21 5 0 and 2x2 1 9x 2 56 5 0 have in
common? Round your answer to the nearest tenth if necessary. 3.5
1.
2.
2
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
3.
2
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
4.
2
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
2
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
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5.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
Name
Class
Date
Enrichment
9-4
Factoring to Solve Quadratic Equations
You have factored to solve quadratic functions. You can also use a table to convert
between the forms ax2 1 bx 1 c and (ax 1 b)(cx 1 d).
Factor 3x2 1 4x 2 4 using the table method.
3x 2
24
212x 2
The First Row
The first term in 3x2 1 4x 2 4 is 3x2 . The third is 24.
Multiplied together, they result in 212x2 .
3x 2
24
212x 2
6x
22x
The Right Column
The two remaining empty squares in the column
farthest to the right must add up to 4x. These two new
terms must multiply to form 212x2 . The terms 6x and
22x will work.
3x 2
3x
24
2
212x 2
6x
22x
The Second Row
The first box in the second row must contain a factor
of 3x2 , and the second box must contain a factor of
24. The product of these two factors is 6x. The first
term can be 3x, and the second can be 2.
3x 2
3x
x
24
2
22
212x 2
6x
22x
The Last Row
The cells in the last row must be factors of their
individual columns, and when multiplied together
equal the third box of their row. What factor of 3x2 ,
times what factor of 24, will equal 22x? The terms x
and 22 will.
Diagonally down and to the right from 3x is 22. So, the first factor is (3x 2 2). Now
start at x and look diagonally up and to the right: 2. The second factor is (x 1 2).
So, 3x2 1 4x 2 4 5 (3x 2 2)(x 1 2).
Solve by factoring using the table method.
1. 6x2 2 17x 1 12 5 0
(x 1 3)(x 2 1)
23; 1
(2x 2 3)(3x 2 4)
4 3
3; 2
4. 22x2 1 6x 1 56 5 0
22(x 2 7)(x 1 4)
24; 7
2. x2 1 2x 2 3 5 0
5. x2 1 18x 1 80 5 0
(x 1 8)(x 1 10)
210; 28
3. 25x2 1 15x 1 90 5 0
25(x 2 6)(x 1 3)
23; 6
6. x2 1 12x 1 20 5 0
(x 1 2)(x 1 10)
210; 22
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38
Name
Class
9-4
Date
Reteaching
Factoring to Solve Quadratic Equations
If the product of two or more numbers is 0, then one of the factors must be 0. You
can use this fact to solve quadratic equations.
Problem
What are the solutions of the equation (4a 1 12)(5a 2 20) 5 0?
Since the product is 0, either (4a 1 12) or (5a 2 20) must equal 0.
4a 1 12 5 0
or
5a 2 20 5 0
4a 1 12 2 12 5 0 2 12
or
5a 2 20 1 20 5 0 1 20
4a 5 212
or
5a 5 20
4a
212
4 5 4
or
5a
20
5 5 5
a 5 23
or
a54
The solutions are 23 and 4.
Exercises
Solve each equation.
1. b(b 1 7) 5 0
2. 8y(2y 2 12) 5 0
0; 27
0; 6
21; 4
8. (4h 2 1)(2h 1 1) 5 0
212 ; 14
2 12 ; 22
10. (s 1 6)(4s 2 6) 5 0
6. (5p 2 10)(2p 1 20) 5 0
2; 210
27; 24
7. (8t 1 4)(3t 1 6) 5 0
26; 32
8; 2
5. (2a 1 14)(3a 1 12) 5 0
4. (m 1 1)(m 2 4) 5 0
3. (d 2 8)(d 2 2) 5 0
9. (8n 2 16)(5n 2 12) 5 0
2; 12
5
11. (5w 2 30)(2w 2 1) 5 0
6; 12
12. (3g 1 1)(2g 2 5) 5 0
213 ; 52
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39
Name
Class
9-4
Date
Reteaching (continued)
Factoring to Solve Quadratic Equations
If you can rewrite a quadratic equation as a product of factors that equals zero, you
can solve the equation. To solve equations in this manner, you must use all your
factoring skills.
Problem
What are the solutions of the equation x2 2 x 5 20?
First rewrite the equation so that one side equals zero.
x2 2 x 5 20
x2 2 x 2 20 5 20 2 20
Subtract 20 from each side.
x2
Simplify.
2 x 2 20 5 0
Now, factor to rewrite the equation as a product of factors equal to zero. Find two
integers whose product is 220 and whose sum is 21. The product of 4 and 25 is
220, and the sum of 4 and 25 is 21.
x2 2 x 2 20 5 0
(x 1 4)(x 2 5) 5 0
x1450
or
x2550
x14245024
or
x25155015
x 5 24
x55
or
The solutions are 24 and 5.
Exercises
Solve each equation by factoring.
13. y2 1 3y 1 2 5 0
24; 5
21; 22
16. 2d2 1 7d 2 4 5 0
1
2;
24
19. s2 1 9s 5 220
24; 25
22. 2h2 2 9h 5 5
212 ; 5
14. a2 2 a 2 20 5 0
15. m2 2 7m 1 6 5 0
1; 6
17. 6t2 1 13t 1 6 5 0
223; 232
18. 5p2 1 29p 2 6 5 0
20. x2 2 5x 5 14
21. b2 1 7b 5 8
22; 7
26
1; 28
23. 3s2 2 13s 5 212
4
3;
1
5;
24. 6v2 1 13v 5 5
1
3;
3
2 52
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40
Name Class 9-5
Date Additional Vocabulary Support
Completing the Square
There are two sets of note cards below that show Kris how to find the solutions of
the equation g2 2 4g 5 45. The set on the left explains the thinking. The set on the
right shows the steps. Write the thinking and the steps in the correct order.
Think Cards
Write Cards
Add 2 to each side.
(g 2 2)2 5 49
Write as two equations.
g 5 9 or g 5 25
Simplify the right side.
g 2 2 5 7 or g 2 2 5 27
b 2
2
Add Q R 5 4 to each side.
g2 2 4g 1 4 5 45 1 4
Find square roots of each
side.
g 2 2 5 4 !49
(g 2 2)2 5 45 1 4
Write g2 2 4g 1 4 as a
square.
Think
Write
2
First, add Q b2 R 5 4 to each side.
Step 1 g2 2 4g 1 4 5 45 1 4
Second, write g2 2 4g 1 4 as
a square.
Step 2
(g 2 2)2 5 45 1 4
Third, simplify the right side.
Step 3
(g 2 2)2 5 49
Next, find square roots of
each side.
Step 4
g 2 2 5 w!49
Step 5
g 2 2 5 7 or g 2 2 5 27
Step 6
g 5 9 or g 5 25
Then, write as two equations.
Finally, add 2 to each side.
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41
Name
Class
Date
Think About a Plan
9-5
Completing the Square
Landscaping A school is fencing in a rectangular area for a
playground. It plans to enclose the playground using fencing on
three sides, as shown at the right. The school has budgeted enough
money for 75 ft of fencing material and would like to make a
playground with an area of 600 ft2 .
a. Let w represent the width of the playground. Write an expression
in terms of w for the length of the playground.
b. Write and solve an equation to find the width w. Round to the
nearest tenth of a foot.
c. What should the length of the playground be?
w
600 ft2
w
What do you know?
1. Let w represent the width of the playground. Write an expression in terms of w
for the length of the playground.
z
z
w 1 w 1 l 5 feet
75
l 5 22w 1 75
2. Write an equation for the area of the playground.
z
z
w ? l 5 ft2
600
What do you need to solve the problem?
3. Substitute the expression for l from Step 1 in the equation from Step 2.
w(22w 1 75) 5 600
How do you solve the problem?
4. Solve the equation in Step 3 to find the width w. Round to the nearest tenth of
a foot. What should the length l of the playground be?
w 5 25.9 ft and l 5 23.2 ft, or w 5 11.6 and l 5 51.8
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42
E
Name
Class
9-5
Date
Practice
Form G
Completing the Square
Find the value of c such that each expression is a perfect-square trinomial.
1. x2 1 4x 1 c 4
2. b2 1 12b 1 c 36
3. g2 2 20g 1 c 100
4. a2 2 7a 1 c 49
4
5. w2 1 18w 1 c 81
6. n2 2 9n 1 c 81
4
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
7. z2 2 19z 5 66
22; 23
8. p2 2 5p 5 24
4; 1
9. b2 1 6b 5 16
28; 2
10. c2 2 4c 5 21
7; 23
11. a2 2 2a 5 15
5; 23
12. v2 1 8v 5 15
29.57; 1.57
13. y2 1 16y 5 17
217; 1
14. x2 1 4x 1 3 5 0
23; 21
15. h2 1 4h 5 1
24.24; 0.24
16. r2 1 8r 1 13 5 0
25.73; 22.27
17. d2 2 2d 2 4 5 0
3.24; 21.24
18. m2 2 24m 1 44 5 0
22; 2
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
19. 3y2 1 5y 5 12
4
3;
23
22. 2c2 1 7c 1 3 5 0
20. 2h2 2 5h 5 21
0.22, 2.28
21. 4k2 1 4k 5 5
0.72; 21.72
23. 3f 2 2 2f 5 1
24. 9x2 2 42x 1 49 5 0
1; 213
23; 212
7
3
25. The rectangle shown at the right has an area of 56 m2.
What is the value of x?
x
4m
3x 1 2
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Name
9-5
Class
Date
Practice (continued)
Form G
Completing the Square
26. What are all of the values of c that will make x2 1 cx 1 49 a perfect square?
14 or 214
27. What are all of the values of c that will make x2 1 cx 1 121 a perfect square?
22 or 222
Solve each equation. If necessary, round to the nearest hundredth. If there is no
solution, write no solution.
28. k2 2 24k 1 4 5 22 23.7; 0.25
29. 4x2 2 20x 1 25 5 0 52
30. 2b2 1 10b 1 15 5 3 22; 23
31. p2 1 3p 1 2 5 21 no solution
32. 5m2 1 10m 2 80 5 75 26.66; 4.66
33. 2a2 2 3a 1 4 5 0 no solution
34. 5a2 2 12a 1 28 5 0 no solution
35. 5t2 2 6t 5 35 22.11; 3.31
36. Writing Discuss the strategies of graphing, factoring, and completing the
square for solving the quadratic equation x2 1 4x 2 6 5 0.
By graphing, the x-intercepts represent the values of x that solve the equation. By
completing the square, you can algebraically find the solution. The given equation
cannot be factored.
37. The height of a triangle is 4x inches and the base is (5x 1 1) inches. The area
of the triangle is 500 square inches. What are the dimensions of the base and
height of the triangle?
27.8 in.; 35.85 in.
38. The formula for finding the volume of a rectangular prism is V 5 lwh. The
height h of a rectangular prism is 12 centimeters. The prism has a volume of
10,800 cubic centimeters. The prism’s length l is modeled by 3x centimeters
and its width w by (2x 1 1) centimeters. What is the value of x? What are the
dimensions of the length and the width?
x = 12 ; l = 36 cm; w = 25 cm
39. Writing In order to solve a quadratic equation by completing the square,
what does the coefficient of the squared term need to be? If the coefficient is
not equal to this, what does your first step need to be to complete the square?
1; divide each term by the coefficient of x2
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44
Name
9-5
Class
Date
Practice
Form K
Completing the Square
Find the value of c such that each expression is a perfect-square trinomial.
1. z2 1 2z 1 c 1
2. h2 1 14h 1 c 49
3. p2 2 11p 1 c 121
4. n2 1 26n 1 c 169
4
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
5. t2 2 17t 5 252 4, 13
6. m2 1 6m 5 7 27, 1
7. f 2 1 3f 5 88 211, 8
8. z2 1 9z 5 36 212, 3
9. a2 1 13a 5 12 213.87, 0.87
10. g2 1 5g 1 4 5 0 24, 21
11. d2 1 7d 1 9 5 0 25.3, 21.7
12. b2 2 5b 2 10 5 0 21.53, 6.53
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
13. 6n2 1 9n 5 12 22.35, 0.85
14. 2t2 2 4t 5 1 20.22, 2.22
15. 3v2 1 9v 1 5 5 0 22.26, 20.74
16. 4c2 2 8c 5 1 20.12, 2.12
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Name
9-5
Class
Date
Practice (continued)
Form K
Completing the Square
17. The rectangle shown at the right has an area of 663 in 2 .
What is the value of x? 13
x in.
4x Ľ1 in.
18. What are all of the values of b that will make x2 1 bx 1 64 a perfect square? w16
19. What are all of the values of b that will make x2 1 bx 1 144 a perfect square? w24
20. The product of two consecutive positive even integers is 168. What are the
integers? 12 and 14
21. Writing Discuss how you could use graphing, factoring, and completing the
square for solving the quadratic equation x2 1 3x 2 2 5 0.
Factoring cannot be used because x2 1 3x 2 2 cannot be factored. Graphing
will give you an answer that is not precise. Completing the square will give
you a precise answer.
22. The height of a triangle is 6x cm and the base is (3x 1 10) cm. The area of the
triangle is 816 cm 2 . What are the dimensions of the base and height of the
triangle?
height = 48 in.; base = 34 in.
23. Writing Does completing the square always give a solution for a quadratic
equation that cannot be factored? Explain.
No, some quadratic equations do not have a solution.
24. Reasoning How do the solutions of the equation x2 2 6x 1 9 5 16 compare
to the solutions of x2 2 6x 1 9 5 25? Explain how you can determine the
relationship without solving both equations.
Solve each equation by factoring the left side as a perfect square trinomial
and taking the square roots of both sides. The solutions are 3 w 4, or 21
and 7. The solutions of the second equation are 3 w 5, or 22 and 8.
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Name
9-5
Class
Date
Standardized Test Prep
Completing the Square
Multiple Choice
For Exercises 1–6, choose the correct letter.
1. What is the value of n such that the expression x2 1 11x 1 n is a perfect
square trinomial? C
A. 11
B. 25
C. 30.25
D. 36
2. What is a solution of x2 1 6x 5 25? G
F. x 5 26
G. x 5 21
H. x 5 1
I. x 5 6
3. Which of the following is a solution of x2 1 4x 2 1 5 0? If necessary, round
to the nearest hundredth. B
A. x 5 20.24
B. x 5 24.24
C. x 5 4.24
D. no solution
4. Which of the following is a solution of x2 1 14x 1 112 5 0? If necessary,
round to the nearest hundredth. I
F. x 5 20.94
G. x 5 14.94
H. x 5 214.94
I. no solution
5. The rectangular poster shown at the right has an area of 5400 cm2 .
What is the value of w? C
A. 245 cm
C. 60 cm
B. 45 cm
D. 90 cm
6. A box shaped like a rectangular prism has a height of 17 in. and a
Actors Wanted
School Play
Tryouts
2w 2 30
Tuesday – 3:30 p.m.
School Auditorium
volume of 2720 in.3 . The length is 4 inches greater than twice the
width. What is the width of the box? G
F. 210 in.
H. 20 in.
G. 8 in.
w
I. 40 in.
Short Response
7. The area of a rectangular television screen is 3456 in.2 . The width of the screen
is 24 inches longer than the length. What is a quadratic equation that
represents the area of the screen? What are the dimensions of the screen?
l2 1 24l 5 3456; 48 in. by 72 in.
[2] Both parts answered correctly
[1] One part answered correctly
[0] Neither part answered correctly
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Name
Class
Date
Enrichment
9-5
Completing the Square
The vertex form of a parabola is y 5 a(x 2 h)2 1 k. The vertex is at the point
(h, k). You can complete the square of an equation of a parabola to determine the
coordinates of its vertex. The value of h is the opposite of the constant in the term
being squared, but the value of k has the same sign as it is does in the expressions.
Example
For the parabola y 5 x2 1 10x 1 7 determine the coordinates of its vertex.
y 1 52 5 x2 1 10x 1 52 1 7
Square half the coefficient of the x-term and
add to both sides.
y 1 25 5 (x2 1 10x 1 25) 1 7
Simplify.
y 1 25 5 (x 1 5)2 1 7
Factor the complete square.
y 5 (x 1 5)2 2 18
Solve for y.
The vertex is at (25, 218).
Practice
1. Graph y 5 x2 1 10x 1 7 to check the answer in the Example.
y
5
Ź12 Ź8 Ź4
O
4
x
Ź5
Ź10
Ź15
For Exercises 2–3, determine the coordinates of the vertex of the parabola by
writing the equation in vertex form. Graph to check.
2. y 5 x2 1 6x 2 3
3. y 5 x2 2 4x 1 12
y
y
4
Ź12 Ź8 Ź4
O
4
x
(2, 8);
y 5 (x 2 2)2 1 8
16
(23, 212);
y 5 (x 1 3)2 2 12
12
Ź4
8
Ź8
4
Ź12
Ź4 Ź2
O
2
4
x
Ź4
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Name
9-5
Class
Date
Reteaching
Completing the Square
You have learned to square binomials. Notice how the coefficient of the a term is
related to the constant value in every perfect-square trinomial.
(a 1 1) 2 5 (a 1 1)(a 1 1) 5 a2 1 2a 1 1 S
22
Q2 R 5 1
(a 2 1) 2 5 (a 2 1)(a 2 1) 5 a2 2 2a 1 1 S
22 2
Q 2 R 51
(a 2 2) 2 5 (a 2 2)(a 2 2) 5 a2 2 4a 1 4 S
24 2
Q 2 R 54
(a 1 3) 2 5 (a 1 3)(a 1 3) 5 a2 1 6a 1 9 S
62
Q2 R 5 9
In each case, half the coefficient of the a term squared equals the constant term.
You can use this pattern to find the value that makes a trinomial a perfect square.
Problem
What is the value of c such that x2 2 14x 1 c is a perfect-square trinomial?
2
214
The coefficient of the x term is 214. Using the pattern, c 5 Q 2 R or 49.
So, x2 2 14x 1 49 is a perfect-square trinomial.
Exercises
Find the value of c such that each expression is a perfect-square trinomial.
1. a2 1 8a 1 c 16
2. x2 2 16x 1 c 64
3. m2 1 20m 1 c 100
4. p2 2 14p 1 c 49
5. y2 2 10y 1 c 25
6. b2 1 18b 1 c 81
7. d2 1 12d 1 c 36
8. n2 2 n 1 c 14
9. w2 1 3w 1 c 9
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4
Name
Class
9-5
Date
Reteaching (continued)
Completing the Square
You can use completing the square to solve quadratic equations.
Problem
What are the solutions of the equation x2 1 2x 2 48 5 0?
First rewrite the equation so that the constant is on one side of the equation and
the other terms are on the other side.
x2 1 2x 2 48 5 0
x2 1 2x 2 48 1 48 5 0 1 48
Add 48 to each side.
x2 1 2x 5 48
Simplify.
2 2
Since Q 2 R 5 1, add 1 to each side.
x2 1 2x 1 1 5 48 1 1
Add 1 to each side.
(x 1 1)2 5 49
Simplify.
x 1 1 5 4 !49
Take the square root of each side.
x 1 1 5 47
Simplify.
x 1 1 5 27
or
x1157
x 1 1 2 1 5 27 2 1
or
x11215721
x 5 28
x56
or
The solutions are 28 and 6.
Exercises
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
10. b2 1 10b 5 75
11. y2 2 18y 5 63
13. a2 1 16a 5 215
14. t2 1 8t 2 9 5 0
29; 1
215; 21
16. m2 2 2m 2 8 5 0
4; 22
15; 5
21; 23
5; 215
12. n2 2 20n 5 275
15. h2 2 12h 2 9 5 0
12.71; 20.71
17. s2 1 6s 1 1 5 0
25.83; 20.17
18. v2 1 4v 2 2 5 0
24.45; 0.45
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Name Class 9-6
Date Additional Vocabulary Support
The Quadratic Formula and the Discriminant
Complete the chart by filling in the missing information about when to use the
given method to solve a quadratic equation.
Method
When to Use
Equation
completing the
square
Use completing the square if the
coefficient of x2 is 1, but you cannot
factor the equation easily.
0 5 x2 2 2x 1 5
factoring
Use factoring if you can factor the
equation easily.
graphing
quadratic formula
square roots
2. Use graphing if you have a
graphing calculator available.
Use the quadratic formula if the
equation cannot be factored easily or
at all.
4. Use square roots if the
equation has no x-term.
1. 0 5 x2 1 8x 1 15
5 (x 1 3)(x 1 5)
0 5 9x2 1 12x 1 4
3. 0 5 2x2 2 4x 2 3
0 5 9x2 2 36
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51
Name
Class
9-6
Date
Think About a Plan
The Quadratic Formula and the Discriminant
Sports Your school wants to take out an ad in the paper
congratulating the basketball team on a successful season, as
shown at the right. The area of the photo will be half the area
of the entire ad. How wide will the border be?
x
7 in.
photo
What do you know?
1. What are the dimensions of the photo and the ad? Let w 5 the width of the
photo and l 5 the length of the photo.
5 in. by 7 in.; (x 1 5)in. by (x 1 7) in.
What do you need to solve the problem?
2. What quadratic equation can you write that best describes the relationship
between the area of the photo and the area of the ad?
1
2 (x
1 5)(x 1 7) 5 35, or x2 1 12x 2 35 5 0
How do you solve the problem?
3. Using the quadratic formula, how will you be able to solve for x, the width of
the border? What is the width of the border?
Substitute 1 for a, 12 for b, and 235 for c; about 2.43 in.
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5 in.
x
Name
9-6
Class
Date
Practice
Form G
The Quadratic Formula and the Discriminant
Use the quadratic formula to solve each equation.
1. 7c2 1 8c 1 1 5 0
2. 2w2 2 28w 5 298
7
3. 2j2 2 3j 5 21
4. 2x2 2 6x 1 4 5 0
2; 1
5. 2n2 2 6n 5 8
4; 21
6. 27d2 1 2d 1 9 5 0
7. 2a2 1 4a 2 6 5 0
23; 1
8. 23p2 1 17p 5 20
9. 4d2 2 8d 1 3 5 0
21; 217
1; 12
21; 97
3 1
2; 2
4; 53
Use the quadratic formula to solve each equation. Round answers to the nearest
hundredth.
10. h2 2 2h 2 2 5 0
20.73; 2.73
11. 5x2 1 3x 5 1
20.84; 0.24
12. 2z2 2 4z 5 22
0.45; 24.45
13. t2 1 10t 5 222
26.73; 23.27
14. 3n2 1 10n 5 5
23.77; 0.44
15. s2 2 10s 1 14 5 0
8.32; 1.68
16. A basketball is passed through the air. The height h of the ball in feet after the
distance d in feet the ball travels horizontally is given by h 5 2d2 1 10d 1 5.
How far horizontally from the player passing the ball will the ball land on the
ground?
about 10.48 ft
Which method(s) would you choose to solve each equation? Justify your
reasoning.
17. h2 1 4h 1 7 5 0
no solution
18. a2 2 4a 2 12 5 0
factoring is easiest
19. 24y2 2 11y 2 14 5 0
quadratic formula
20. 2p2 2 7p 2 4 5 0
factor
21. 4x2 2 144 5 0
use square roots
22. f 2 2 2f 2 35 5 0
complete the square
23. Writing Explain how the discriminant can be used to determine the number
of solutions a quadratic equation has.
If the discriminant is S 0, there are two real solutions. If the discriminant 5 0,
there is one solution. If the discriminant is R 0, there are no real solutions.
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Name
Class
9-6
Date
Practice (continued)
Form G
The Quadratic Formula and the Discriminant
Find the number of real-number solutions of each equation.
24. x2 2 8x 1 7 5 0
two
25. x2 2 6x 5 0
two
26. 2x2 2 5x 1 16 5 0
27. 23x2 2 4x 2 8 5 0
no real solutions
28. 7x2 1 12x 2 21 5 0
two
29. 2x2 1 4x 1 2 5 0
one
no real solutions
Use any method to solve each equation. If necessary, round answers to the
nearest hundredth.
30. 5m2 2 3m 2 15 5 0
2.06; 21.46
31. 9y2 1 6y 5 212
no solution
32. 4a2 5 36
3; 23
33. 6t2 2 96 5 0
4; 24
34. z2 1 7z 5 210
22; 25
35. 2g2 1 4g 1 3 5 0
4.65; 20.65
Find the value of the discriminant and the number of real-number solutions of
each equation.
36. x2 1 11x 2 10 5 0
161; two
37. x2 1 7x 1 8 5 0
17; two
38. 3x2 1 5x 2 9 5 0
133; two
39. 22x2 1 10x 2 1 5 0
92; two
40. 3x2 1 6x 1 3 5 0
0; one
41. 6x2 1 x 1 12 5 0
2287; no real solutions
42. The weekly profit of a company is modeled by the function w 5 2g2 1 120g 2 28.
The weekly profit, w, is dependent on the number of gizmos, g, sold. If the
break-even point is when w 5 0, how many gizmos must the company
sell each week in order to break even?
120 gizmos
43. Reasoning The equation 4x2 1 bx 1 9 5 0 has no real-number solutions. What
must be true about b?
212 R b R 12
44. Open-Ended Describe three different methods to solve x2 2 x 2 56 5 0. Tell
which method you prefer. Explain your reasoning.
Factor: (x 2 8)(x 1 7) 5 0 using the zero products property to find that x = 8 or
x 5 27; graph and find x –intercepts at x = 8 and x 5 27; use the quadratic
formula to find solutions at 8 and 27; I prefer to factor. It is quickest.
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54
Name
9-6
Class
Date
Practice
Form K
The Quadratic Formula and the Discriminant
Use the quadratic formula to solve each equation.
1. 3z2 1 z 2 4 5 0 243, 1
2. 2d2 1 9d 5 5 25, 12
3. 2y2 1 12y 1 10 5 0 25, 21
3
4. 2t2 2 5t 2 12 5 0 22, 4
5. 3c2 2 13c 1 4 5 0 13, 4
6. 15b2 1 22b 1 8 5 0 223, 245
Use the quadratic formula to solve each equation. Round answers to the nearest
hundredth.
7. y2 2 4y 2 4 5 0 20.83, 4.83
8. 3r2 1 5r 5 1 21.85, 0.18
9. h2 1 12h 5 216 210.47, 21.53
10. 5v2 1 3v 5 1 20.84, 0.24
11. A football is passed through the air and caught at ground level for a touchdown.
The height h of the ball in feet is given by h 5 2d2 1 12d 1 6, where d is the
distance in feet the ball travels horizontally. How far from the player passing the
ball will the ball be caught? about 12.48 ft
Which method(s) would you choose to solve each equation? Justify your
reasoning.
12. a2 1 3a 2 11 5 0
quadratic formula, completing
the square, or graphing; the
coefficient of the x2 -term is 1, but
the equation cannot be factored.
13. 9d2 2 100 5 0
square roots; there is no
x-term.
14. 6h2 2 11h 2 3 5 0
quadratic formula, the equation
cannot be factored.
15. n2 2 n 2 6 5 0
factoring; the equation is
easily factorable.
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Name
9-6
Class
Date
Practice (continued)
Form K
The Quadratic Formula and the Discriminant
Find the number of real-number solutions of each equation.
16. x2 2 10x 1 9 5 0 2
17. 25x2 2 2x 2 14 5 0
no real solutions
18. x2 1 10x 1 25 5 0 1
19. x2 2 4x 5 0 2
Use the quadratic formula to solve each equation. If necessary, round answers
to the nearest hundredth.
20. 4r2 2 100 5 0 w5
21. a2 2 2a 5 99 29, 11
22. 7g2 2 2g 2 10 5 0 21.06, 1.35
1 2
23. 15k2 2 7k 5 2 25, 3
Find the value of the discriminant and the number of real-number solutions of
each equation.
24. x2 1 7x 1 5 5 0 29, 2
25. x2 1 4x 1 10 5 0
224; no real solutions
26. 23x2 1 9x 2 2 5 0 57, 2
27. 5x2 1 11x 1 8 5 0
239; no real solutions
28. The daily production of a company is modeled by the function
p 5 2w2 1 75w 2 1200. The daily production, p, is dependent on the
number of workers, w, present. If the break-even point is when p 5 0, what
are the least and greatest number of workers the company must have present
each day in order to break even? 23; 51
29. Reasoning The equation 3x2 1 bx 1 3 5 0 has one real solution. What
must be true about b? b 5 w6
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Name
Class
9-6
Date
Standardized Test Prep
The Quadratic Formula and the Discriminant
Multiple Choice
For Exercises 1–6, choose the correct letter.
1. Which expression gives the solutions of 25 1 2x2 5 26x? C
A.
2 4 "4 2 (4)(6)(25)
12
C.
26 4 "36 2 (4)(2)(25)
4
B.
25 4 "25 2 (4)(2)(6)
210
D.
6 4 "36 2 (4)(2)(5)
4
2. What are the approximate solutions of 2x2 2 x 1 10 5 0? I
F. 22, 2.5
G. 21.97, 2.47
H. 22.5, 2
I. no solution
3. What are the approximate solutions of 7x2 1 4x 2 9 5 0? B
A. 21.42, 0.85
C. 20.88, 1.5
B. 21.5, 0.88
D. no solution
4. Which method is the best method for solving the equation
8x2 2 13x 1 3 5 0? I
F. square roots
H. graphing
G. factoring
I. quadratic formula
5. How many solutions are there for 5x2 1 7x 2 4 5 0? C
A. 0
B. 1
C. 2
D. 3
6. The perimeter of a rectangle is 54 cm. The area of the same rectangle is
176 cm2 . What are the dimensions of the rectangle? F
F. 11 cm by 16 cm
H. 5.5 cm by 32 cm
G. 8 cm by 22 cm
I. 4 cm by 44 cm
Short Response
7. The flight of a baseball that has been hit when it was 4 feet off the ground is
modeled by the function h 5 216t2 1 75t 1 4 where h is the height of the
baseball in feet after t seconds. Rounding to the nearest hundredth, how long
will it take before the ball lands on the ground? Show your work.
4.74 s
[2] Both parts answered correctly.
[1] One part answered correctly.
[0] Neither part answered correctly.
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Name
9-6
Class
Date
Enrichment
The Quadratic Formula and the Discriminant
You have used the discriminant to find the number of solutions to a quadratic
equation. You can also use the discriminant to determine the number of
x-intercepts of the graph of the related function.
Discriminant
Positive Discriminant
b2 2 4ac . 0
Discriminant is Zero
b2 2 4ac 5 0
y
Example
Negative Discriminant
b2 2 4ac , 0
y
y
x
x
Number of
x-intercepts
of graph
of related
function
The graph has two
x-intercepts.
The graph has one
x-intercept.
x
The graph has no
x-intercepts.
Practice
Use the discriminant of the related quadratic equation to determine the
number of x-intercepts of the graph of the function.
1. y 5 x2 1 4x 1 5
2. y 5 x2 2 x 2 2
two
none
3. y 5 x2 2 2x 1 1
4. y 5 x2 2 4x 1 13
one
none
5. y 5 2x2 1 11x 2 5
6. y 5 4x2 2 17x 2 15
two
two
7. y 5 x2 2 9x
8. y 5 3x2 2 7x 1 5
none
two
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9-6
Date
Reteaching
The Quadratic Formula and the Discriminant
If a quadratic equation is written in the form ax2 1 bx 1 c 5 0, the solutions can
be found using the following formula.
x5
2b 4 "b2 2 4ac
2a
This formula is called the quadratic formula.
Problem
What are the solutions of x2 1 7x 5 60? Use the quadratic formula.
First rewrite the equation in the form ax2 1 bx 1 c 5 0.
x2 1 7x 5 60
x2 1 7x 2 60 5 60 2 60
Subtract 60 from each side.
x2 1 7x 2 60 5 0
Simplify.
Therefore, a 5 1, b 5 7, and c 5 260.
x5
2b 4 "b2 2 4ac
2a
x5
27 4 "72 2 4(1)(260)
2(1)
x5
27 4 "289
2
x5
27 4 17
2
The two solutions are
27 2 17
27 1 17
or 212 and
or 5.
2
2
Exercises
Use the quadratic formula to solve each equation.
1. x2 2 19x 1 70 5 0
14; 5
225; 27
4. x2 2 10x 5 75
212; 11
7. 20x2 1 11x 5 3
3. 2x2 1 37x 2 19 5 0
219; 0.5
5. x2 1 x 5 132
15; 25
234 ; 15
2. x2 1 32x 1 175 5 0
6. 6x2 1 13x 5 28
23.5; 1.3
8. 4x2 1 24x 5 235
23.5; 22.5
9. 15x2 1 20 5 40x
2; 23
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59
Name
Class
Date
Reteaching (continued)
9-6
The Quadratic Formula and the Discriminant
In the quadratic equation, the expression under the radical sign, b2 2 4ac, is
called the discriminant. Consider the quadratic formula.
x5
2b 4 "b2 2 4ac
2a
• If b2 2 4ac is a negative number, the square root cannot be found in the real
numbers. There are no real-number solutions of the equation. The graph of
the quadratic does not cross the x-axis.
• If b2 2 4ac equals 0, x 5
2b 4 !0 2b
or 2a . There is only one solution of the
2a
equation. The vertex of the quadratic is on the x-axis.
• If b2 2 4ac is a positive number, there are two solutions of the equation,
2b 2 "b2 2 4ac
2b 1 "b2 2 4ac
and
. The graph of the quadratic intersects
2a
2a
the x-axis twice.
Problem
What is the number of solutions of x2 1 13 5 25x?
First rewrite the equation in the form ax2 1 bx 1 c 5 0.
x2 1 13 5 25x
x2 1 5x 1 13 5 0
Add 5x to each side.
Therefore, a 5 1, b 5 5, and c 5 13.
b2 2 4ac 5 52 2 4(1)(13)
5 227
Since b2 2 4ac is a negative number, there are no real-number solutions of the
equation.
Exercises
Find the number of solutions of each equation.
10. 4x2 1 12x 1 9 5 0
one
11. x2 2 12x 1 32 5 0
two
12. x2 2 10x 1 1 5 0
two
13. 3x2 1 6x 1 8 5 0
14. 3x2 2 5x 5 26
15. x2 1 100 5 20x
no real solutions
16. 5x2
2 7x 5 2
two
no real solutions
17. 9x2
1 4 5 12x
one
one
18. 3x2 1 5x 5 2
two
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60
Name Class Date Additional Vocabulary Support
9-7
Linear, Quadratic, and Exponential Models
Concept List
data of exponential function
exponential function
graph of quadratic function
data of linear function
data of quadratic
function
graph of linear
function
quadratic function
graph of exponential
function
linear function
Choose the concept from the list above that best represents the item in
each box.
2.
1. y 5 mx 1 b
4
3.
y
x
0
1
2
3
4
2
x
−4
−2
O
2
4
−2
y
1
3
5
7
9
−4
linear function
graph of linear function
4.y 5 a ? bx
5.
exponential function
data of exponential
function
quadratic function
7.
8.
9.
6. y 5 ax2 1 bx 1 c
HSM11ALTR_0907_T00102
x
y
HSM11ALTR_0907_T00101
–2
1
–1
2
0
4
1
8
2
16
4
y
x
−4
O
x
y
–2
–1
0
1
2
–4
–2.25
–1
–0.25
0
4
HSM11ALTR_0907_T00103
2
−2
data of linear function
2
4
−2
y
2
x
−4
−2
O
2
4
−2
−4
−4
graph of quadratic function
HSM11ALTR_0907_T00104
data of quadratic function
HSM11ALTR_0907_T00105
graph of exponential
function
HSM11ALTR_0907_T00106
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61
Name
9-7
Class
Date
Think About a Plan
Linear, Quadratic, and Exponential Models
Zoology A conservation organization collected the data on the
number of frogs in a local wetlands. Which kind of function best
models the data? Write an equation to model the data.
Year
Number of
Frogs
0
120
1
101
2
86
3
72
4
60
What do you know?
Estimated Population
1. Let x 5 year and y 5 number of frogs.
Graph the points in the table.
y
120
90
60
30
0
x
0
1
2 3
Year
4
2. How will graphing the points in the table help you determine which function
best models the data?
The shape of the graph indicates which model best fits the data.
What do you need to solve the problem?
3. How will finding the differences or ratios between the data points help you
determine which function best models the data?
The pattern for differences or ratios indicates which model but fits the data. Common
difference: linear; common secondary difference: quadratic; common ratio: exponential
How do you solve the problem?
4. Write an equation that best models the data.
y 5 120 ? 0.84x
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Name
Class
Date
Practice
9-7
Form G
Linear, Quadratic, and Exponential Models
Graph each set of points. Which model is most appropriate for each set?
1. (23, 28), (21, 22), (0, 1), (1, 4), (3, 10)
linear;
2. (22, 0.75), (21, 1.5), (0, 3), (1, 6)
exponential;
y
Ź4 Ź2
y
12
6
6
4
2
O
4
2
x
Ź6
Ź4 Ź2
Ź12
O
2
4
x
Ź2
3. (22, 1), (21, 0), (0, 1), (1, 4), (2, 9)
quadratic; check graphs
4. (22, 211), (21, 25), (0, 23), (1, 25), (2, 211)
quadratic; check graphs
5. (24, 0), (22, 21), (0, 22), (2, 23), (4, 24)
linear; check graphs
6. (21, 20.67), (0, 22), (1, 26), (2, 218)
exponential; check graphs
7. (23, 10), (21, 2), (0, 1), (1, 2), (3, 10)
quadratic; check graphs
8. (22, 4), (21, 2), (0, 0), (1, 22), (2, 24)
linear; check graphs
Which type of function best models the data in each table? Use differences
or ratios.
9.
quadratic 10.
linear
11.
exponential
x
y
0
Ľ12
0
3
0
3
1
Ľ11
1
Ľ2
1
12
2
Ľ8
2
Ľ7
2
48
3
Ľ3
3
Ľ12
3
192
4
4
4
17
4
768
x
y
x
y
12. Which type of function best models the ordered pairs (21, 6), (0, 1), (1, 2), and
(2, 9)? Use differences or ratios. quadratic
13. Which type of function best models the ordered pairs (21, 20.25),
(0, 20.5), (1, 21), and (2, 22)? Use differences or ratios. exponential
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Name
Class
Date
Practice (continued)
9-7
Form G
Linear, Quadratic, and Exponential Models
Which type of function best models the data in each table? Write an equation to
model the data.
14.
17.
x
y
0
Ľ7
1
15.
x
y
Ľ4
32
Ľ1
Ľ3
2
5
3
linear;
y 5 27 1 6x
16.
x
exponential;
y 5 2 ? 0.5x
y
0
4
16
1
0
Ľ2
8
2
Ľ12
11
Ľ1
4
3
Ľ32
4
17
0
2
4
Ľ60
x
y
x
y
x
y
Ľ1
22
Ľ2
Ľ1
0
Ľ1
0
15
Ľ1
Ľ2
1
Ľ2
1
10
0
Ľ4
2
Ľ3
2
7
1
Ľ8
3
Ľ4
3
6
2
Ľ16
4
Ľ5
18.
quadratic;
y 5 x2 2 6x 1 15
19.
exponential;
y 5 24 ? 2x
quadratic;
y 5 24x2 1 4
linear;
y 5 2x 2 1
Which type of function best models the data in each ordered pair? Write an equation
to model the data.
20. (23, 33), (21, 21), (0, 15), (1,9), (3, 23)
21. (22, 216), (21, 28), (0, 24), (1, 22), (2, 21)
exponential; y 5 4 ? 0.5x
linear; y 5 26x 1 15
1
1
1
22. (22, 27), (21, 9), (0, 3), (1, 1), (2, 3)
23. (22, 22), (21, 23.5), (0, 24), (1, 23.5), (2, 22)
quadratic; y 5 12x2 2 4
exponential; y 5 13 ? 3x
24. (26, 5), (23, 4.5), (0, 4), (3, 3.5), (6, 3)
linear; y 5
216x
25. (21, 10), (0, 3), (1, 0), (2, 1)
14
quadratic; y 5 22x2 2 5x 1 3
26. The population of a city for years since 2000 is shown below. Which kind of function
best models the data? Write an equation to model the data. exponential; y 5 1500 ? 2x
Years since 2000
Population
0
2
4
1500
6000
24,000
6
8
96,000 384,000
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64
Name
Class
Date
Practice
9-7
Form K
Linear, Quadratic, and Exponential Models
Graph each set of points. Which model is most appropriate for each set?
1. (23, 21), (22, 0), (21, 1), (0, 2), (1, 3)
2. (0, 3), (1, 1), (2, 0), (3, 1)
y
y
4
4
2
2
x
Ź4 Ź2
2
O
4
linear
x
Ź4 Ź2
Ź2
Ź2
Ź4
Ź4
3. (22, 20.25), (21, 20.5), (0, 21), (1, 22)
quadratic
y
4
4
2
2
x
2
O
4
x
exponential
Ź4 Ź2
2
O
Ź2
Ź2
Ź4
Ź4
5. (26, 6), (24, 4), (22, 2), (0, 0)
4
quadratic
6. (22, 0.25), (21, 0.5), (0, 1), (1, 2)
y
y
8
4
4
2
x
Ź8 Ź4
4
4. (24, 0), (22, 2), (0, 3), (2, 2), (4, 0)
y
Ź4 Ź2
2
O
4
O
8
x
linear
Ź4 Ź2
2
O
Ź4
Ź2
Ź8
Ź4
4
exponential
Which type of function best models the data in each table? Use differences or
ratios.
7.
x
y
0
1
2
3
4
0
Ľ2
Ľ8
Ľ18
Ľ32
8.
quadratic
x
y
0
1
2
3
4
1
Ľ3
Ľ9
Ľ27
Ľ81
exponential
9. Which type of function best models the ordered pairs (21, 1), (0, 22), (1, 1),
and (2, 10)? Use differences or ratios. quadratic
10. Which type of function best models the ordered pairs (21, 2.5), (0, 1), (1, 20.5),
and (2, 22)? Use differences or ratios. linear
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65
Name
Class
Date
Practice (continued)
9-7
Form K
Linear, Quadratic, and Exponential Models
Which type of function best models the data in each table? Write an equation to
model the data.
11.
13.
x
y
0
1
2
3
4
Ľ2
Ľ4
Ľ8
Ľ16
Ľ32
x
y
0
1
2
3
4
1
1.5
3
5.5
9
12.
exponential;
y 5 22(2x)
14.
quadratic;
y 5 12 x2 1 1
x
y
0
1
2
3
4
Ľ2
Ľ5
Ľ8
Ľ11
Ľ14
x
y
Ľ2
Ľ1
0
1
2
12
6
3
1.5
0.75
linear; y 5 23x 2 2
exponential;
y 5 3 Q 12 R x
Which type of function best models the data in each ordered pair? Write an equation
to model the data.
4
4
16. (22, 236), (21, 212), (0, 24), (1, 23), (2, 29)
15. (21, 4), (0, 5), (1, 4), (2, 1), (3, 24)
quadratic; y 5 2x2 1 5
exponential; y 5 24 Q 13 R x
17. (0, 26), (4, 25), (6, 24.5), (8, 24), (12, 23) 18. (21, 26), (0, 21), (2, 23), (3, 210), (4, 221)
linear; y 5 0.25x 2 6
quadratic; y 5 22x2 1 3x 2 1
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66
Name
9-7
Class
Date
Standardized Test Prep
Linear, Quadratic, and Exponential Models
Multiple Choice
For Exercises 1–4, choose the correct letter.
1. Which kind of function best models the set of data points (21 ,22), (0, 6),
(1, 210), (2, 226), (3, 242)? A
A. linear
B. quadratic
C. exponential
D. none of the above
2. Which kind of function best models the set of data points (23, 18), (22, 6),
(21, 2), (0, 11), (1, 27)? G
F. linear
G. quadratic
H. exponential
I. none of the above
3. What function can be used to model data pairs that have a common ratio? C
A. linear
B. quadratic
C. exponential
D. none of the above
4. The attendances at the high school basketball games seemed
G. a 5 25g 1 100
a
Attendance
to be affected by the success of the team. The graph at the right
models the attendance over the first half of the season. Which
function would also represent the data shown in the graph where
a represents the attendance and g represents the number of
games the team has won? G
F. a 5 25(3)g
H. a 5 25g2 1 100
400
300
200
100
g
0
0 2 4 6 8 10
Games Won
I. a 5 225g2 1 100
Short Response
5. The data in the table show the population growth of a city since
Year
Population
0
5275
1
10,550
common ratio of 2 each year.
2
21,100
[2] Both parts answered correctly.
[1] One part answered correctly.
[0] Neither part answered correctly.
3
42,200
4
84,400
the year 2000. What kind of function models the data? How do you
know? exponential, because the population is multiplied by a
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67
Name
9-7
Class
Date
Enrichment
Linear, Quadratic, and Exponential Models
A regression curve is a mathematical curve that summarizes the general tendency
of the relationship between the variables. The most typical type of regression is
linear regression. However, there are other types of regression as well.
Match each type of regression described with its graph and equation.
1. The ends of a cubic regression curve
point in opposite directions. C
A.
y 5 ax 1 b
2. Exponential regression curves just
decrease or just increase. E
B.
y 5 ax2 1 bx 1 c
3. Linear regression is a line. Lines
increase or decrease. A
C.
y 5 ax3 1 bx2 1 cx 1 d
4. Quadratic regression is a parabola.
D.
Parabolas have U-shaped graphs. B
y 5 ax4 1 bx3 1 cx2 1 dx 1 e
5. The ends of a quartic regression curve
E.
point in the same direction. These
curves are M-shaped or W-shaped. D
y 5 abx
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68
Name
Class
Date
Reteaching
9-7
Linear, Quadratic, and Exponential Models
Data can resemble a linear function, a quadratic function, or an exponential
function. Recall the general shapes of these functions.
Linear Function
Quadratic Function
y
y
2
x
O
Ź2
Exponential Function
2
y
4
4
2
2
x
Ź2
O
Ź2
x
2
Ź2
O
2
Problem
Which model is most appropriate for the data points
(0.5, 1.75), (1, 1), (1.5, 1.75), (2, 4) and (2.5, 7.75)?
Graph the data points.
y
6
Notice that the points are not in a straight line. The points do
not have an exponential shape. A quadratic function would best
represent the data because the graph appears to be U-shaped.
4
2
x
O
Ź2
2
Exercises
Graph each set of points. Which model is most appropriate for each set?
1. (0, 0.25), (1, 0.75),
2. (0.5, 0.5), (1, 3),
3. (1, 1.5), (1.5, 1.75),
(1.5, 1.3), (2, 2.25),
(1.5, 4.5), (2, 5),
(2, 2), (2.5, 2.25),
(2.5, 3.9), (3, 6.75)
(2.5, 4.5), (3.5, 0.5)
(3, 2.5), (4, 3)
exponential
quadratic
linear
y
y
y
8
8
8
6
6
6
4
4
4
2
2
2
x
O
2
4
6
8
x
O
2
4
6
8
x
O
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69
2
4
6
8
Name
Class
Date
Reteaching (continued)
9-7
Linear, Quadratic, and Exponential Models
You can find the best model for a function using a table of values.
Linear Function
Quadratic Function
Exponential Function
y 5 22x 1 5
y 5 x2 1 7
y 5 3x
x
y
x
Ľ2
11
à1
à1
à1
à1
x
y
Ľ2
9
Ľ1
7
0
5
1
3
2
1
Ľ2
à1
Ľ2
à1
Ľ2
à1
Ľ2
à1
Ľ1
8
0
7
1
8
2
11
Ľ3
Ľ1
à1
à3
à2
à1
à2
à1
à2
à1
1
3
2
9
Ľ2
Ľ1
à1
ñ3
ñ3
ñ3
ñ3
For each increase of 1 for
the x values, the y values
have a common ratio.
For each increase of 1 for
the x values, the y values
change at different rates.
But, the differences have
a common difference.
For each increase of 1 for
the x values, the y values
have a common difference.
0
y
1
9
1
3
1
Exercises
Which kind of function best models the data in each table? Use differences or ratios.
4.
x
y
0
1
25
1
5
1
1
5
2
25
Ľ2
Ľ1
5.
exponential
6.
x
y
Ľ2
Ľ5
Ľ1
Ľ2
0
1
Ľ1
1
4
2
7
0
linear
x
Ľ2
1
2
7.
x
y
Ľ2
11
Ľ1
8.
9.
x
y
Ľ2
Ľ13
1
Ľ1
0
Ľ5
1
2
y
1
50
1
10
1
2
5
2
25
2
x
y
Ľ2
9
Ľ6
Ľ1
7
0
Ľ1
0
5
Ľ7
1
2
1
3
Ľ5
2
3
2
1
quadratic
quadratic
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70
exponential
linear
Name Class 9-8
Date Additional Vocabulary Support
Systems of Linear and Quadratic Equations
The column on the left shows the steps used to find the solutions of a system of
equations. Use this column to answer each question in the column on the right.
P roblem
1. Read the title of the example. What
Solve by Graphing
process are you going to use to solve
the problem?
What are solutions of the system?
Solve by graphing.
Solve by graphing
y 5 x2 1 2x 1 1
y 5 2x 1 1
Graph both equations in the same
coordinate plane.
2. What does graphing both equations in
the same coordinate plane mean?
Graph the two equations on the
y
same
coordinate plane.
4
2
x
−2
O
2
−2
Identify the point(s) of intersection, if
any.
3. What does intersection mean?
Intersection is where the two graphs
HSM11ALTR_0908_T00101
The
points of intersection are (23, 4)
and (0, 1).
cross.
Solution.
4. What do solutions of the system
mean?
The solutions of the system are
(23, 4) and (0, 1).
The solutions of the system are the
points
where the two graphs intersect.
Write true or false for each statement.
T
_____
5. Systems of linear and quadratic equations can have two solutions.
T
_____
6. Systems of linear and quadratic equations can have one solution.
T
_____
7. Systems of linear and quadratic equations can have no solutions.
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71
Name
Class
9-8
Date
Think About a Plan
Systems of Linear and Quadratic Equations
Business The number of customers at a coffee shop can be modeled by the
function y 5 0.25x2 2 5x 1 80, where x is the number of days since the beginning
of the month. The number of customers at another coffee shop can be modeled by
a linear function. Both shops have the same number of customers on days 10 and
20. What function models the number of customers at the second shop?
What do you know?
1. Using the function y 5 0.25x2 2 5x 1 80, find the values of y when x 5 10
and x 5 20.
When x 5 10, y 5
55
.
When x 5 20, y 5
80
.
What do you need to solve the problem?
2. How can you use these two data points to write a linear function that models
the number of customers at the second coffee shop?
Use the two points to find the slope of the line. Then use one of the points to find the
value of b in y 5 mx 1 b.
How do you solve the problem?
3. Write the linear function that models the data for the second coffee shop.
y 5 2.5x 1 30.
4. Check your function in Step 3. Explain the method you used.
Substitute both of the points into the equation:
55 5 2.5(10) 1 30
80 5 2.5(20) 1 30
55 5 25 1 30
80 5 50 1 30
55 5 55
80 5 80
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72
Name
9-8
Class
Date
Practice
Form G
Systems of Linear and Quadratic Equations
Solve each system by graphing.
1. y 5 x2 1 2
2. y 5 x2
y5x12
3. y 5 x2 2 5
y 5 2x
(0, 2); (1, 3)
y5x23
(0, 0); (2, 4)
4. y 5 x2 1 1
5. y 5 x2 2 4x 2 2
y5x11
y 5 2x 2 2
(0, 22); (3, 25)
(0, 1); (1, 2)
(21, 24); (2, 21)
6. y 5 x2 2 6x 2 7
y5x11
(21, 0); (8, 9)
Solve each system using elimination.
7. y 5 x2
8. y 5 x2 2 4
y5x12
y 5 2x 2 2
(22, 0); (1, 23)
(21, 1); (2, 4)
10. y 5 2x2 1 4x 2 3
y 5 2x 1 1
11. y 5 2x2 1 2x 1 4
y 5 2x 1 4
(0, 4); (3, 1)
(1, 0); (4, 23)
9. y 5 x2 2 2x 1 2
y 5 2x 2 2
(2, 2)
12. y 5 x2 2 x 2 6
y 5 2x 2 2
(21, 24); (4, 6)
13. The weekly profits of two different companies selling similar items that
opened for business at the same time are modeled by the equations shown
below. The profit is represented by y and the number of weeks the companies
have been in business is represented by x. According to the projections, what
week(s) did the companies have the same profit? What was the profit of both
companies during the week(s) of equal profit?
Company A: y 5 x2 2 70x 1 3341
Company X: y 5 50x 1 65 weeks 42 and 78; wk 42: $2165 profit; wk 78: $3965 profit
14. The populations of two different cities are modeled by the equations shown
below. The population (in thousands) is represented by y and the number of
years since 1970 is represented by x. What year(s) did the cities have the same
population? What was the population of both cities during the year(s) of equal
population?
Baskinville: y 5 x2 2 22x 1 350
Cryersport: y 5 55x 2 950 yrs 1995 and 2022; in 1995: 425,000 people; in 2022:
1,910,000 people
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Name
9-8
Class
Date
Practice (continued)
Form G
Systems of Linear and Quadratic Equations
Solve each system using substitution.
15. y 5 x2 1 x 2 60
y 5 2x 2 4
16. y 5 x2 2 3x 1 7
17. y 5 x2 2 2x 2 5
(2, 5); (5, 17)
(0, 25); (3, 22)
y 5 4x 2 3
(27, 218); (8, 12)
18. y 5 2x2 2 2x 2 4
7x 1 y 5 2
19. y 5 x2 1 6x
x2y54
y5x25
20. y 5 x2 1 4x 2 15
y 2 25 5 x
(24, 28); (21, 25)
(2, 212); (3, 219)
(28, 17); (5, 30)
Solve each system using a graphing calculator.
21. y 5 x2 1 5x 1 13
y 5 25x 1 3
22. y 5 x2 2 x 1 82
y 5 22x 1 50
(21.13, 8.64); (28.87, 47.36)
24. y 5 x2 2 2x 1 2.5
y 5 2x 2 1.25
(1.5, 1.75); (2.5, 3.75)
23. y 5 x2 2 12x 1 150
y 5 15x 2 20
(10, 130); (17, 235)
no solution
25. y 5 x2 2 0.9x 2 1
y 5 0.5x 1 0.76
26. y 5 x2 2 68
(20.8, 0.36); (2.2, 1.86)
y 5 25x 1 25.75
(7.5, 211.75); (212.5, 88.25)
27. Reasoning What are the solutions of the system y 5 2x2 2 11 and
y 5 x2 1 2x 2 8? Explain how you solved the system.
Set the equations equal:
2x2 2 11 5 x2 1 2x 2 8
Simplify to get 0 on one side: x2 2 2x 2 3 5 0
Factor:
(x 2 3) (x 1 1) 5 0
The solutions are (21, 29) and (3, 7).
28. Writing Explain why a system of linear and quadratic equations can only
have 0, 1, or two possible solutions.
The solutions for the system are the points where the graphs intersect. They can
intersect at 0, 1, or 2 points. There is no way to intersect a line
y
and parabola at more than two points.
29. Reasoning The graph at the right shows a quadratic function
and the linear function x 5 b.
Ź4
a. How many solutions does this system have? one solution
b. If the linear function were changed to y 5 b, how many
solutions would the system have? none
c. If the linear function were changed to y 5 b 1 3, how many solutions
would the system have? one, at the parabola’s vertex.
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74
2
x
O
2
Name
Class
Date
Practice
9-8
Form K
Systems of Linear and Quadratic Functions
Solve each system by graphing.
2. y 5 x2 1 2x
1. y 5 x2 2 3x
y 5 3x
y5x15
y
y
4
4
2
2
x
Ź4 Ź2
2
O
4
(5, 10) and (21, 4)
Ź4 Ź2
Ź2
Ź2
Ź4
Ź4
x
(0, 0) and (1, 3)
x
(24, 27) and
(21, 24)
4
4. y 5 x2 1 6x 1 1
3. y 5 x2
y5x23
y 5 22x
y
y
4
8
2
4
x
Ź4 Ź2
2
O
O
2
4
(22,4) and (0, 0)
Ź8 Ź4
O
Ź2
Ź4
Ź4
Ź8
4
8
Solve each system using elimination.
5. y 5 x2
y 5 3x
6. y 5 x2 2 5x
(0, 0) and (3, 9)
7. y 5 x2 1 6x 2 8
y5x22
y5x28
(4, 24) and (2, 26)
8. y 5 x2 1 20x 1 80 (210, 220) and
(29, 219)
(1, 21) and (26, 28)
y 5 x 2 10
9. The sales of two different products are modeled by the equations shown
below. The sales are represented by y and the number of weeks the products
have been selling is represented by x. According to the projections, what
week(s) did the products have the same amount of sales? What were the sales
of both products during the week(s) of equal sales?
Product 1: y 5 x2 2 17x 1 89
Product 2: y 5 17x 1 25 weeks 2 and 32; 59 and 569
10. The population of two different villages are modeled by the equations shown
below. The population (in thousands) is represented by y and the number of
years since 1975 is represented by x. What year(s) did the villages have the
same population? What was the population of both cities during the year(s) of
equal population?
Lewiston: y 5 x2 2 30x 1 540
Lockport: y 5 20x 1 15 years 1990 and 2010; 315,000 and 715,000
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75
Name
Class
9-8
Date
Practice (continued)
Form K
Systems of Linear and Quadratic Functions
Solve each system using substitution.
12. y 5 x2 2 x 2 5
11. y 5 x2 2 3x 2 27
y5x26
y 5 2x 2 1
(23, 29) and (7, 1)
(21, 23) and (4, 7)
14. y 5 x2 2 6
13. y 5 x2 2 4x 2 15
y 5 27x 1 12
y 5 23x 1 5
(24, 17) and (5, 210)
(29, 75) and (2, 22)
Solve each system using a graphing calculator.
15. y 5 x2 1 x 2 60
16. y 5 x2 2 6x 2 35
y5x14
y 5 x 1 25
(28, 24) and (8, 12)
(12, 37) and (25, 20)
17. y 5 x2 2 x 1 0.5
18. y 5 x2 1 0.15x 2 0.04
y 5 x 2 0.25
y 5 0.2x 1 0.1
(1.5, 1.25) and (0.5, 0.25)
(0.4, 0.18) and (20.35, 0.03)
19. Writing What are the solutions of the system y 5 3x2 1 2x 2 20 and
y 5 2x2 1 6x 1 1? Explain how you solved the system.
(23, 1) and (7, 141); See student’s work for explanation.
20. Reasoning The graph at the right shows a quadratic function
y
and the linear function x 5 b.
a. How many solutions does this system have? 1
b. Will the number of solutions be the same for any value of b?
Explain.
Yes; any value of b will give a vertical line that
intersects the parabola at one point.
c. If the linear function were changed to y 5 b, would the number of
solutions be the same for any value of b?
No, the number of solutions will change for
different values of b.
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76
2
x
2
2
2
Name
Class
Date
Standardized Test Prep
9-8
Systems of Linear and Quadratic Equations
Multiple Choice
For Exercises 1–4, choose the correct letter.
1. Which system of equations represents the graph shown? D
A. y 5 x 1 3
y5
x2
C. y 5 x 1 3
29
y5
12
y 5 2x2 2 18
B. y 5 x 2 3
x2
18 y
D. y 5 x 2 3
29
y5
2x2
2 18
6
Ź9 Ź6 Ź3
Ź6
3
Ź12
Ź18
2. What is the solution of the system of equations shown below?
y5x22
y 5 x2 2 8x 1 6 I
F. (21, 23) and (28, 210)
H. (0, 22) and (5, 3)
G. (2, 0) and (28, 210)
I. (1, 21) and (8, 6)
3. What is the solution of the system of equations shown below?
y 5 x2 2 5x 1 18
y 5 4x 1 4 C
A. (22, 24) and (27, 224)
B. (0, 4) and (2, 12)
C. (2, 12) and (7, 32)
D. (4, 20) and (5, 24)
4. An architect makes a drawing of a parabolic-shaped arch with a linear support
intersecting it in two places. The parabola can be modeled by the function
y 5 x2 2 5x 1 10. The line intersects the parabola when x 5 2 and x 5 4.
What is the equation of the line?
H
F. y 5 x 2 6
G. y 5 x 2 2
H. y 5 x 1 2
I. y 5 x 1 6
Short Response
5. Graph the following system of equations. How many solutions does this
system have? Explain your reasoning.
y 5 2x2 1 2 no solutions; the graphs don’t intersect.
y 5 2x 2 2
[2] Both parts answered correctly.
[1] One part answered correctly.
[0] Neither part answered correctly.
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77
6
x
9
Name
Class
Date
Enrichment
9-8
Systems of Linear and Quadratic Equations
You have solved systems of equations by graphing. You can also solve systems of
inequalities by graphing. All of the points that lie in the solution region of both
inequalities are in the solution system.
Graph y S 2 and y S 3x 2 2.
4
y
2
x
Ź4
O
Ź2
2
The solution is the region where
the solutions to the inequalities overlap.
4
Ź2
Ź4
Practice
Graph each system of inequalities.
1. y $ 22 and x , 23
2. x , 2 and y # 20.5x 1 3
y
Ź4 Ź2
y
4
4
2
2
2
O
4
x
Ź4 Ź2
Ź2
Ź4
Ź4
3. y . 21 and y $ 5x 2 2
x
y
4
4
2
2
O
4
4. x # 22 and y . 4
y
Ź4 Ź2
2
O
Ź2
2
4
x
Ź4 Ź2
O
Ź2
Ź2
Ź4
Ź4
2
4
x
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78
Name
Class
Date
Reteaching
9-8
Systems of Linear and Quadratic Equations
You can solve a system of a quadratic equation and a linear equation by graphing.
As with systems of linear equations, all points where the two graphs coincide are
solutions to the system. A system of a quadratic equation and a linear equation
can have 2, 1, or 0 solutions.
Problem
What are the solutions of the system
y 5 x2 2 4x 2 1 and y 5 25? Solve by graphing.
y
x
O
2
4
Ź2 y âx2Ź4 x Ź1
Graph y 5 x2 2 4x 2 1 and y 5 25. The graph of
y 5 25 is a horizontal line.
Ź4
(2, Ź5) y âŹ5
The two graphs coincide at (2, 25). The solution of the
system is (2, 25).
Problem
What are the solutions of the system
y 5 2x2 1 1 and y 5 x 1 4? Solve by graphing.
y
4
y âxà4
Graph y 5 2x2 1 1 and y 5 x 1 4.
There are no points where the two graphs coincide. There are no
solutions to this system of equations.
Ź2
Ź4
2
O
2x
y âx2à1 Ź2
Exercises
Graph each system of equations. Solve the system.
1. y 5 x2 2 x 2 2
2. y 5 2x2 1 4x 2 5
y5x11
y5x25
y
6
2
4
Ź2
2
O
y5x23
y
y
6
Ź2
3. y 5 x2 2 4x 1 4
O
2
4
6
4
x
2
Ź2
2
4
6
Ź2
x
Ź4
(21,0); (3,4)
O
Ź2
Ź6
2
4
6
Ź2
(0,25); (3,22)
No solution
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79
x
Name
Class
9-8
Date
Reteaching (continued)
Systems of Linear and Quadratic Equations
You can solve system of a quadratic equation and a linear equation algebraically
just as you solved systems of linear equations algebraically.
Problem
What are the solutions of the system y 5 2x2 1 4x 2 1 and y 5 2x 1 3?
Use substitution.
y 5 2x2
2x 1 3 5 2x2
2x 1 3 2 3 5 2x2
2x 5 2x2
2x 1 x 5 2x2
0 5 2x2
1
1
1
1
1
1
4x
4x
4x
4x
4x
5x
2
2
2
2
2
2
1
1
123
4
41x
4
Substitute 2x 1 3 for y.
Subtract 3 from each side.
Simplify.
Add x to each side.
Simplify.
Use any method you learned in this chapter to solve the quadratic equation.
x5
2b 4 "b2 2 4ac
2a
Quadratic formula
x5
25 4 "52 2 4(21)(24)
2(21)
a 5 21, b 5 5, and c 5 24.
x5
25 4 !9
22
Simplify.
x5
25 2 !9
22
x54
25 1 !9
22
or
x5
or
x51
Find y for x 5 4.
y 5 2x 1 3
y 5 24 1 3 5 21
Find y for x 5 1.
y 5 2x 1 3
y 5 21 1 3 5 2
The solutions are (4, –1) and (1, 2).
Exercises
Solve each system algebraically.
4. y 5 2x2 1 4x 2 1
y 5 2x 1 3
(1, 2); (4, 21)
7. y 5
(0, 1); (3, 4)
y 5 2x 2 3
(2, 1); (4, 5)
2 2x 1 1
y5x11
x2
5. y 5 x2 2 4x 1 5
8. y 5
2x2
1x14
y5x13
(21, 2); (1, 4)
6. y 5 x2 1 6x 1 7
y5x13
(24, 21); (21, 2)
9. y 5 x2 1 2x 1 1
y5x13
(22, 1); (1, 4)
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80
Name
Class
Date
Chapter 9 Quiz 1
Form G
Lessons 9-1 through 9-4
y
4
Do you know HOW?
2
1. Graph the quadratic function y 5 x2 2 2.
Ź4 Ź2
2
O
x
4
Ź2
Ź4
Graph each function. Label the axis of symmetry and the vertex.
2. y 5 6x2 2 12x
3. f(x) 5 x2 1 x 2 12
y
y
2
O
Ź4 Ź2
4
2
4
x
Ź8 Ź4
O
Ź2
Ź4
Ź4
Ź8
Ź6
(1, Ź6)
Ź12
x=0
x=1
4
8
x
(0, 12)
4. A ball is thrown into the air with an initial upward velocity of 60 ft/s. Its height
h in feet after t seconds is given by the function h 5 216t2 1 60t 1 6.
a. After how many seconds will the ball hit the ground? about 4 s
b. What will the height be at t 5 3 seconds? 42 ft
Solve each equation by finding square roots.
5. x2 2 121 5 0 211; 11
6. 5x2 2 245 5 0 27; 7
7. Solve 4x2 2 36 5 0 by graphing the related function. 3; 23
Solve by factoring.
8. m2 1 8m 1 7 5 0
9. c2 5 8c
27; 21
10. n2 1 2n 2 24 5 0
26; 4
0; 8
Do You UNDERSTAND?
11. Reasoning Explain why either a or b must be 0 if ab 5 0, but neither a nor b
must be 4 when ab 5 4.
The only way for a product to be equal to 0 is if one of the factors is 0;
it is not possible to multiply two nonzero numbers to get 0. For nonzero
products, such as 4, there are several pairs of factors, such as 1 and 4 and
10 and 0.4. So it is not possible to conclude the values of a and b when the
product is not 0.
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81
Name
Class
Date
Chapter 9 Quiz 2
Form G
Lessons 9-5 through 9-8
Do You Know HOW?
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
1. r2 1 6r 5 16 28; 2
2. m2 2 12m 1 1 5 0 0.08; 11.92
Solve each equation using the quadratic formula. If necessary, round answers
to the nearest hundredth.
3. x2 2 4x 2 7 5 0 21.32; 5.32
4. 2x2 2 5x 2 12 5 0 21.5; 4
Solve each equation using any method. Explain why you chose the method you
used. If necessary, round to the nearest hundredth.
5. x2 1 6x 1 5 5 0
6. 3x2 2 12x 5 21
7. f 2 1 12f 5 0
0.09; 3.91 quadratic formula
212; 0
25; 21 factoring was easy
because answer wasn’t an
factoring was easy
integer
Find the number of real-number solutions of each equation.
8. 5x2 2 4x 1 6 5 0 no solution
9. 3a2 2 4a 2 5 5 0 two solutions
10. Which kind of function best models the data in the table? Write
an equation to model the data. linear; y 5 2x 1 5
11. What are the solutions of the system? (21, 4); (3, 8)
x
y
0
5
1
7
2
9
3
11
4
13
y 5 x2 2 x 1 2
y5x15
Do You UNDERSTAND?
12. Vocabulary Explain why the quantity b2 2 4ac is called the discriminant. This
quantity allows you to discriminate between the numbers of soluton to any quadratic
equation.
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82
Name Class Date Chapter 9 Test
Form G
Do You Know HOW?
Match each graph with its function.
1. y 5 22x2 1 2 A
A.
4
2
−4 −2
y
2. y 5 2x2 D
3. y 5 2x2 C
4. y 5 3x2 2 4 B
B. C. D.
4
2
x
O
−4 −2
2 4
y
x
O
4
2
y
−4
x
O
−4 −2
−2
2 4
4
2
y
−4 −2
2 4
x
O
2 4
−4
Find the equation of the axis of symmetry and the coordinates of the vertex of
the graph of each function.
HSM11A1TR_0901_T08301
HSM11A1TR_0901_T08302
HSM11A1TR_0901_T08303
HSM11A1TR_0901_T08304
3 2
1 2
2
5. y 5 24x2 1 3
x 5 0; (0,3)
6. y 5 x 1 5x 2 12 7. y 5 2 x 2 6x 1 5 8. y 5 4 x 1 8x
x 5 2; (2, 21)
x 5 216; (216, 264)
x 5 22.5
(22.5, 218.25)
Graph each function.
2
9. y 5 3 x2 10. y 5 2x2 1 3
y
−4 −2
y
4
4
2
2
O
2
4
x
−4 −2
O
−2
−2
−4
−4
5
11. Solve the system of equations. (2 12 , 22), (2, 5)
HSM11A1TR_0909_ANT004
y 5 2x2 2
3
y 5 3x 2 1
2
4
x
HSM11A1TR_0909_ANT005
Find the number of real-number solutions of each equation.
12. 0 5 8x2 one
13. 0 5 4x2 1 9
none
14. 0 5 23x2 1 x 2 415. 0 5 x2 2 5x
none
two
Find the value of n such that each expression is a perfect square trinomial.
16. p2 1 10p 1 n
25
17. y2 2 60y 1 n
18. x2 2 14x 1 n
49
900
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83
Name Class Date Chapter 9 Test (continued)
Form G
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
19. x2 2 18x 5 19 21; 19
20. 4a2 1 8a 2 20 5 0 23.45; 1.45
Find the number of real-number solutions of each equation.
21. x2 1 3x 5 4
22. 22x2 2 5x 5 0
23. 4x2 2 3x 5 22
two
two
24. x2 1 12 5 0
none
none
Solve each equation. If necessary, round to the nearest hundredth.
25. (x 2 5)(2x 1 1) 5 0
212 ; 5
26. x2 5 10x
28. 6x2
29. x2
2 8x 2 30 5 0
2123 ; 3
27. x2 2 7x 5 212
3; 4
0; 10
30. 2x2 1 5x 2 63 5 0
2 81 5 0
27; 4.5
29; 9
Do You UNDERSTAND?
31. Open-Ended Write an equation of a parabola that has two x-intercepts and a
minimum vertex. Include a graph of the parabola. Answers may vary. Sample: y 5 x2 2 2
Model each problem with a quadratic equation. Then solve.
1
32. The volume of a square pyramid is given by the formula V 5 3 hx2, where h is
the height of the pyramid and x is the length of one side of the base. A pyramid
with a height of 15 ft has a volume of 2880 ft 3 . What is the length of one side of
the base? 24 ft
33. The area of a rectangular soccer field is 5000 yd 2 . The length of the field is
twice the width. Find the dimensions of the field. width: 50 yd; length: 100 yd
Which model is most appropriate for the data shown in each graph?
34. 6
y
35. 6
y
36.
4
4
4
2
2
2
x
−2
6
O
x
−2
2
quadratic
O
x
−2
2
exponential
HSM11A1TR_0901_T08401
y
HSM11A1TR_0901_T08402
2
linear
HSM11A1TR_0901_T08403
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84
O
Name Class Date Chapter 9 Quiz 1
Form K
Lessons 9-1 through 9-4
Do you know HOW?
Find the equation of the axis of symmetry and the coordinates of the vertex of
the graph of each function.
1. y 5 25x2 1 1 x 5 0; (0, 1)
2. y 5 x2 2 4x 1 4 x 5 2; (2, 0)
Graph each function. Label the axis of symmetry and the vertex.
3. y 5 2x2 1 8x
4. y 5 3x2 2 x 2 10
y
y
8
8
4
4
x
−8 −4
O
4
x
−4 −2
8
O
−4
−4
(−2, −8) −8
−8
x=0
(0, −10)
x = −2
2
4
Solve each equation by finding square roots. If the equation has no real
solution, write no solution.
5.
x2 2 81 5 0 w9
HSM11A1TR_09_T003
6.
3x2 2 192 5 0 w8
HSM11A1TR_09_T005
Solve by factoring.
7. x2 5 26x 0, 26
8. x2 2 7x 2 8 5 0 21, 8
9. How many x-intercepts will the graph of each function have?
a. y 5 3x2 1
b. y 5 2x2 1 3 0
c. y 5 x2 2 6x 2
Do you UNDERSTAND?
10. Vocabulary What are the roots of an equation? Given an example of a
quadratic equation and its roots.
The roots of an equation are the solutions of the equation. Answers may vary.
Sample: 2x2 2 8 5 0; w2
1
11. Compare and Contrast How are the graphs of y 5 5x2 and y 5 5 x2
different? How are they similar?
The graph of y 5 5x2 is narrower than the graph of y 5 15 x2 . The graphs both
open up and have the same vertex, (0, 0), and axis of symmetry, x 5 0.
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85
Name Class Date Chapter 9 Quiz 2
Form K
Lessons 9-5 through 9-8
Do you know HOW?
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
1. n2 2 5n 5 3 5.54, 20.54
2. v2 1 3v 2 5 5 0 1.19, 24.19
Use the quadratic formula to solve each equation. If necessary, round to the
nearest hundredth.
3. 6g2 1 g 2 2 5 0 0.5, 20.67
4. a2 1 7a 2 9 5 0 28.11, 1.11
Find the number of solutions of each equation.
5. 2h2 2 3h 1 2 5 0 0
6. b2 1 5b 2 3 5 0 2
Find the value of n such that each expression is a perfect square trinomial.
7. a2 1 18a 1 n 81
8. t2 2 22t 1 n 121
9. b2 1 20b 1 n 100
Solve each equation. If necessary, round to the nearest hundredth.
10. d2 2 d 5 30 25, 6
11. 15x2 2 23x 1 4 5 0 0.2, 1.33
Do you UNDERSTAND?
12. Open-Ended Write a set of data points that you could model with a linear
function. What function models the data?
Answers may vary. Sample: (1, 5), (2, 9); f (x) 5 4x 1 1
13. Reasoning What are the possible number of solutions of a system consisting
of a linear equation and a quadratic equation? What does the number of
solutions tell you about the graphs of the equations?
A system of a linear equation and a quadratic equation can have 0, 1, or 2
solutions. The number of solutions is the number of times the graphs of the two
equations intersect.
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86
Name Class Date Chapter 9 Test
Form K
Do you know HOW?
Find the equation of the axis of symmetry and the coordinates of the vertex of
the graph of each function.
1. y 5 3x2 2 2 x 5 0; (0, 22)
2. y 5 x2 2 6x 1 5 x 5 3; (3, 24)
Graph each function. Label the axis of symmetry and the vertex.
3. f (x) 5 x2 1 2x 1 1
4. y 5 x2 2 5x 1 4
y
y
8
4
(−1, 0)
−4 −2
4
2
x
x
O
2
−8 −4
4
O
−2
x = −1
−4
−4
−8
4
8
(2.5, −2.25)
x = 2.5
5. A water balloon is tossed into the air with an upward velocity of
25 ft/s. Its height h(t) in ft after t seconds is given by the function
h(t) 5 216t2 1 25t 1 3.
HSM11A1TR_09_T006
HSM11A1TR_09_T004
a. After how many seconds will the balloon hit the
ground? 1.67 sec
b. What will the height be at t 5 1 second? 12 ft
Solve each equation by finding square roots. If the equation has no real
solution, write no solution.
6. x2 2 121 5 0 w11
7. 4x2 2 144 5 0 w6
Solve by factoring.
8. z2 1 10z 1 21 5 0 23, 27
9. t2 5 5t 0, 5
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
10. t2 1 6t 2 11 5 0 27.47, 1.47
11. x2 2 3x 5 21 0.38, 2.62
Use the quadratic formula to solve each equation. If necessary, round to the
nearest hundredth.
12. p2 2 11p 5 21 0.09, 10.91
13. 2x2 1 10x 5 23 24.68, 20.32
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Name Class Date Chapter 9 Test (continued)
Form K
What type of function best models the data in each table? Write an equation to
model the data.
14. 15.
x
x
y
quadratic; y 5 0.25x2
0
0
1
0.25
2
y
22
213
21
28
1
0
23
3
2.25
1
2
4
4
2
7
linear; y 5 5x 2 3
Solve each system using substitution.
(2, 21), (4, 1)
16. y 5 x 2 3
y5
x2
17. y 5 2x 1 2
y5
2 5x 1 5
2x2
(21, 0), (5, 12)
1 6x 1 7
Do you UNDERSTAND?
18. Writing Describe how you know by looking at the equation of a quadratic
function whether the graph will open up or down.
If the coefficient of the x2 term is positive, the graph opens up. If the
coefficient of the x2 term is negative, the graph opens down.
19. Reasoning Can you use the axis of symmetry to make graphing a quadratic
equation easier? Explain.
Yes, plot several points on one side of the axis of symmetry. Then reflect
those points across the axis of symmetry to graph the other half of the
quadratic equation.
20. Open-Ended Write a quadratic equation that has only one real-number
solution.
Answers may vary. Sample: 4x2 1 4x 1 1 5 0
21. Reasoning Find a nonzero value of k such that kx2 2 48x 1 64 5 0 has only
one solution. What is the solution?
k 5 9; 83 or 2.67
22. Writing Explain how the value of the discriminant, b2 2 4ac, can be used to
predict the number of solutions an equation has.
If b2 2 4ac S 0, the equation has two real-number solutions. If b2 2 4ac 5 0, the
equation has one real-number solution. If b2 2 4ac R 0, the equation has no
real-number solutions.
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Name
Class
Date
Performance Tasks
Chapter 9
Give complete answers.
TASK 1
You and your friend need to graph quadratic functions of the form y 5 ax2 and
y 5 ax2 1 c. Your friend asks you to write some hints to help her graph these
types of equations.
a. Explain the role of a.
z a z affects the width of the parabola. 3x2 is narrower than x2 , which is narrower than
1 2
3 x . If a is negative, the parabola opens down; if a is positive, it opens up.
b. Explain the maximum and minimum.
A parabola that goes up has a minimum at (0, c); a parabola that opens down has a
maximum at (0, c)
c. What is the vertex?
The maximum or minimum point on the graph, (0, c).
d. Explain the role of c.
c shifts the graph of y 5 ax2 up or down z c z units. Up if c S 0, down if c R 0.
[4] Student shows understanding of the task, completes all portions of the task
appropriately, and fully supports work with appropriate explanations.
[3] Student shows understanding of the task, completes all portions of the task
appropriately, and supports work with appropriate explanations with a minor error.
[2] Student shows understanding of the task. but needs to explain better.
[1] Student shows minimal understanding of the tast or offers little explanation.
[0] Student shows no understanding of the task and offers no explanation.
TASK 2
a. Write and graph a quadratic function in standard form, y 5 ax2 1 bx 1 c, that
opens downward. Identify the axis of symmetry, vertex, and the y-intercept.
Answers may vary. Sample: y 5 2x2 1 2x 1 1; x 5 1; (1, 2); 1
b. List two possible real-life situations that can be modeled by using a quadratic
function written in standard form, y 5 ax2 1 bx 1 c.
Answers may vary. Sample: height as a function of time for a dropped or thrown
object; height as a function of horizontal distance for a thrown object.
[4]
[3]
[2]
[1]
[0]
Student gives clear and correct calculations and explanations.
Student gives calculations and explanations that may contain some minor errors.
Student answers one part correctly and the other part has major errors.
Student gives calculations or explanations that contain major errors or omissions.
Student makes little or no effort.
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Name
Class
Date
Performance Tasks (continued)
Chapter 9
TASK 3
You are planning to open a frozen yogurt stand. You would like to offer cones in
three sizes: small, medium, and large. Use the following formula for the volume
of a cone and information to find the radius of each cone. Round to the nearest
hundredth.
V 5 13 pr2h
a. Small cone: V 5 8 in. 3 , h 5 4 in. 1.38 in.
b. Medium cone: V 5 12 in. 3 , h 5 5 in. 1.51 in.
c. Large cone: V 5 16 in. 3 , h 5 6 in. 1.60 in.
[4]
[3]
[2]
[1]
[0]
Student gives clear and correct calculations and explanations.
Student gives calculations and explanations that may contain some minor errors.
Student answers one part correctly and the other part has major errors.
Student gives calculations or explanations that contain major errors or omissions.
Student makes little or no effort.
TASK 4
Find the value of the discriminant and the number of solutions. Verify your results
by solving each quadratic equation using the quadratic formula. Show all work.
a. 2y2 1 7y 5 23 25; two; 23, 212
b. p2 2 8p 1 16 5 0 0; one; 4
c. 3x2 5 2x 2 5 256; none; no solution
[4] Student shows understanding of the task, completes all portions of the task
appropriately with no errors in computation, and fully supports work with
appropriate explanations.
[3] Student shows understanding of the task, completes all portions of the task
appropriately with no error in computation, and supports work with appropriate
explanations.
[2] Student shows understanding of the task. but makes errors in computation
resulting in incorrect answer(s), or needs to explain better.
[1] Student shows minimal understanding of the tast or offers little explanation.
[0] Student shows no understanding of the task and offers no explanation.
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90
Name Class Date Cumulative Review
Chapters 1–9
Multiple Choice
For Exercises 1–14, choose the correct letter.
1. How many real solutions does the equation x2 1 4x 1 1 5 0 have? C
A.0
B. 1
C. 2
D. 3
2. What is u 7.4u ? I
F. 27.4
H. 7
G. 27
I. 7.4
3. What is the y-intercept of the line that is parallel to 2x 1 3y 5 4 and contains
the point (6, 22)? C
A. 22
B. 1
C. 2
D. 3
1
4. What percent of 1 is 2 ? G
F. 25%
G. 50%
H. 75%
I. 100%
5. Which expression shows (2x2 y24)3(23x24 y5)22 in simplified form? B
8x2
8x14
2x14
72
A. 2 B. 22 C. 2 22 D. 2 2
9y
9y
3y
x y
6. What is the standard form of the product (4x 2 3)(7x 1 2)? G
F. 28x2 2 4 2 29x 1 6 G. 28x2 2 13x 2 6 H. 11x2 2 6
I. 3x2 2 29x 2 6
7. Simplify 5x3 1 x 2 1 2 (x2 1 x 1 3). A
A. 5x3 2 x2 2 4
B. 5x3 1 x2 1 2x 1 2 C. 4x3 1 2x 1 2
D. 4x3 2 x2 2 4
8. What is the value of the discriminant of 3x2 1 3x 1 6 5 0? F
F. 263
G. 54
H. 63
I. 81
9. Which ordered pair is a solution of the system below? B
y 5 x2 1 8x 2 2
y 5 2x 2 7
A. (25, 17)
B. (21, 29)
C. (1, 25)
D. (5, 3)
10. Which number line shows the solution set to the inequality x 2 5 . 26? H
F. H.
G.
–5 –4 –3 –2 –1
0
1
2
3
4
5
–5 –4 –3 –2 –1
0
1
2
3
4
5
HSM11A1TR_0901_T09101
I.
–5 –4 –3 –2 –1
0
1
2
3
4
5
–5 –4 –3 –2 –1
0
1
2
3
4
5
HSM11A1TR_0901_T09102
HSM11A1TR_0901_T09104
HSM11A1TR_0901_T09103
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91
Name Class Date Cumulative Review (continued)
Chapters 1–9
1
11. A line perpendicular to y 5 4 x 1 3 passes through the point (0, 6). Which
other point lies on the line? B
A. (4, 22)
B. (3, 26)
C. (0, 26)
D. (22, 2)
12. Between what two consecutive integers is 2!52? H
F. 25 and 26
G. 26 and 27
H. 27 and 28
I. 28 and 29
13. What is the equation of the axis of symmetry of y 5 2x2 1 4x 2 3? A
A. x 5 21
B. x 5 0
C. x 5 1
D. x 5 2
14. What is the common ratio of the geometric sequence 108, 36, 12, 4, … ? H
1
F. 3
G. 23
H. 3 I. 108
15. Simplify 8 1 6 ? 4 4 2 1 2. 22
16. If y varies directly with x and y 5 70 when x 5 14, find x when y 5 110. 22
17. A swimsuit has been marked down from an original price of $75 to $56.25. By
what percent of the original price has the suit been marked down? 25%
18. The perimeter of a rectangle is 60 m, and its length is twice its width. Find the
length of the rectangle. 20 m
x
4.5
19. What is the solution to the proportion 24 5 12 ? 9
20. The cost of four lunches and six dinners is $122. The cost of five lunches and
eight dinners is $160. Find the cost of one lunch. $8
21. Writing Explain the following statement: All functions are relations, but not
all
relations are functions. A function is a relation in which each input has exactly one
output. A relation in which an input has two or more outputs, such as y 5 wx, is not a
function.
22. Extended Response Graph y 5 6x2 2 2x. Label the vertex, axis of
symmetry, and x-intercepts of the graph of the equation.
y
1
1
2
(0, 0)
−1 −1 O
2 −1
2
−1
(
1
, 0)
3
1
1 x
2
1 ∙1
( ,
)
6 6
1
x5
6
[4] Parabola, vertex, intercepts, and axis of symmetry shown
clearly
[3] All parts shown with only one minor area
[2] Most parts shown correctly, but at least one element
incorrect or omitted
[1] At least one part of response completed correctly
[0] No parts of response completed correctly
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HSM11A1TR_0909_ANT006
92
TEACHER INSTRUCTIONS
Chapter 9 Project Teacher Notes: Full Stop Ahead
About the Project
In this project, students will calculate stopping distances and discover
relationships among speed, reaction time, and stopping distance.
Introducing the Project
• Ask students to estimate how long it takes to stop a car traveling on dry
pavement. Encourage students to consider what factors affect stopping
distance. Responses may include speed, road conditions, and reaction time.
• Gather statistics to demonstrate actual stopping distances for vehicles. You
may want to have the class measure the actual distances so that students can
better visualize data.
• Ask students to look at Activity 1. Explain that they will use the given formula
to calculate safe stopping distances for different speeds.
• Have students graph the results of their calculations.
• Challenge students to compare their calculated results with their earlier
estimations and draw a conclusion about the differences.
Activity 1: Graphing
Students use a given equation to calculate safe stopping distances. They make
tables and graphs to display the results.
Activity 2: Calculating
Students will evaluate the given formulas to complete the table. They will also
come to conclusions about safe driving distances.
Activity 3: Reasoning
Students use the formula from Activity 1 to calculate the maximum speed at which
the car should travel in order to stop in 150 ft.
Activity 4: Communicating
Students work in groups to plan skits in which they demonstrate the facts they
learned about safe distances in driving.
Finishing the Project
You may wish to plan a project day on which students share their completed
projects. Encourage groups to explain their processes as well as their results.
• Have students review their equations, graphs, and explanations.
• Ask groups to share their insights that resulted from completing the project,
such as any shortcuts they found for using formulas or making graphs.
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Name
Class
Date
Chapter 9 Project: Full Stop Ahead
Beginning the Chapter Project
What is a safe distance between cars traveling on the highway? After you apply
brakes to stop your car, how far will your car travel before coming to a full stop?
How do accident investigators determine whether cars involved in accidents were
traveling at safe speeds? There are many variables that affect how quickly a car can
stop. These variables include the car’s speed, the driver’s reaction time, the type of
road, the weather conditions and, of course, the effectiveness of the brakes.
As you work through the activities, you will use formulas to estimate safe speeds
under various conditions. You will make a graph to illustrate the relationship
between speed and stopping distance. Then, you will plan a skit with your
classmates to illustrate what you have learned about safe highway driving.
List of Materials
• Calculator, graph paper
Activities
Activity 1: Graphing
To reduce the likelihood of an accident when driving, you should consider how
far your car will travel before safely coming to a stop for the speed at which you
are traveling. Assume you are traveling on a dry road and have an average reaction
time. The formula d 5 0.044s2 1 1.1s gives you a safe stopping distance d in feet,
where s is your speed in mi/h. Make a table of values for speeds of 10, 20, 30, 40,
50, and 60 mi/h. Then, graph the function.
Activity 2: Calculating
Suppose a car left a skid mark d feet long. The formulas will estimate the speed s
in miles per hour at which the car was traveling when the brakes were applied.
• Use the formulas to complete the table. Round to the nearest mile per hour.
Traveling Speed
Dry Road
s 5 !27d
Wet Road
s 5 !13.5d
Skid Mark Length (d)
Estimated Speed (s)
Dry Road
Wet Road
60 ft
120 ft
• Why do you think the estimates of speed do not double when the skid marks
double in length? Based on these results, what conclusions can you make
about safe distances between cars?
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Name
Class
Date
Chapter 9 Project: Full Stop Ahead (continued)
Activity 3: Reasoning
Suppose you are driving on a dry road with 150 ft (about 10 car lengths) between
your car and the car in front of you. Use the formula from Activity 1 to find the
maximum speed you should be traveling in order to leave a safe stopping distance.
Activity 4: Communicating
Work with a group of your classmates to plan a skit that will demonstrate what you
have learned about safe distances in driving. Illustrate the relationships among
reaction times, road conditions, speeds, and stopping distances.
Finishing the Project
The answers to the four activities should help you complete your project. Gather
together all the data you compiled as you worked on the project. Include the
equations you used and your graphs. Discuss your conclusions about safe driving
speeds, stopping distances, and road conditions with your classmates. Then, as a
group, plan and rehearse your skit.
Reflect and Revise
Present your skit to a small group of classmates. After you have heard their
comments, decide if your presentation is clear and convincing. If needed, make
changes to improve your skit before presenting it to the rest of the class.
Extending the Project
If you have access to the Internet, explore some of the forums and user groups that
are related to driving and motor vehicles.
You may also want to contact a highway patrol officer or a registry of motor vehicle
official for more information about the habits of drivers. Ask them what errors or
violations are most common.
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Name
Class
Date
Chapter 9 Project Manager: Full Stop Ahead
Getting Started
Read the project. As you work on the project, you will need a calculator, graph
paper, materials on which you will record your calculations, and materials to make
accurate and attractive graphs.
Checklist
Suggestions
☐ Activity 1: calculating safe
stopping distance
☐ Consider what scale to choose for your graph.
☐ Activity 2: estimating speed
☐ Compare your answers with someone else’s.
☐ Activity 3: calculating
maximum speed
☐ Consider whether your solution is reasonable.
☐ Activity 4: presenting a skit
☐ Think of other relationships to represent.
☐ skit presentation
☐ How can your skit effectively illustrate the danger of
driving too closely behind the car in front of you? Will
your work on this project change the way you drive?
What factors affect reaction time?
Scoring Rubric
4
Your calculations are correct. The graph is neat, accurate, and clearly shows
the relationship between the variables. The graph has appropriate scales.
The skit convinces viewers of the relationships among reaction time, road
conditions, speed, and stopping distance.
3
Your calculations are mostly correct, but contain minor errors. The graph is
neat, and mostly accurate, with minor errors in scale. The skit illustrates a
relationship between different driving conditions and stopping distances.
2
Your calculations contain major errors. The graph contains inaccuracies.
The skit should be expanded to make a convincing argument.
1
A few elements of the project are accurate and limited understanding of the
subject.
0
Major elements of the project are incomplete or missing.
Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.
Teacher’s Evaluation of Project
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