THE CHARLES A. DANA CENTER
THE UNIVERSITY OF TEXAS AT AUSTIN
Algebra TEKS
Assessment
Supplement
a Texas SSI Publication
This publication is based upon work supported by the National Science Foundation
under Cooperative Agreement #ESR-9250036. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and
do not necessarily reflect the views of the National Science Foundation.
Copyright © 1998 by Texas Statewide Systemic Initiative.
The Charles A. Dana Center, The University of Texas at Austin, Austin, Texas.
Permission to photocopy is granted for educational purposes. Permission must be
sought for commercial use of any or all of this document.
This book was designed and produced by John Budz & Vee Sawyer of
Firefly Multimedia in Austin, Texas (firefly@jumpnet.com).
It was printed by Mpress, Inc. of Austin, Texas.
ii
Algebra Action Team
ASSESSMENT WORKING GROUP
Barrie Madison, Chair
David D. Molina, Texas SSI Staff Liaison
Bill Hadley, Consultant
Robbie Bonneville
Cindy Boyd
Armando Cisneros
Lucy Flores-Sanchez
Susan T. Funk
Diana Garcia
Juan Manuel Gonzalez
Mary Alice Hatchett
Lewisville ISD
The Charles A. Dana Center
Pittsburgh Public Schools
La Joya ISD
Abilene ISD
Austin ISD
Edinburg ISD
Ysleta ISD
United ISD
Laredo ISD
Round Rock ISD
Sallie Langseth
Deer Park ISD
Lori Mitchell
Lewisville ISD
Diane Reed
Ysleta ISD
Jane Silvey
ESC Region VII
Liz Smith
Martinsville ISD
Susan Thomas
Alamo Heights ISD
Terry Whistler
Austin ISD
iii
table of contents
Introduction
l
v
Texas Essential Knowledge and Skills for Algebra I
l
viii
Foundations for Functions
b1 Understanding Functions
l
2
b2 Properties and Attributes of Functions
b3 Representing Situations Using Algebra
l
l
6
10
b4 Using Algebraic Skills to Solve Problems
l
14
Linear Functions
c1
Representations of Linear Functions
c2 Slope and Intercepts
l
l
18
22
c3 Formulating and Solving Equations and Inequalities
l
c4 Formulating and Solving Systems of Linear Equations
Quadratic and Other Nonlinear Functions
d1 Graphs and Parameters of Quadratic Functions
d2 Solving Quadratic Equations
l
40
d3 Non Linear and Non Quadratic Functions
iv
l
44
l
36
28
l
32
introduction
T
he state goal that all students successfully complete Algebra 1 has
tremendous implications and impact on how teachers assess student mastery of Algebra 1 summatively as well as day-to-day. Created with “Algebra for All” in mind, the Algebra 1 Texas Essential Knowledge and Skills (TEKS)
and their supporting Performance Descriptions have determined the content of
the course. The potential importance of the Algebra 1 End-of-Course Exam on
campus/district accountability has focused attention on the assessment of student performance. The purpose of this document is to provide a model of how
to assess the thinking skills addressed in the Texas Essential Knowledge and
Skills in Algebra 1, and in doing so to be confident that students will be successful on the Algebra 1 End-of-Course Exam.
With increased opportunities for professional development, many teachers
are implementing lessons emphasizing concept development and relevance of
content. Yet, assessment of these lessons may remain traditionally focused on
skills and concepts that may not require students to prove their ability to solve
problems requiring high levels of thinking. This document attempts to bridge
this gap by providing sample assessments that require students not to demonstrate their ability to only manipulate symbols, but also to use and/or apply important mathematical concepts. The assessment items support the implementation of the Texas Essential Knowledge and Skills and their Performance
Descriptions and are connected to the Algebra 1 End-of-Course Exam. The document also includes recommendations for optimizing Algebra 1 End-of-Course
results and a bibliography of additional resources for teachers.
This document was written and tested by the Assessment Working Group of
the Texas SSI Algebra Action Team. The Team consisted of classroom teachers,
school administrators, regional service center representatives, district facilitators, and higher education personnel.
Purpose of the Assessment Document
For teachers, the document provides
• examples of TEKS-based assessment items for use in Algebra 1,
• sample assessment items that can be an integral part of teaching,
learning, and student evaluation,
v
• sample assessment items that complement and enhance teacher understanding of the Algebra 1 TEKS,
• sample performance-based assessment items,
• sample assessment items for teachers to use as models in creating similar
assessment items, and
• connections between the Algebra 1 TEKS and the Algebra 1
End-of-Course Exam.
Students successful on the sample assessment items in this document
• will demonstrate a deeper understanding of the mathematics,
• will more readily make connections within mathematics and to other
disciplines,
• will perform better on standardized tests, such as the Algebra 1
End-of-Course Exam, and
• will be able to solve a problem in multiple ways.
Document Format
The Assessment Document follows the sequence of the Algebra 1 TEKS. For
each Knowledge and Skill statement, there is a comprehensive/global assessment item. For each Performance Description, there is a shorter item that supports the type of thinking required for the student to be successful on the
Knowledge and Skill problem.
Finally, Algebra 1 End of Course questions from released tests are connected to each Knowledge and Skill statement. If students can successfully
complete the suggested assessments for the complete TEKS statement (Knowledge and Skill and the Performance Descriptions), they should be prepared for
the corresponding End-of-Course Exam questions.
Use of the Document
This document is intended to be used by individual teachers and/or campus teams as a model for the kind of assessments that are appropriate in today’s
Algebra 1 classroom. It also can be useful in the professional development of
teachers and can be incorporated as part of the appropriate TEXTEAMS training
such as The Algebra Institute.
vi
Recommendations for Optimizing
End-of-Course Results
In the Algebra 1 Classroom,
• use graphing calculators as an integral component of the course.
• teach test-taking skills—especially how to complete grid-in answers.
• use sample assessments from this document as part of ongoing classroom
assessment.
• use items from released tests as part of ongoing classroom assessment.
For the testing environment,
• have the students’ algebra teachers administer the examination.
• do not set real or implied time limits.
• have a tutorial session immediately before the test that addresses BIG
MATHEMATICAL IDEAS and test-taking tips.
• provide healthy snacks before students enter to take the examination.
• encourage frequent breaks.
• encourage students to answer items in the test booklet and then transfer
answers to the bubble sheet.
• have teachers verify that the students have recorded an answer to every
question before accepting students’ answer sheets.
Bibliography
Mathematics Assessment, from National Council of Teachers of Mathematics.
Waves of Learning Issue V Academic Assessment, by Carolyn S. Carr, Ph.D., Eastern
Washington University, from Texas Assocation for Supervision and Curriculum
Development.
Assessing Student Outcomes, by Robert J. Marzano, Debra Pickering, and Jay
McTighe, from Association for Supervision and Curriculum Development.
vii
Texas Essential Knowledge
and Skills for Algebra 1
From
Texas Administrative Code
Chapter 111. Texas Essential
Knowledge and Skills in
Mathematics
Subchapter C. High School
§111.32 Algebra 1 (one credit)
(a)
BASIC UNDERSTANDINGS.
(1) Foundation concepts for high school mathematics. As presented in Grades K-8,
the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students will continue to build on this foundation as they expand their
understanding through other mathematical experiences.
(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical
role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to
study relationships among quantities.
(3) Function concepts. Functions represent the systematic dependence of one quantity on another. Students use functions to represent and model problem situations and to analyze and interpret relationships.
(4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in
many situations to set up equations and use a variety of methods to solve these
equations.
viii
(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and
technology, including, but not limited to, powerful and accessible hand-held
calculators and computers with graphing capabilities and model mathematical
situations to solve meaningful problems.
(6) Underlying mathematical processes. Many processes underlie all content areas
in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication,
connections within and outside mathematics, and reasoning, as well as multiple
representations, applications and modeling, and justification and proof.
(b)
FOUNDATIONS FOR FUNCTIONS:
KNOWLEDGE AND SKILLS AND PERFORMANCE DESCRIPTIONS.
(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student describes independent and dependent quantities in
functional relationships.
(B) The student gathers and records data, or uses data sets, to determine
functional (systematic) relationships between quantities.
(C) The student describes functional relationships for given problem situations and writes equations or inequalities to answer questions arising from the situations.
(D) The student represents relationships among quantities using concrete
models, tables, graphs, diagrams, verbal descriptions, equations, and
inequalities.
(E) The student interprets and makes inferences from functional relationships.
ix
(2) The student uses the properties and attributes of functions.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student identifies and sketches the general forms of linear (y = x)
and quadratic (y = x 2) parent functions.
(B) For a variety of situations, the student identifies the mathematical
domains and ranges and determines reasonable domain and range
values for given situations.
(C) The student interprets situations in terms of given graphs or creates
situations that fit given graphs.
(D) In solving problems, the student collects and organizes data, makes
and interprets scatterplots, and models, predicts, and makes decisions and critical judgments.
(3) The student understands how algebra can be used to express generalizations
and recognizes and uses the power of symbols to represent situations.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student uses symbols to represent unknowns and variables.
(B) Given situations, the student looks for patterns and represents generalizations algebraically.
(4) The student understands the importance of the skills required to manipulate
symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in
problem situations.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student finds specific function values, simplifies polynomial expressions, transforms and solves equations, and factors as necessary
in problem situations.
(B) The student uses the commutative, associative, and distributive
properties to simplify algebraic expressions.
x
(c)
LINEAR FUNCTIONS:
KNOWLEDGE AND SKILLS AND PERFORMANCE DESCRIPTIONS.
(1) The student understands that linear functions can be represented in different
ways and translates among their various representations.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student determines whether or not given situations can be represented by linear functions.
(B) The student determines the domain and range values for which linear functions make sense for given situations.
(C) The student translates among and uses algebraic, tabular, graphical,
or verbal descriptions of linear functions.
(2) The student understands the meaning of the slope and intercepts of linear functions and interprets and describes the effects of changes in parameters of linear
functions in real-world and mathematical situations.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations.
(B) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.
(C) The student investigates, describes, and predicts the effects of
changes in m and b on the graph of y = mx + b.
(D) The student graphs and writes equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.
(E) The student determines the intercepts of linear functions from
graphs, tables, and algebraic representations.
(F) The student interprets and predicts the effects of changing slope and
y-intercept in applied situations.
(G) The student relates direct variation to linear functions and solves
problems involving proportional change.
xi
(3) The student formulates equations and inequalities based on linear functions,
uses a variety of methods to solve them, and analyzes the solutions in terms of
the situation.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student analyzes situations involving linear functions and formulates linear equations or inequalities to solve problems.
(B) The student investigates methods for solving linear equations and inequalities using concrete models, graphs, and the properties of
equality, selects a method, and solves the equations and inequalities.
(C) For given contexts, the student interprets and determines the reasonableness of solutions to linear equations and inequalities.
(4) The student formulates systems of linear equations from problem situations,
uses a variety of methods to solve them, and analyzes the solutions in terms of
the situation.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student analyzes situations and formulates systems of linear
equations to solve problems.
(B) The student solves systems of linear equations using concrete models, graphs, tables, and algebraic methods.
(C) For given contexts, the student interprets and determines the reasonableness of solutions to systems of linear equations.
(d)
QUADRATIC AND OTHER NONLINEAR FUNCTIONS:
KNOWLEDGE AND SKILLS AND PERFORMANCE DESCRIPTIONS.
(1) The student understands that the graphs of quadratic functions are affected by
the parameters of the function and can interpret and describe the effects of
changes in the parameters of quadratic functions.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student determines the domain and range values for which quadratic functions make sense for given situations.
(B) The student investigates, describes, and predicts the effects of
changes in a on the graph of y = ax 2.
xii
(C) The student investigates, describes, and predicts the effects of
changes in c on the graph of y = x 2 + c.
(D) For problem situations, the student analyzes graphs of quadratic
functions and draws conclusions.
(2) The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student solves quadratic equations using concrete models, tables, graphs, and algebraic methods.
(B) The student relates the solutions of quadratic equations to the roots
of their functions.
(3) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations.
FOLLOWING ARE PERFORMANCE DESCRIPTIONS.
(A) The student uses patterns to generate the laws of exponents and applies them in problem-solving situations.
(B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic
methods.
(C) The student analyzes data and represents situations involving exponential growth and decay using concrete models, tables, graphs, or
algebraic methods.
Source: The provisions of this §111.32 adopted to be effective
September 1, 1998, 21 TexReg 7371.
xiii
Algebra TEKS
Assessment
Supplement
b1
Foundations for Functions
The student understands that a function
“represents
a dependence of one quantity on
another and can be described in a variety of
ways.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
The time it takes a student to walk home from school is related to
the distance between home and school.
1. Identify which quantity is independent and which quantity is dependent.
2. Sketch a reasonable graph that describes this situation.
3. If Jackie walks at a rate of 3 miles per hour, complete the table shown.
Distance in Miles
3
5
?
7
?
z
z
z
z
z
z
Time in Minutes
60
?
120
?
?
4. Graph the data shown in the table.
5. Write an equation to represent the relationship.
6. If Jackie increases her rate by 1 mph, how far was she from home if it took her
three hours to walk that distance?
2
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
At the time this document was published, there were not any released items that
would apply.
Foundations for Functions
3
performance questions
Performance Descriptions and the type of assessment
items students should be able to perform
(b)(1)(A)
The student describes independent and dependent quantities in functional
relationships.
1. For a given job, the number of hours worked and the amount of money
earned are related. Identify which quantity is independent and which is
dependent. Defend your answer. Note: either quantity could be dependent
depending on the student’s response.
(b)(1)(B)
The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.
1. Using a metric tape, measure the diameter and circumference of at least 5
different circles. Record the data in a table and describe the functional
relationship between the 2 quantities.
(b)(1)(C)
The student describes functional relationships for given problem situations
and writes equations or inequalities to answer questions arising from the situations.
1. Membership in a CD club is $5.00 and each CD costs $10.95. Alex has
saved $85.00. Write an inequality that he could use to find the number of
CDs he can purchase and not exceed his savings.
4
Algebra TEKS Assessment Supplement
(b)(1)(D)
The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
1. Write a story or describe a situation that the graph below could describe.
Label the axes to fit your story.
The student interprets and makes inferences from functional relationships.
1. Three students drew the graphs below to represent the relationship
between the number of 32 cent stamps purchased and the total cost.
Which graph is correct and why?
stamps
cost
cost
cost
(b)(1)(E)
stamps
stamps
Foundations for Functions
5
b2
Foundations for Functions
The student uses the properties and
“attributes
of functions.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
The class is assigned the task of rolling marbles down a ramp from a
height of 10 cm, 15 cm, 30 cm, and 50 cm. The marbles must be
released from the edge of the ramp. Students measure the distance
the marble rolls once it leaves the end of the ramp.
1. Make a table of values for the distance the marble travels, in cm, when
released from the various heights. Graph the data.
2. Find the domain and range for this situation.
3. Predict the distance the marble will travel if released from a height of 8 cm,
12 cm, and 20 cm.
4. Make a table of the values from the time the marble is released until it comes
to a stop when releasing the marble from various heights. Graph the data.
5. Find the domain and range for this situation.
6. Predict the time it will take the marble to come to a stop if released from a
height of 8 cm, 12 cm, and 20 cm.
7. Explain what you think will happen if ping pong balls are used instead of
marbles.
8. What do you think will happen if marbles are rolled down a ramp to a carpeted surface? a tiled floor? dirt?
6
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. What is the range of the function
f (x) = x2 – 3
when the domain is {–5, –3, –1}?
A {7, 3, –1}
B {–28, –12, –4}
C {–13, –9, –5}
D {22, 6, –2}
E {–7, –3, 1}
2. The graph shows the relationship between the number of boxes of
candy sold and the amount of profit made.
profit
$90
$70
$50
$30
$10
0
20 40 60 80 100
no. of boxes
How many boxes of candy must be sold to yield a $250 profit?
F 50
G 100
H 125
J
175
K 250
Foundations for Functions
7
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(b)(2)(A)
The student identifies and sketches the general forms of linear (y = x) and
quadratic (y = x 2) parent functions.
1. Which of the following is a graph of y = x2
A
B
4
4
3
3
2
2
1
1
-4 -3 -2 -1
-1
1 2 3 4
-4 -3 -2 -1
-1
-2
-3
-4
-4
D
4
4
3
3
2
2
1
1
-4 -3 -2 -1
-1
1
2 3 4
-4 -3 -2 -1
-2
(b)(2)(B)
2 3 4
-2
-3
C
1
-1
1
2 3 4
-2
-3
-3
-4
-4
For a variety of situations, the student identifies the mathematical domains and
ranges and determines reasonable domain and range values for given situations.
1. This graph represents a diver’s distance from the surface of the water at a
given time.
100
80
60
40
20
20 40 6080 100
State the domain and range of this graph.
8
Algebra TEKS Assessment Supplement
(b)(2)(C)
The student interprets situations in terms of given graphs or creates situations that fit given graphs.
1. Which of the situations below would fit this graph?
8
6
4
2
-3
-2
-1
1
-2
-4
-6
Situation 1: In Alaska the temperature was 4 degrees below zero (0) and
was increasing at a rate of 3 degrees per hour.
Situation 2: Sue was putting books on the shelf where 3 books were
already stacked. She put four more books on the shelf every five minutes.
Situation 3: Jim has 4 sets of baseball cards. He plans to add 3 new sets
every week.
(b)(2)(D)
In solving problems, the student collects and organizes data, makes and interprets
scatterplots, and models, predicts, and makes decisions and critical judgements.
You have been given 5 different squares. Measure the length of the diagonal
and the length of a side in metric (cm) and record the data in the table below.
Square
A
B
C
D
E
z
z
z
z
z
z
z
Length of
Side
z
z
z
z
z
z
z
Length of
Diagonal
1. Plot the data. Draw a reasonable line of best fit for this data.
2. Enter the data in a graphing calculator and find the linear regression equation.
3. If you have a square with a diagonal of 150 cm, predict the length of a side.
4. Predict the length of the diagonal of a square if the length of a side is
19 cm.
5. What does the slope of the line represent in this situation?
Foundations for Functions
9
b3
Foundations for Functions
student understands how algebra can
“beThe
used to express generalizations and
recognizes and uses the power of symbols to
represent situations.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
The following chart was created to describe how toothpicks can
be used to build a row of squares. Using the pictures in the visual
column, complete the chart.
task
# of
Squares
1
2
3
...
...
12
n
10
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Visual
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Process
4
4 + 3
?
?
?
?
?
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Written Description
it takes 4 toothpicks
to make 1 square
it takes 7 toothpicks
to make 2 squares
Algebra TEKS Assessment Supplement
?
?
?
?
?
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
Total #
of Toothpicks
4
7
?
?
?
?
?
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. Mark earns $4.50 per hour. He worked 3 hours more this week
than last week. If h is the number of hours he worked last week,
which equation shows the amount, t, that Mark earned for both
weeks?
F t = 2(4.50)(h + 3)
G t = 4.50h + 3
H t = 4.50(2h + 3)
J
t = 4.50(h + 3)
4.50(h 1 3)
K t 5 }}}}}{
2
2. Everett works at the lake renting boats to visitors. Last weekend he
rented 4 more sailboats than rowboats. He rented 10 boats in all.
Which equation could be used to find the number of rowboats, r,
he rented last weekend?
A r + 4 = 10
B 4r = 10
C 4(r + 1) = 10
D r – 4 = 10
E r + (r + 4) = 10
Foundations for Functions
11
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(b)(3)(A)
The student uses symbols to represent unknowns and variables.
John inherits his father’s baseball card collection containing 100 cards and
joins a card collection club that sends him 5 new cards each month.
1. Write an expression telling how many cards he will have in his collection
after m months.
2. Write an equation to find out in how many months John will have 195
cards.
12
Algebra TEKS Assessment Supplement
(b)(3)(B)
Given situations, the student looks for patterns and represents generalizations algebraically.
1. A florist designs flower arrangements using roses and carnations. A small
arrangement uses 1 rose surrounded by 8 carnations. The medium
arrangement uses 2 roses surrounded by 10 carnations. The large arrangement uses 3 roses surrounded by 12 carnations. If the pattern continues,
complete the table below.
Arrangement
small
medium
large
extra large
jumbo
...
super size
Texas size
z
z
z
z
z
z
z
z
z
Roses
1
2
3
4
?
15
20
z
z
z
z
z
z
z
z
z
Carnations
8
10
12
?
16
?
?
Foundations for Functions
13
b4
Foundations for Functions
student understands the importance
“ofThe
the skills required to manipulate
symbols in order to solve problems and uses
the necessary algebraic skills required to
simplify algebraic expressions and solve
equations and inequalities in problem
situations.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
A rectangle has a length that is 6 m longer than the width.
w + 6
w
1. Write an equation that represents the perimeter in terms of w. Write an
equation that represents the area in terms of w.
2. If the width is 50 m, what is the area?
3. For what value(s) of w will the area be equal to 520 m2?
4. For what value(s) of w will the perimeter be less than 60 m?
5. For what value of w is the area equal to the perimeter?
14
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. Nick’s rectangular bedroom has a length of (2x – 5) and a width of
(x + 1). Which equation describes the area, A, of Nick’s bedroom
in terms of x?
F A = 3x – 4
G A = 6x – 8
H A = 2x2 – 3x – 5
J
A = 2x2 – 5x + 1
K A = 2x2 –7x – 5
2. The sides of a triangle have lengths 2x – 1, x + 3, and 3x – 4.
Which of the following describes the perimeter, P, of the triangle in
terms of x?
A P = 5x – 8
B P = 6x + 9
C P = 6x3 – 12
D P = 6x – 2
E P = 5x – 1
Foundations for Functions
15
3. The Meadowbrook High School band rented a bus for a trip to a
football game. The bus company charged $475, plus $0.45 per
mile over 200 miles. If the bus trip cost $529, how many miles
was the trip?
4. What is the solution to the equation
3(2x – 1) – 4x = –5
5. Stan is carrying a load of 50 boxes of books in his truck. Some of
the boxes weigh 20 pounds each, and the rest of the boxes weigh
10 pounds each. If all the boxes weigh a total of 900 pounds, how
many 20-pound boxes are in Stan’s load?
16
Algebra TEKS Assessment Supplement
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(b)(4)(A)
The student finds specific function values, simplifies polynomial expressions,
transforms and solves equations, and factors as necessary in problem
situations.
1. The number of degrees for the sum, s, of the interior angles of a polygon
with n sides is s = 180(n – 2). How many sides would a polygon have if
the sum of the angles is 1800 degrees?
(b)(4)(B)
The student uses the commutative, associative, and distributive properties to
simplify algebraic expressions.
1. In mathematics class the teacher gave the following problem.
n + 2
n
n
n + 2
Find the area of the shaded region.
Sally’s answer was (n + 2)2 – n 2 and John’s answer was 2(n + 2) + 2n.
Show that both answers are correct.
Foundations for Functions
17
c1
Linear Functions
The student understands that linear
“functions
can be represented in different
ways and translates among their various
representations.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
One of the following situations represents a linear function.
1. Situation 1: “Refrigerator Jones,” the 346 pound fullback for the Texas Oilers,
was told to lose 70 pounds. He was successful at this. However, during the 5
weeks of his summer vacation, he gained weight at the rate of 3{4 of a pound
every 2 days. Which table represents situation 1?
z
z
z
z
z
Vacation Day
0
2
4
6
Weight
Vacation Day
276
0
276.75
2
277.5
4
278.25
6
z
z
z
z
z
Weight
346
346.75
347.5
348.25
2. Situation 2: A biology class is studying fruit flies for four days. They start with
10 fruit flies. Fruit fly populations doubled every 3 hours. Which table represents situation 2?
Hours
0
3
6
9
z
z
z
z
z
Flies
Hours
10
0
20
3
40
6
80
9
z
z
z
z
z
3. Which situation produced a linear function? Graph the function.
4. What is a reasonable domain and range for the linear function?
18
Algebra TEKS Assessment Supplement
Flies
10
20
30
40
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. What is the range of the function
f (x) = x 2 – 3
when the domain is {–5, –3, –1}?
A {7, 3, –1}
B {–28, –12, –4}
C {–13, –9, –5}
D {22, 6, –2}
E {–7, –3, 1}
2. When Pedro arrived at his cousin’s home in North Dakota, there
were 5 inches of snow on the ground. The next day snow started
falling again at a rate of 2 inches per hour. The graph below shows
the amount of snow on the ground.
snow on ground
11
9
number 7
of
inches 5
3
1
0
1 2 3 4 5 6
number of hours
The equation is
s = 2h + 5
where s is the total amount of snow on the ground and h is the
number of hours. Which graph on the following page best represents the total amount of snow on the ground if it had snowed at a
rate of 3 inches per hour?
Linear Functions
19
A
B
11
11
9
9
7
7
5
5
3
3
1
1
0
1
2
3
4
5
6
C
1
2
3
4
5
6
0
1
2
3
4
5
6
D
11
11
9
9
7
7
5
5
3
3
1
1
0
1
2
3
0
1
2
3
4
5
6
E
12
10
8
6
4
2
20
0
4
5
6
Algebra TEKS Assessment Supplement
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(c)(1)(A)
The student determines whether or not given situations can be represented
by linear functions.
1. Juanita put $2.00 each week into an account for graduation expenses. Her
father occasionally adds $2.00. How could you change this situation so it
could be represented by a linear function?
(c)(1)(B)
The student determines the domain and range values for which linear functions make sense for given situations.
1. John borrowed $40 and is paying it back at a rate of $4 per week. He
makes the following table and uses the equation m = 4w, where m is the
amount of money and w is the number of weeks.
Time in weeks
10
9
8
7
6
z
z
z
z
z
z
Money owed
40
36
32
28
24
What is the domain in this situation? What is the range?
(c)(1)(C)
The student translates among and uses algebraic, tabular, graphical, or verbal
descriptions of linear functions.
1. Given the function y = 2x + 3, describe a situation that could be represented by the function.
Linear Functions
21
c2
Linear Functions
The student understands the meaning of
“the
slope and intercepts of linear functions
and interprets and describes the effects of
changes in parameters of linear functions
in real-world and mathematical
situations.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
An aircraft begins its descent from an altitude of 1000 ft at a rate of
120 ft/min.
1. Write an equation that represents the altitude of the aircraft at time (t ).
2. Sketch the graph that represents this problem situation.
3. What is the slope of this line and what does it represent in this situation?
4. Find the x-intercept and explain what it means in the problem situation.
5. Sketch a graph of the glide path of a Cesna aircraft with an initial altitude of
3000 ft and a rate of descent of 60 ft/min.
6. What is the slope of this line and what does it represent in this situation?
7. After how many minutes will the first aircraft touch down on the ground?
8.
If a DC-10 landed in 20 minutes and its rate of descent was 90 ft/min, at
what altitude did the aircraft begin its descent?
9. If the equation A = 40t + 3600 is given as the model for the altitude of an
MD-80 at time (t ), describe what is occurring.
22
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. Which point lies on the line having as its equation 2x + y = 8?
8
6
4
2
-8 -6 -4
-2
2 4 6 8
-4
-6
-8
F Point K
G Point L
H Point M
J
Point N
K Point P
2. The graph of the function y = 2{3 x – 1 is shown below. If the line is shifted
3 units up, which of the following would describe the new line?
3
2
1
-4 -3 -2 -1
-1
1
2 3 4
-2
-3
-4
F y = 5{3 x – 1
G y = 2{3 x + 2
H y = 2{3 x(x + 2)
J
y = 2{3 x + 3
K y = 2{3 x(x + 3)
Linear Functions
23
3. Which equation describes a line parallel to the graph of
y = –2x + 3?
A y = – {12 x – 4
B y = –2x – 1
C y = 2x + 9
D y = {12 x + 6
E y = 2x – 3
4. What equation best describes the graph below?
F y = {12 x + 1
G y = –2x + 2
H y = – {12 x + 2
J
y = – {12 x + 1
K y = 2x + 1
24
Algebra TEKS Assessment Supplement
5. Which graph below best represents the equation of a line with a
slope of –2 and a y-intercept of 7?
A
B
C
D
E
Linear Functions
25
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(c)(2)(A)
The student develops the concept of slope as rate of change and determines
slopes from graphs, tables, and algebraic representations.
1. Stephen F. Austin High School is going to charter a bus for a school trip. A
bus company provided the following table of fees.
z
z
z
z
z
Number of Students
1
2
3
4
Total Cost
$216
$222
$228
$234
From the table determine the rate of change for the graph that would represent the data.
(c)(2)(B)
The student interprets the meaning of slope and intercepts in situations using
data, symbolic representations, or graphs.
1. The following is the graph of y = 3x + 2. Describe a situation this portion
of the graph could model. Explain what is represented by the slope and yintercept in this situation.
8
6
4
2
-4
-2
2
4
6
8
-2
-4
26
Algebra TEKS Assessment Supplement
(c)(2)(C)
The student investigates, describes, and predicts the effects of changes in m
and b on the graph of y = mx + b.
10
20
40
–10
–20
–30
–40
–50
–60
depth
–70
–80
–90
–100
(c)(2)(D)
60
time
80 100
1. A scuba diver starts ascending from
a depth of 100 ft at a rate of 60 ft
per minute. The graph represents
the diver’s depth at particular times
in seconds. How would the graph
change if the diver began his or her
ascent from a depth of 60 ft? How
would the graph change if the diver
began his or her ascent from 100 ft
at a rate of 50 ft per minute?
The student graphs and writes equations of lines given characteristics such as
two points, a point and a slope, or a slope and y-intercept.
1. Find the equation of a line through the points (–2, 3) and (6, –5).
(c)(2)(E)
The student determines the intercepts of linear functions from graphs, tables,
and algebraic representations.
1. Where does the graph of y = –2x + 5 cross the y-axis? Where does it cross
the x-axis? How did you find the answers?
(c)(2)(F)
The student interprets and predicts the effects of changing slope and y-intercept in applied situations.
15
1. The graph at the left shows the
amount a person earns at a rate
of $3 per hour. The equation is
a = 3h where a is the amount in
dollars and h is the number of
hours worked. How would a
raise to $6 an hour change the
graph?
13
11
9
7
amount
earned 5
3
1
1
(c)(2)(G)
3
5
7 9 11 13 15
hours worked
The student relates direct variation to linear functions and solves problems
involving proportional change.
1. In the function y = 4x, how much does y change when x is
increased by 4?
Linear Functions
27
c3
Linear Functions
The student formulates equations and
“inequalities
based on linear functions, uses
a variety of methods to solve them, and
analyzes the solutions in terms of the
situation.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
The 38th Annual State Fair starts on October 1, 1997, and lasts 4
weeks. Bill, who is 6 years old, wants to ride the Texas Cyclone
roller coaster. Bill is presently 3 feet, 4 inches tall. Safety rules state
that the minimum height for riding the Cyclone is 48 inches. Bill
grows at a rate of –41 inch per month.
1. Use a table to generate a function representing the situation.
Time in
Months
0
1
2
3
4
...
8
...
N
z
z
z
z
z
z
z
z
z
z
z
Process
z
z
z
z
z
z
z
z
z
z
z
Bill’s Height
(inches)
40
41
42
2. What equation can be used to find out when Bill will reach the minimum
height for the ride?
3. What is the solution to the equation? Explain how to solve it.
4. In how many months will Bill be 44.75 inches tall?
5. If the pattern continues, in what year will Bill be able to ride the Texas
Cyclone?
6. Is this a reasonable model for Bill’s height? Why or why not?
28
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they
can answer EOC questions like...
1. A student committee must decide between a band that costs $300
plus 40% of the ticket sales and a disc jockey that costs $450. The
committee plans to charge $3 per ticket. Which inequality can be
used to determine the number of tickets, t, that must be sold in
order for the band to be the better value?
A 0.40(300) + 3t < 450
B 300 + 0.40(3)t < 450
C 0.40(300 + 3t) < 450
D 300 + 0.40(3)t > 450
E 0.40(300 + 3t) > 450
2. Mark earns $4.50 per hour. He worked 3 hours more this week
than last week. If h is the number of hours he worked last week,
which equation shows the amount, t, that Mark earned for both
weeks?
F t 5 2(4.50) (h + 3)
G t 5 4.50h + 3
H t 5 4.50(2h + 3)
J
t 5 4.50(h + 3)
4.50(h 1 3)
K t 5 }}}}}{
2
Linear Functions
29
3. The Meadowbrook High School band rented a bus for a trip to a
football game. The bus company charged $475, plus $0.45 per
mile over 200 miles. If the bus trip cost $529, how many miles
was the trip?
4. Mark was buying a stereo that was on sale for 1{4 off the original
price, x. Which equation below could be used to find the amount,
y, that Mark would pay, not including tax?
A y = x – 1{4
B y = x – 1{4x
C y = 1{4 x
D y = x – 3{4 x
E y=x–4
5. The math club sold 64 large chocolate chip cookies for $0.75 each
and has 80 cookies left. At what sale price would the remaining
cookies be sold to have an overall average price of $0.50 per
cookie?
F $0.25
G $0.30
H $0.38
J
$0.63
K $0.90
30
Algebra TEKS Assessment Supplement
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(c)(3)(A)
The student analyzes situations involving linear functions and formulates
linear equations or inequalities to solve problems.
1. The drama club is going to have some posters printed to announce their
opening Christmas special. The print shop charges $3 per poster plus a
$10 fee for a master copy. The drama club budget allows them to spend
no more than $200. Write an inequality that could be used to solve this
problem.
(c)(3)(B)
The student investigates methods for solving linear equations and inequalities
using concrete models, graphs, and the properties of equality, selects a
method, and solves the equations and inequalities.
1. Given the equation, 3(x + 5) + 3(x + 5) + 4(x + 5) = 100, solve this equation and show your work.
(c)(3)(C)
For given contexts, the student interprets and determines the reasonableness
of solutions to linear equations and inequalities.
1. Carlos has $100 and decides to spend $25 each week for entertainment
purposes. His 8-year-old sister, Margarita, believes he will run out of
money in 3 weeks. State whether she is correct or not, and explain why.
Linear Functions
31
c4
Linear Functions
The student formulates systems of linear
“equations
from problem situations, uses a
variety of methods to solve them, and
analyzes the solutions in terms of the
situation.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
A set of twin boys were born, with Troy weighing 6 lbs and Tyrone
weighing 3.5 lbs. Troy’s weight increases 1.2 lbs per month, and
Tyrone’s weight increases 1.4 lbs per month.
1. If this rate of weight gain continues, in how many months will the two boys
weigh the same?
2. If this rate of weight gain continues, how much would each boy weigh
at age 16?
3. Is this a reasonable model for the weights of these boys at the age of 16? Why
or why not?
32
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they
can answer EOC questions like...
1. Marie has 24 coins in quarters and dimes. The total value is $4.95.
Which system of equations below will determine the number of
quarters, q, and the number of dimes, d, she has?
F d + q = 4.95
0.10d + 0.25q = 24
G d + q = 24
0.25d + 0.10q = 4.95
H d + q = 24
0.35dq = 4.95
J
d + q = 24
d + q = 4.95
K d + q = 24
0.10d + 0.25q = 4.95
2. Harry bought 9 movie tickets for a total of $45. Adult tickets cost
$6 each and child tickets cost $4.50 each. How many adult tickets
did he buy?
F 2
G 3
H 4
J
5
K 6
3. Bill bought some neon fish at $2 each and some angelfish at $3
each for his new aquarium. If Bill bought a total of 20 fish and
spent a total of $45, how many angelfish did he buy?
Linear Functions
33
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(c)(4)(A)
The student analyzes situations and formulates systems of linear equations to
solve problems.
1. Two (2) CDs and 5 tapes cost $68.65. Four (4) CDs and 8 tapes cost
$119.40. Write a system of equations you could use to find the cost
of 1 CD.
(c)(4)(B)
The student solves systems of linear equations using concrete models,
graphs, tables, and algebraic methods.
1. The tables below describe a linear system. Solve the system.
x
0
2
4
x
–2
0
2
34
z
z
z
z
z
z
z
z
y
–30
–26
–22
y
5
3
1
Algebra TEKS Assessment Supplement
(c)(4)(C)
For given contexts, the student interprets and determines the reasonableness
of solutions to systems of linear equations.
1. There are two 500-gallon water tanks. One is full and is to be emptied at
a rate of 2.5 gallons per minute. The other is empty and is to be filled at a
rate of 5 gallons per minute. The valves on both tanks are opened at the
same time. The graph shows this situation. What does the point of intersection, a, mean in this situation? What happens if the valves are left open
for 3 hours?
1000
gallons
800
600
400
200
0
100
200
time in minutes
gallons emptied
gallons filled
Linear Functions
35
d1
Quadratic & Other Nonlinear Functions
“
The student understands that the graphs of
quadratic functions are affected by the
parameters of the function and can interpret
and describe the effects of changes in the
parameters of quadratic functions.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
The student council wants to build two flower gardens for its community project. They want to build one at Avalon Retirement Home
and another at Country Manor Retirement Home. The garden at
Avalon has a length that is 4 times its width, and the garden at
Country Manor has 3 more square feet than the area of the garden
at Avalon.
1. Write the function representing the area of Avalon’s garden and graph it.
2. Write the function representing the area of Country Manor’s garden and graph
it on the same axis.
3. How are the graphs and functions for the areas of the two gardens alike? How
are they different?
4. If the student council has at most 120 feet of landscaping timbers for the fencing for the garden at Avalon, what would be a reasonable domain and range
for the area?
5. If the graph of the Avalon garden is shifted up 4 units, what conclusion can be
drawn about its area?
36
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they
can answer EOC questions like...
At the time this document was published, there were not any released items
that would apply.
Quadratic and Other Nonlinear Functions
37
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(d)(1)(A)
The student determines the domain and range values for which quadratic
functions make sense for given situations.
1. The path of a ball that is thrown straight up in the air is modeled by the
function H = 75t – 16t 2. H is the height in feet and t is the time in seconds. What is a reasonable domain and range for this situation?
(d)(1)(B)
The student investigates, describes, and predicts the effects of changes in a
on the graph of y = ax2.
1. Given the graphs of y = 2x2, y = 5x2, y = 8x2, y = –2x2, y = –5x2, and
y = –8x2, explain how the difference in the functions relates to the difference in graphs. How are the graphs alike and why? How are the graphs
different and why?
38
Algebra TEKS Assessment Supplement
(d)(1)(C)
The student investigates, describes, and predicts the effects of changes in c
on the graph of y = x2 + c.
1. The graph of y = 7x 2 is shifted up 10 units. Write the equation of the new
graph.
(d)(1)(D)
For problem situations, the student analyzes graphs of quadratic functions
and draws conclusions.
1. A ball is dropped from the top of a building. The graph below gives the
distance the ball is above the ground at time, t. The general function for
the distance, h, is h = –16t 2 + c, where c is the height of the building.
About how high is the ball 2 seconds after it is dropped?
300
height
above 200
ground
100
0
1
2
3
4
5
time in seconds
Quadratic and Other Nonlinear Functions
39
d2
Quadratic & Other Nonlinear Functions
The student understands there is more
“than
one way to solve a quadratic equation
and solves them using appropriate
methods.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
The following table was generated using a
function of the form y = ax 2 + bx + c.
x
–1
0
1. Find the value(s) of x for which
ax 2 + bx + c = 0.
1
2
2. Find the value(s) of x for which
ax 2 + bx + c = 3.
3
4
3. Below is the graph of another function of
the form y = ax 2 + bx + c.
10
0
–10
–20
–30
–2
0
2
4
6
8
10
Find the roots of the equation ax 2 + bx + c = 0.
4. Below is the graph of y = x 2 – 8x + 15.
15
10
5
0
2
4
6
8
-5
Find x if x 2 – 8x + 15 = 3.
40
Algebra TEKS Assessment Supplement
5
z
z
z
z
z
z
z
z
y
15
8
3
0
–1
0
3
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. The area of a rectangular flag is 15 square feet. The length of the
flag is 2 feet longer than the width. What are the dimensions of the
flag?
F 2.5 ft by 6 ft
G 3 ft by 5 ft
H 2 ft by 7.5 ft
J
5 ft by 7 ft
K 7 ft by 8 ft
2. The area of the front of a cabinet is 18 square feet. The width is
3 feet longer than the height. What are the dimensions of the
cabinet?
A 7.5 ft by 4.5 ft
B 15 ft by 3 ft
C 6 ft by 3 ft
D 9 ft by 6 ft
E 7 ft by 4 ft
3. The equation that describes the path of a rocket after it is shot into
the air is
h = 48t – 6t 2
where h is the height, in feet, above ground level after t seconds.
After how many seconds will the rocket be at a height of 90 feet?
A t = 15
B t = 3 and t = 5
C t = 8 and t = 15
D t = 18 and t = 30
E t=8
Quadratic and Other Nonlinear Functions
41
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(d)(2)(A)
The student solves quadratic equations using concrete models,
tables, graphs, and algebraic methods.
1. Below is a graph and a partial table for the function of y = 2x 2 + 2x – 12.
Solve the equation 2x 2 + 2x –12 = 0.
x
–2
–1
0
1
2
z
z
z
z
z
z
y
–8
–12
–12
–8
0
15
10
5
-4
-2
2
4
-5
-10
-15
42
Algebra TEKS Assessment Supplement
(d)(2)(B)
The student relates the solutions of quadratic equations to the roots of their
functions.
1. The following is a graph of a quadratic function. What are its roots?
6
4
2
-6
-4
-2
2
4
6
-2
-4
-6
Quadratic and Other Nonlinear Functions
43
d3
Quadratic & Other Nonlinear Functions
The student understands there are
“situations
modeled by functions that are
neither linear nor quadratic and models
the situations.
”
With a complete understanding of this Knowledge and Skills
statement, students should be able to perform the following
assessment task.
task
Jimmy has just received his driver’s license. He went to Slick Sam’s
Used Cars and was offered the following payment plan for a
$10,000 truck.
Payment Plan: Pay $0.01 on the first day, $0.02 on the second day, $0.04 on
the third day, $0.08 on the fourth day, and so on for 21 days.
1. Complete the following table
Day
1
2
3
4
5
6
7
8
9
10
z
z
z
z
z
z
z
z
z
z
z
Payment
0.01
0.02
0.04
0.08
?
?
?
?
?
?
2. Make a scatter plot for this plan. Does this scatter plot represent a linear function, quadratic function, or neither? Explain your thinking.
3. How much would Jimmy’s payment be on the 21st day?
44
Algebra TEKS Assessment Supplement
end of course exam questions
When students can perform a task like this, they can
answer EOC questions like...
1. An animal population that doubles every 6 months can be
described by the equation
p = n ? 2 2t
where p is the population after t years and n is the original number
of animals. If 2 of these animals were introduced into an area,
what would be the estimated population after 3 years?
F 24
G 64
H 128
J
512
K 4096
2. Carla earns $6.40 per hour. She gets a 5% raise each year. The
amount she will earn per hour, x, is given by the formula
x = w (1 + r )y
where w is her current wage per hour, r is the rate of increase, and
y is the number of years. To the nearest cent, how much will she
earn per hour, x, after 2 years on the same job?
A $6.50
B $7.06
C $8.50
D $13.44
E $14.40
Quadratic and Other Nonlinear Functions
45
performance descriptions
Performance Descriptions and the type of assessment
items students should be able to perform
(d)(3)(A)
The student uses patterns to generate the laws of exponents and applies
them in problem-solving situations.
Find the missing exponent.
1. x2 ? x5 ? x7 ? x n = x18
z5
2. }{{
5 z2
zn
3. (b4)n = b24
4. 3n = 1
(d)(3)(B)
The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.
The table below shows various rates and the respective times it takes a
motor bike to cover a distance of 40 miles.
Rate (mph)
4
5
10
20
40
z
z
z
z
z
z
Time (hours)
10
8
4
2
1
1. How fast in miles per hour does the motorbike have to travel to cover this
distance in 30 minutes?
2. How long would it take the motorbike to travel the same distance traveling at the rate of 60 mph?
46
Algebra TEKS Assessment Supplement
(d)(3)(C)
The student analyzes data and represents situations involving exponential
growth and decay using concrete models, tables, graphs, or algebraic
methods.
1. The number of lily pads in a pond triples every year. If there are 5 lily
pads in the pond this year, how many lily pads will there be next year? in
9 more years?
Quadratic and Other Nonlinear Functions
47