(4th ed) Preface Table of Contents (Units 0-3) - no CW no WS · 3 (Topic 2) Intercepts and Graphing...

Authors

Jennifer Boehlke, Belinda Eerdmans, Jean Hanson, Mark Hedin, Mary Jo Hughes, Jennifer Lehman,

Anne Roehrich, Stephanie Pogalz, Loren Tenold, Renee Voltin

Contributors

Sarah DeBoer, John FitzSimons, Jonathan Kell, Phyllis Kisch, Patrick Pangborn, Matt Rowe, Julie Rydberg

Editors (Electronic and Print)

Bruce DeWitt and Renee Voltin

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4th

Edition, Printed: July, 2016

ii PREFACE

Preface

Mathematics has the extraordinary power to reduce complicated problems to simple rules and

procedures. Therein lies the danger in teaching mathematics: it is possible to teach the subject as

nothing but the rules and procedures – thereby losing sight of both the mathematics and of its practical

value. The revision process into the fourth edition of Intermediate Algebra, produced by teachers from

Anoka-Hennepin and math consultants from the local area, strives to refocus the teaching of

mathematics on concepts as well as procedures.

Balancing Skills and Conceptual Understanding

These materials stress conceptual understanding and multiple ways of representing mathematical ideas.

Our goal is to provide students with a clear understanding of the ideas of functions as a solid foundation

for subsequent courses in mathematics and other disciplines. When we designed this material, we

started with a clean slate; we focused on the key concepts, emphasizing depth of understanding.

Skills are developed in the context of problems and reinforced in a variety of settings, thereby

encouraging retention. This balance of skills and understanding enables students to realize the power of

mathematics in modeling.

Guiding Principles: Varied Problems and the Rule of Four

Since students usually learn most when they are active, the exercises in a text are of central importance.

In addition, research has shown that multiple representations encourage students to reflect on the

meaning of the material. Consequently, we have been guided by the following principles.

• Our problems are varied and some are challenging. Many cannot be done by following a

template in the text.

• The Rule of Four promotes multiple representations. Each concept and function is represented

symbolically, numerically, graphically and verbally.

• The components of these Intermediate Algebra materials make every effort to tie together

conceptual knowledge and procedural fluency. Functions as models of change is a central

theme, and algebra is integrated where appropriate.

• Problems involving real data are included to prepare students to use mathematics in other fields.

• To use mathematics effectively, students need skill in both symbolic manipulation and the use of

technology. The exact proportions of each may vary widely, depending on the preparation of the

student and the wishes of the instructor.

Intermediate Algebra iii

Intermediate Algebra

Unit 1: Linear Inequalities and Systems of Linear Inequalities In this unit we will learn how to graph inequalities in one and two variables and explore the region on

the graph that represents the solutions to the inequality. We will learn to solve systems of two or more

inequalities by graphing and locating the region on the graph that represents the solutions all the

inequalities have in common. We will apply our work with systems of linear inequalities to solve

linear programming problems that involve maximizing or minimizing an outcome.

Identified Learning Targets:

1.1 I can demonstrate understanding of how to represent a region on a graph with an

inequality.

1.2 I can demonstrate understanding of real-world situations that can be modeled as

linear equations or linear inequalities.

1.3 I can represent real-world situations as a linear programming problem and

demonstrate an understanding of how to find reasonable solutions.

Unit 2: Functions Prior to a deeper study of specific functions other than linear functions, students learn how to recognize

when a relationship is a function and to evaluate a function for a given input. Students recognize and

interpret key features of graphs and of tables of input/output values and relate the features to what is

occurring in real world situations. Key features include: intercepts; the domain and range; intervals

where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;

and the average rate of change over an interval.


2.1 I can demonstrate understanding of the definition of a function and can determine

when relations are functions given a graph, table or real-world situation.

2.2 I understand the meaning of function notation and can evaluate a function for a given

input.

2.3 I can demonstrate understanding of the significant features of a function represented

by a graph, a table, or an equation and the relationship these features have to

real-world situations.

Unit 3: Exponential Functions

Students learn about exponential functions by comparing the situations and equations for exponential

functions to those for linear functions. Students recognize, use, and create tables, graphs and situations

modeling exponential growth and decay. Students are able to evaluate exponential functions when

given inputs that are rational numbers and to relate the meaning of a rational exponent to the context of

the situation. Students use tables and graphs to solve exponential equations and translate between

representations.


3.1 I can demonstrate understanding about exponential functions and compare situations

and equations for exponential functions to those for linear functions.

3.2 I can use tables and graphs to solve exponential equations including real-world

situations and translate between representations.

3.3 I can evaluate exponential functions in the form y = a·bx and relate the meaning to

the context of a real-world situation.

3.4 I can demonstrate understanding of the significant features of a graph of an

exponential function and their relationship to real-world situations.

iv PREFACE

Table of Contents

� This book is organized in a workbook style manner. Each unit has the following structure:

o The reference and resource material for the entire unit (i.e. Unit 1: pages 11-21)

o A collection of practice problems to demonstrate understanding of a concept (i.e. Unit 1: pages 22-51)

� You can access the online resources, available through the Anoka-Hennepin District Moodle website at

anoka.k12.mn.us/intermediatealgebra and click on “Log in as a guest”

o Electronic access to the entire textbook (pdf format)

o Electronic access to the worked out solutions to the practice problems

o Web-based video reference to many of the topics that are addressed throughout this course.

� A graphing calculator is available through online access at Desmos.com or by downloading the free

Desmos app available for most electronic devices.

Unit 0: Prerequisite Skills/Concepts

Practice Problems

page 2 (Topic 1) Slope and Slope Intercept Form

3 (Topic 2) Intercepts and Graphing in Standard Form

3 (Topic 3) Stained Glass Window Equations – More Graphing in Slope–Intercept and

Standard Form

4 (Topic 4) Rewrite Equations into Slope–Intercept Form

4 (Topic 5) Solving Linear Systems by Graphing

5 (Topic 6) Solve Linear System Real-Life Situations by Graphing

6 (Topic 7) Solving Linear Systems Using Substitution and Elimination

7 (Topic 8) Solving Linear Systems by Choosing the Best Method

8 (Topic 9) Solving Linear System Real-Life Situations Using Substitution and

Elimination

9 (Topic 10) Solving Inequalities

Unit 1: Linear Inequalities and Systems of Linear Inequalities

page 11 Unit 1 Reference and Resource Material

Practice Problems

page 22 1.1A Introduction to Linear Inequalities

24 1.1B Graphing Linear Inequalities

27 1.1C Graphing Linear Inequalities in Standard Form and Slope-Intercept Form

29 1.1D Graphing Linear Inequalities in Any Form using Graphing Technology

30 1.2 Writing Equations and Inequalities

32 1.3A Modeling Real-World Situations with Equations and Inequalities

34 1.3B Graphing Systems of Linear Inequalities to Find a Feasible Region

37 1.3C Linear Programming – Finding Vertices Graphically

39 1.3D Linear Programming – Finding Vertices Algebraically

44 1.3E Solving Linear Programming Problems

47 1.3F Using Graphing Technology for Linear Programming

49 Unit 1 Review Material

Intermediate Algebra v

Unit 2: Functions


Practice Problems

page 71 2.1A Functions and Situations

72 2.1B Classifying Functions

74 2.2A Evaluating Functions Using Function Notation

77 2.2B Graphing Functions

81 2.3A Significant Features of Functions – Part I

84 2.3B Significant Features of Functions – Part II

87 2.3C Significant Features of Functions – Part III

88 2.3D Rate of Change

89 2.3E Relations as Inverses

92 2.3F Using Graphing Technology


Unit 3: Exponential Functions


Practice Problems

page 111 3.1 Compare and Contrast Exponential and Linear Functions

114 3.2A Explore – Exponential Growth/Decay: Tables, Graphs and Real-World

situations

118 3.2B Solve – Exponential Growth/Decay: Tables, Graphs and Real-World

Situations

120 3.3 Evaluate Exponential Functions

122 3.4 Significant Features of an Exponential Function Graph


Appendices

Appendix A - Glossary

page 130 Glossary – Units 1 – 3

Appendix B - Selected Answers to Practice Problems

page 133 Unit 0 – Prerequisites Skills/Concepts

137 Unit 1 – Inequalities and Systems of Linear Inequalities

147 Unit 2 – Functions

155 Unit 3 – Exponential Functions

vi PREFACE

To Students: How to Learn from this Flexbook

� This book may be different from other math textbooks that you have used, so it may be helpful to

know about some of the differences in advance. At every stage, this book emphasizes the

meaning (in practical, graphical or numerical terms) of the symbols you are using. There is

much less emphasis on “plug-and-chug” and using formulas, and much more emphasis on the

interpretation of these formulas than you may expect. You will often be asked to explain your

ideas in words or to explain an answer using graphs.

� The book contains the main ideas of intermediate algebra concepts in plain English. Success in

using this book will depend on reading, questioning, and thinking hard about the ideas presented.

It will be helpful to read the text in detail, not just the worked out examples.

� There are few examples in the text that are exactly like the practice problems, so practice

problems can’t be done by searching for similar-looking “worked out” examples. Success with

the practice problems will come by grappling with the ideas of advanced algebra.

� Many of the problems in the book are open-ended. This means that there is more than one

correct approach and more than one correct solution. Sometimes, solving problem relies on

common sense ideas that are not stated in the problem explicitly but which you know from

everyday life.

� This book assumes that you have access to a calculator or computer that can graph functions and

find (approximate) roots of equations. There are many situations where you may not be able to

find an exact solution to a problem, but can use a calculator or computer to get a reasonable

approximation. An answer obtained this way can be as useful as an exact one. However, the

problem does not always state that a calculator is required, so use your own judgment.

� This book attempts to give equal weight to four methods for describing functions: graphical (a

picture), numerical (a table of values), algebraic (a formula) and verbal (words). Sometimes it’s

easier to translate a problem given in one form into another. For example, you might replace the

graph of a parabola with its equation, or plot a table of values to see its behavior. It is important

to be flexible about your approach; if one way of looking at a problem doesn’t work, try another.

� Students using this book will find discussing these problems in small groups helpful. There are

great many problems which are not cut-and-dried; it can help to attack them with the other

perspectives your colleagues can provide. If group work is not feasible, see if your instructor can

organize a discussion session in which additional problems can be worked on.

� You are probably wondering what you’ll get from the book. The answer is, if you put in a solid

effort, you will get a real understanding of functions as well as a real sense of how mathematics

is used in the age of technology.

PREREQUISITE SKILLS/CONCEPTS

This section includes practice on many of the skills and concepts that were

developed and assessed in a previous math course. The problems may be

assigned to you by your teacher or you may choose to strengthen your skills

by working through various problems that appear within this unit. Answers

for this section are included within the answer key section of this text.

Practice Problems

page 2 (Topic 1) Slope and Slope Intercept Form

3 (Topic 2) Intercepts and Graphing in Standard Form

3 (Topic 3) Stained Glass Window Equations – More Graphing in

Slope–Intercept and Standard Form

4 (Topic 4) Rewrite Equations into Slope–Intercept Form

4 (Topic 5) Solving Linear Systems by Graphing

5 (Topic 6) Solve Linear System Real-Life Situations by Graphing

6 (Topic 7) Solving Linear Systems Using Substitution and Elimination

7 (Topic 8) Solving Linear Systems by Choosing the Best Method

8 (Topic 9) Solving Linear System Real-Life Situations Using Substitution and

Elimination

9 (Topic 10) Solving Inequalities

UNIT

0

Unit 0 (Topic 1) Slope and Slope-Intercept Form

2 UNIT 0 – PREREQUISITE SKILLS/CONCEPTS

#1 – 2: Identify the slope and y-intercept of each line.

1) 10y x= + 2) 2 7y x= − +

#3 – 4: Find the slope of each line.

3) 4)

slope: slope:

#5 – 7: Graph the equation on a coordinate grid. Use a separate coordinate grid for each equation.

5) 4

15

y x= + 6) 2 3y x= − 7) 5

22

y x= − +

#8 – 11: Find the slope for the line passing through the given points. Then tell whether the line rises, falls,

is horizontal, or is vertical.

8) ( ) ( )7,3 , 2,3− − 9) ( ) ( )6, 6 , 6, 6− −

10) ( ) ( )4, 2 , 18,1− 11) ( ) ( )9,8 , 9, 2− −

#12 – 14: Determine which line is steeper. Be sure to show your thinking.

12) Line 1: through ( ) ( )4,1 and 8, 6− 13) Line 1: through ( ) ( )1, 9 and 6, 6− − −

Line 2: through ( ) ( )2, 4 and 1, 8− − − Line 2: through ( ) ( )7, 23 and 0, 2− − −

14) Line 1: through ( ) ( )0,3 and 4, 17− −

Line 2: through ( ) ( )1,13 and 6, 22− −

#15 – 16: Given the following information, answer each question.

15) The top of the Leaning Tower of Piza is about 55.9 meters above the ground. As of 1997, its top was

leaning about 5.2 meters off-center. Approximate the slope of the tower.

16) You are camping at the Grand Canyon. When you pitch your tent at1:00 pm, the temperature is 81°F.

When you wake up at 6:00 am the temperature is 47°F. What is the average rate of a change in the

temperature? Estimate the temperature when you went to sleep at 9:00 pm.

Unit 0 (Topic 2) Intercepts and Graphing in Standard Form

(Topic 1) Slope and Slope-Intercept Form 3

(Topic 2) Intercepts and Graphing in Standard Form (Topic 3) Stained Glass Window Equations

#1 – 3: Match the equation with its graph.

1) 4 8x y− = − 2) 3 6 9x y+ = − 3) 2 3 12x y− = −

A. B. C.

#4 – 9: Identify the x-intercept and y-intercept. Then, draw a coordinate grid and graph the equation.

4) 2 8x y+ = 5) 3 4 10x y+ = −

6) 3 3x y− = 7) 5 6 30x y− = −

8) 6y = 9) 5x = −

#10 – 15: Graph the equation using any method. Which method did you choose and why (slope and slope-

intercept, or intercepts and standard form)?

10) 3 7y x= + 11) 8x = −

12) 2 7 14x y− = 13) 5

22

y x= −

14) 5 10 30x y+ = 15) 3

4y =

Unit 0 (Topic 3) Stained Glass Window Equations – More Graphing in Slope-

Intercept and Standard Form

Using AT LEAST 12 equations from the list below, create your unique stained glass window by graphing these

lines on a sheet of graph paper. Do not graph using a table. Practice graphing using the quick method (from

either standard form or slope-intercept form). When you are finished graphing the equations, use markers to color

each section to create your stained glass window.

7 2y x+ = 5 0x + = 0 1 x= + 12y x= − +

7x = 3 6 18x y− + = − 9y = − 2 8y x= − +

0 2 y= + 4 8x y− + = 6 0y − = 4 5x y− − = −

5y x= + 8x = − 4 2 16x y− = − 1 y=

9 0x− + = 1

64

y x= − 9x y− = + 5 0y− − =

1

33

y x= − − 0 8 y= − 2 0x y+ = 2 x=

Unit 0 (Topic 4) Rewrite Equations into Slope-Intercept Form


#1 – 12: Rewrite the equation in slope intercept form. (Solve for y.)

1) 2 11y x− − = 2) 3 55x y− = − 3) 5 2 20x y− =

4) 2 6 6x y+ = − 5) 5 3 15x y− + = − 6) 8 2 14x y− =

7) 10 7x y+ = 8) 3 6x y− = 9) 5 6 2x y− = −

10) 4 9 30x y+ = 11) 3 4 10x y− + = − 12) 9 4 16y x− = −

#13 – 18: Solve the inequality for the variable y.

13) 5x y+ < 14) 9 2 18x y+ ≤ − 15) 9 4 36x y− >

16) 3 12 4x y− + < − 17) 2 3 4x y− ≥ − 18) 2 5 10x y− − <

Unit 0 (Topic 5) Solving Linear Systems by Graphing

#1 – 2: Check whether the ordered pair is a solution of the system.

1) 4 25

3 2 16

x y

x y

− =

− − = − ( )6, 1− 2)

2 52

9 10

x y

x y

− =

− = − ( )2, 8− −

#3 – 8: Graph the linear system and tell how many solutions it has. If there is one solution, name it.

3)

15

2

13

2

y x

y x

= −

= +

4) 5

3 15

y x

x y

= − −

+ = −

5) 4 24 4

6 1

y x

y x

− = +

= − − 6)

2 7

2 8

x y

y x

− =

= +

7)

33

4

3 6

y x

y x

= +

= −

8) 6 2 8

20 4 16

x y

x y

− =

+ =

Unit 0 (Topic 6) Solve Linear Systems Real-Life Situations by Graphing

(Topic 4) Rewrite Equations into Slope-Intercept Form (Topic 5) Solving Linear Systems by Graphing 5

(Topic 6) Solve Linear System Real-Life Situations by Graphing

#1 – 7: Write a system of equations. Graph the system of equations on a coordinate grid and use the graph

to answer each of the questions.

1) You want to burn 400 Calories during 40 minutes of exercise. You burn about 8 Calories per minute

inline skating and 16 Calories per minute swimming. How long should you spend doing each activity?

2) Your school is planning a 5 hour outing at the community park. The park rents bicycles for $9 per hour

and inline skates for $3 per hour. The total budget per person is $27. How many hours should students

spend doing each activity?

3) Ariel bought several bags of caramel candy and taffy. The number of bags of taffy was 5 more than the

number of bags of caramels. Taffy bags weigh 8 ounces each, and caramel bags weigh 16 ounces each.

The total weight of all of the bags of candy was 400 ounces. How many bags of candy did she buy?

4) Smith Chevrolet charges $50 per hour for labor on car repairs. Lopez Ford charges a diagnosis fee of

$30 plus $40 per hour for labor. If the labor on an engine repair costs the same at either shop, how long

does the repair take?

5) Thirty coins, all dimes and nickels, are worth $2.60. How many nickels are there?

6) A landscaper installs some trees and bushes at a bank. He installs 25 plants for a total cost of $1,500.

How many trees (t) and how many bushes (b) did he install if each tree costs $100 and each bush costs

$50?

7) Pat was in a fishing competition at Lake Pisces. She caught some bass and some trout. Each bass

weighed 3 pounds and each trout weighed 1 pound. Pat caught a total of 30 pounds of fish. She got 5

points in the competition for each bass, but since trout are endangered species in Lake Pisces, she lost 1

point for each trout. Pat scored a total of 18 points. How many bass and trout did Pat catch?

Unit 0 (Topic 7) Solving Linear Systems Using Substitution and Elimination


#1 – 5: Solve the system using the substitution method.

1) 2 3 5

5 9

x y

x y

+ =

− = 2)

2 6

4 2 5

x y

x y

− + =

− =

3) 2 3

4 5 3

x y

x y

− + =

− = − 4)

2 2

7 3 20

x y

x y

+ =

− = −

5) 3 4

9 3 12

x y

x y

− =

− + = −

#6 – 10: Solve the system using the elimination (linear combination) method.

6) 3 2 6

6 3 6

x y

x y

+ =

− − = − 7)

6 5 4

7 10 8

x y

x y

− + =

− = −

8) 7 4 3

2 5 7

x y

x y

− = −

+ = − 9)

9 6 0

12 8 0

x y

x y

− + =

− + =

10) 6 2

18 3 4

x y

x y

− = −

− + =

Unit 0 (Topic 8) Solving Linear Systems by Choosing the Best Method

(Topic 7) Solving Linear Systems Using Substitution and Elimination 7

(Topic 8) Solving Linear Systems by Choosing the Best Method

#1 – 9: Solve each linear system using substitution or elimination.

1) 5 7 11

5 3 19

x y

x y

− + =

− + = 2)

3

2 2 6

x y

x y

− =

− + = −

3) 2 5 10

3 4 15

x y

x y

− =

− + = − 4)

3 11

5 2 16

x y

x y

− + =

− − = −

5) 4 6 11

6 9 3

x y

x y

− − =

+ = − 6)

4 2

3 8 1

x y

x y

− = −

− + = −

7) 12 3 16

36 9 32

x y

x y

+ =

− − = 8)

5 17

2 10 34

x y

x y

− + =

− = −

9)

19

3

2 2 6

x y

x y

+ =

− + = −

10) Explain how you can tell whether the system has infinitely many solutions or no solution without trying

to solve the system.

a) 5 2 6

10 4 12

x y

x y

− =

− + = − b)

2 8

6 3 12

x y

x y

− + =

− + =

Unit 0 (Topic 9) Solving Linear System Real-Life Situations Using

Substitution and Elimination


1) Suppose a teacher, Mr. Turner, has a class with b boys and g girls.

a) Mr. Turner noticed that 23b g+ = . What does that tell you about his class?

b) If 3b g= − , what statement can you make about the number of boys and girls?

c) How many girls are in Mr. Turner’s class? Explain how you know.

#2 – 4: Write a system of equations and solve the system using the substitution method.

2) Elsie took all of her cans and bottles from home to the recycling plant. The number of cans was one

more than four times the number of bottles. She earned $0.10 for each can and $0.12 for each bottle, and

ended up earning $2.18 in all. How many cans and bottles did she recycle?

3) The Fabulous Footballers scored an incredible 55 points at last night’s game. Interestingly, the number

of field goals was one more than twice the number of touchdowns. The Fabulous Footballers earned 7

points for each touchdown and 3 points for each field goal. Determine how many touchdowns and field

goals the Fabulous Footballers earned last night.

4) The Math Club sold roses and tulips for Valentine’s Day. The number of roses sold was 8 more than

4 times the number of tulips sold. Tulips were sold for $2 each and roses for $5 each. The club made

$414. How many roses were sold?

#5 – 8: Write a system of equations and solve the system using the elimination (linear combination)

method.

5) Selling frozen yogurt at a fair, you make $565 and use 250 cones. A single-scoop cone cost $2 and a

double-scoop cone costs $2.50. How many of each type of cone did you sell?

6) Two ships are sailing on the same coordinate system. One ship is following a path described by

2 3 6x y+ = , and the other is following a path described by 2 3 9x y− = . Find the point at which the two

7) Tickets for your school’s play are $3 for students and $5 for non-students. On opening night 937 tickets

were sold and $3943 is collected. How many tickets were sold to students? To non-students?

8) A store advertises two types of cell phones, one selling for $67 and the other for $100. If the receipts

from the sale of 36 phones totaled $2,940, how many of each type were sold?

#9 – 12: Write a system of equations and solve the system using substitution or elimination.

9) An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some

are worth five points. How many two point questions are on the test?

10) At last night’s volleyball game, the number of hotdogs sold was three fewer than twice the number of

corndogs. A hotdog costs $3 and a corndog costs $1.50. If $201 was collected, how many corndogs

were sold?

11) You buy 7 bags of gummy bears and 3 bags of chocolate for a total of $22. Your friend buys 3 bags of

gummy bears and 1 bag of chocolate and her total is $8. What is the price of a bag of gummy bears?

12) A car and a truck leave Rockford at the same time, heading in opposite directions. When they are 350

miles apart, the car has gone 70 miles farther than the truck. How far has the car traveled?

Unit 0 (Topic 10) Solving Inequalities

(Topic 9) Solving Linear System Real-Life Situations Using Substitution and Elimination 9

(Topic 10) Solving Inequalities

#1 – 2: Decide whether the given number is a solution of the inequality.

1) 7 12 8; 4x − < 2) 3 2 6; 3x− < ≤

#3 – 8: Solve the inequality. Then graph your solution on a number line.

3) 4 5 25x + > 4) 5 2 27x− ≥ 5) 3

7 22

x − <

6) 4 1 14x − > 7) 4.7 2.1x− > −7.9 8) ( )2 4 8x− >

#9 – 12: Solve the compound inequality. Then graph your solution on a number line.

9) 5 7 11x− ≤ − ≤ 10) 5 6 0n− ≤ − − ≤

11) 2 5 4 2x or x+ ≤ − ≥ 12) 7 6 1 5x− < − <

13) You have $50 and are going to an amusement park. You spend $25 for the entrance fee and $15 for

food. You want to play a game that costs $0.75 per time.

a) Write and solve an inequality to find the possible number of times you can play the game.

b) If you play the game the maximum number of times, will you have spent the entire $50? Explain.

14) A professor announces that course grades will be computed by taking 40% of a student’s project score

(0-100 points) and adding 60% of the student’s final exam score (0-100 points). If a student gets an 84

on the project, what scores can she get on the final exam to get a course grade of at least 90?

15) The international standard for scientific temperature measurement is the Kelvin scale. A Kelvin

temperature can be obtained by adding 273.15 to a Celsius temperature. The daytime temperature on

Mars ranges from 89.15°C to –31.15°C.

a) Write the daytime temperature range on Mars as a compound inequality in degrees Celcius.

b) Rewrite the compound inequality in degrees Kelvin.

16) Write an inequality that has no solution. Show or explain why it has no solution.

10 UNIT 1 – LINEAR INEQUALITIES AND SYSTEMS OF LINEAR INEQUALITIES

LINEAR INEQUALITIES AND SYSTEMS OF INEQUALITIES

In this unit we will learn how to graph inequalities in one and two variables and explore

the region on the graph that represents the solutions to the inequality. We will learn to

solve systems of two or more inequalities by graphing and locating the region on the

graph that represents the solutions all the inequalities have in common. We will apply

our work with systems of linear inequalities to solve linear programming problems

that involve maximizing or minimizing an outcome.


Practice Problems

1.1 I can demonstrate

understanding of how

to represent a region

on a graph with an

inequality.

page 22 1.1A Introduction to Linear Inequalities

24 1.1B Graphing Linear Inequalities

27 1.1C Graphing Linear Inequalities in Standard Form and Slope-

Intercept Form

29 1.1D Graphing Linear Inequalities in Any Form using Graphing

Technology

1.2 I can demonstrate

understanding of

real-world situations

that can be modeled

as linear equations or

linear inequalities.

30 1.2 Writing Equations and Inequalities

1.3 I can represent real-

world situations as a

linear programming

problem and

demonstrate an

understanding of how

to find reasonable

solutions.

32 1.3A Modeling Real-World Situations with Equations and

Inequalities

34 1.3B Graphing Systems of Linear Inequalities to Find a Feasible

Region

37 1.3C Linear Programming – Finding Vertices Graphically

39 1.3D Linear Programming – Finding Vertices Algebraically

44 1.3E Solving Linear Programming Problems

47 1.3F Using Graphing Technology for Linear Programming


UNIT

1


1.1 I can demonstrate understanding of how to represent a region on a graph with an

inequality.

1.2 I can demonstrate understanding of real-world situations that can be modeled as linear

equations or linear inequalities.

1.3 I can represent real-world situations as a linear programming problem and demonstrate

an understanding of how to find reasonable solutions.

Unit 1 Reference and Resource Material

Reference and Resource Material 11

1.1 I can demonstrate understanding of how to represent a region on a graph with

an inequality.

A linear inequality in two variables takes the

form y mx b< + or y mx b> +

( ) and as welly mx b y mx b≤ + ≥ + . Linear

Inequalities are closely related to graphs of

straight lines. A straight line has the equation

. When we graph a line in the

coordinate plane, we can see that it divides the

plane in two half-planes.

The solution to a linear inequality includes all the points in one of the

half-planes. If the inequality is written in slope intercept form, we can

tell which half of the plane the solution is by looking at the inequality sign.

> The solution is the half plane above the line, but not the points on the line itself.

≥ The solution is the half plane above the line and also all the points on the line.

< The solution is the half plane below the line, but not the points on the line itself.

≤ The solution is the half plane below the line and also all the points on the line.

A point “above the line” means, that for a given x-coordinate, the points with y-values greater than the y-value of

the point on the line would be located “above the line”.

For a strict inequality, we draw a dashed line to show that the points on the line are not part of the solution.

For an inequality that includes the equal sign, we draw a solid line to show that the points on the line are part of

the solution.

Here is what you should expect linear inequality graphs to look like.

The solution of

y mx b≥ + is the half

plane above the

boundary line and all

the points on the line.

The solution of

y mx b> + is the

half plane above the

boundary line. The

dashed line shows

that the points on

the line are not part

of the solution.

The solution of

y mx b≤ + is the half

plane below the

boundary line and all

the points on the line.

The solution of

y mx b< + is the

half plane below the

boundary line. The

dashed line shows

that the points on

the line are not part

of the solution.

m by x= +

Unit 1 Resources

y mx b≥ +

y mx b≤ +

y mx b> +

y mx b< +



Graph Linear Inequalities in One Variable in the Coordinate Plane

We have graphed inequalities in one variable on the number line. We can also graph inequalities in one variable

on the coordinate plane. We just need to remember that when we graph an equation of the type we get a

vertical line and when we graph an equation of the type we get a horizontal line.

1.1 Example 1:

Graph the inequality x > 4 on the coordinate plane.

Solution

First, let’s remember what the solution to x > 4 looks like on the number line.

The solution to this inequality is the set of all real numbers x

that are bigger than four but not including four. The solution is

represented by a line. Note that this has an open circle on the

number line.

In two dimensions we are also concerned with values of y, and

the solution to 4x > consists of all coordinate points for

which the value of x is bigger than four. The solution is

represented by the half plane to the right of 4x = .

The boundary line at 4x = is dashed because the equal sign is

not included in the inequality and therefore, points on the line

are not included in the solution of the inequality.

1.1 Example 2:

Graph the inequality y ≤ 6 on the coordinate plane.

Solution

The solution is all coordinate points for which the value of y is less

than or equal than 6. This solution is represented by the half plane

below the line .

The line is solid because the equal sign is included in the

inequality sign and the points on the line are included in the solution.

Graph Linear Inequalities in Two Variables

The general procedure for graphing inequalities in two variables is as follows.

Step 1: Graph several points of the boundary line using a method that is appropriate to the form the inequality

is written (for example, plotting two or more points using a table of values, using slope and

y-intercept if the inequality is in slope-intercept form, finding x- and y-intercepts if the inequality is

in standard form). Draw the line, either dashed or solid based on the inequality given. Draw a dashed

line if the equal sign is not included and a solid line if the equal sign is included.

Step 2: Select a point not on the boundary line. Substitute the x- and y-values from the point chosen into the

inequality to determine if it creates a true or false statement when simplifying the inequality.

Step 3: Based on evaluation of the whether or not the chosen point in step 2 is a solution (creates a true

statement of inequality), shade the half-plane of all coordinate points that make the inequality true.

x a=

y b=

6y =

6y =



-5 5

-5

5

x

y

1.1 Example 3: Graph the inequality y ≥ 2x – 3.

Solution Step 1: Graph the boundary line (y = 2x – 3) by making a table of values or

using the slope and y-intercept information from the equation.

x y = 2x – 3 y (x, y)

–1 ( )2 1 3 5− − = − –5 (–1, –5)

0 ( )2 0 3 3− = − –3 (0, –3)

1 ( )2 1 3 1− = − –1 (1, –1)

Using these 3 points, create the boundary line. The boundary line

will be solid because the value of the expressions on each side of the inequality can be equal to

one another. All coordinate points on the boundary line will make the inequality true.

Step 2: Test a point to determine the solution space. Choose a point in one of the two half-planes to

determine if it is a solution to the inequality y ≥ 2x – 3. We will choose the point (1, 4) and test

the inequality using this point.

x y Test

1 4 ( )

?

?

?

2 3

4 2 1 3

4 1

y x≥ −

≥ −

≥ −

True

statement

Step 3: Because the point (1, 4) makes the inequality TRUE, then all

coordinate pairs in that half-plane will create a TRUE inequality

statement, thus identifying the points in the solution space. Shade

the half-plane that contains all points that make the inequality true.

1.1 Example 4: Graph the inequality 5 2 4x y− < .

Solution

Step 1: Graph the boundary line. Because the inequality is written in

standard form, let’s find the x- and y-intercepts of the boundary

line of the inequality 5 2 4x y− < .

x y HOW Summary (x, y)

0 ( )

5 2 45 0 2 4

2

x yyy

− =

− =

= −

When

x = 0, y = –2

( )0, 2−

0 ( )

5 2 45 2 0 4

4

5

x yx

x

− =

− =

=

When

40,

5y x= =

4, 0

5

Using these two points, create the boundary line. The boundary line will be dashed because

the value of the expression cannot be exactly equal to 4.

Step 2: Test a point to determine the solution space. Let’s test (1, 4) again.

x y Test

1 4 ( ) ( )

?

?

?

5 2 4

5 1 2 4 4

3 4

x y− <

− <

− <

False

statement

Step 3: Because the point (1, 4) makes the inequality FALSE, then all

coordinate pairs in the half-plane NOT containing (1, 4) will create

a TRUE inequality statement, thus identifying the points in the solution space. Shade the

half-plane that contains all points that make the inequality true.

-5 5

-5

5

x

y

-5 5

-5

5

x

y

-5 5

-5

5

x

y



1.1 Example 5: Graph the inequality 4 53

xy + ≤ − + .

Solution Step 1: Graph the boundary line. Because the inequality is written in form that is close to slope-

intercept form, isolate the y variable to determine the slope and y-intercept (or make a table of

values to determine several points on the boundary line.

4 531

13

xy

y x

+ ≤ − +

≤ − +

Using these information from this equation, the y-intercept is

(0, 1) and the slope is 1

3m = − . The boundary line will be solid

because the value of the expressions on each side of the

inequality can be equal to one another. All points on the

boundary line will make the inequality true.

Step 2: Test a point to determine the solution space. Let’s test (3, 4). An x-coordinate that is a multiple

of 3 was chosen to make working with the division by 3 result in an integer value.

Step 3: Because the point (3, 4) makes the inequality FALSE, then all

coordinate pairs in the half-plane NOT containing (3, 4) will

create a TRUE inequality statement, thus identifying the points in

the solution space. Shade the half-plane that contains all points that make the inequality true.

Vocabulary

• An inequality is a mathematical sentence built from expressions using one or more of the

symbols <, >, ≤, or ≥. Like an equation, it is a relationship between two quantities that are not

necessarily equal.

• A linear inequality is an inequality that involves a linear function.

• The linear function in a linear inequality forms the boundary line to the solutions which lie in a shaded

region.

o If the line is included as part of the solution it is a solid line.

o If the line is not included a part of the solution it is plotted as a dashed line.

• The slope-intercept form of a linear equation is � = �� + �.

• The standard form of a linear equation is �� + �� = .

Video Resources:

• CK-12 Foundation: Linear Inequalities in Two Variables

• http://www.montereyinstitute.org/courses/Algebra1/U05L2T1_RESOURCE/index.html

• http://www.montereyinstitute.org/courses/Algebra1/U05SIM_RESOURCE/index.html

x y Test

3 4 ( )

?

?

?

?

4 533

4 4 53

8 1 5

8 4

xy + ≤ − +

+ ≤ − +

≤ − +

≤

False

statement

-5 5

-5

5

x

y

-5 5

-5

5

x

y



1.2 I can demonstrate understanding of real-world situations that can be modeled as linear equations and

linear inequalities

Solve Real-World Problems Using Linear Inequalities

In this section, we see how linear inequalities can be used to solve real-world applications.

1.2 Example 1:

A pound of coffee blend is made by mixing two types of coffee beans. One type costs $9 per pound and

another type costs $7 per pound. Find all the possible mixtures of weights of the two different coffee

beans for which the blend costs $8.50 per pound or less.

Solution

Let’s apply our problem solving plan to solve this problem.

Step 1: Let x = the weight (pounds) of the $9 per pound coffee beans

Let y = the weight (pounds) of the $7 per pound coffee beans

Step 2: The two types of coffee beans will be blended together. The cost of a pound of coffee blend

can be determined as follows:

($9) (weight of the $9 per pound bean) + ($7) (weight of the $7 per pound bean)

($9) (x) + ($7) (y)

9x + 7y

We are looking for the mixtures that cost $8.50 or less.

We write the inequality 9 7 8.50x y+ ≤ .

Step 3: To find the solution set, graph the inequality 9 7 8.50x y+ ≤ .

Because the inequality is in standard form, determine the x- and y-intercepts of the boundary

line.

x y 9x + 7y = 8.50 Summary (x, y)

0 ( )

9 7 8.5

9 0 7 8.5

7 8.5

1.21

x y

y

y

y

+ =

+ =

=

=

When

x = 0, y = 1.21

( )0,1.21

0 ( )

9 7 8.5

9 7 0 8.5

9 8.5

0.94

x y

x

x

x

+ =

+ =

=

=

When

0, 0.94y x= = ( )0.94, 0

Step 4: Graph the inequality.

The line will be solid. We shade below the line because

the point (0, 0) will create a TRUE inequality statement

( ) ( )9 0 7 0 8.50 + ≤

Notice that we show only the first quadrant of the

coordinate plane because the weight values should be

positive.

The shaded region tells you all the possibilities of the

two bean mixtures that will give a total less than or

equal to $8.50.



1.2 Example 2:

Julian has a job as an appliance salesman. He earns a commission of $60 for each washing machine

he sells and $130 for each refrigerator he sells. How many washing machines and refrigerators must

Julian sell in order to make $1000 or more in commission?

Solution: Let’s apply our problem solving plan to solve this problem.

Step 1: Let x = the number of washing machines Julian sells

Let y = the number of refrigerators Julian sells

Step 2: The total commission is given by the expression .

We are looking for total commission of $1000 or more. We write the inequality.

.

Step 3: To find the solution set, graph the inequality .

Because the inequality is in standard form, determine the x- and y-intercepts.

x y 60x + 130y = 1000 Summary (x, y)

0 ( )

60 130 1000

60 0 130 1000

130 1000

7.69

x y

y

y

y

+ =

+ =

=

=

When

0, 7.69x y= = ( )0,1.21

0 ( )

60 130 1000

60 130 0 1000

60 1000

16.67

x y

x

x

x

+ =

+ =

=

=

When

0, 16.67y x= = ( )0.94, 0

Step 4: Graph the inequality. The boundary line will be solid

because the total commission could be exactly $1000.

We shade above the line because the point (0, 0) creates

a false inequality, so shade the half-plane that makes the

inequality true. ( ) ( )60 0 130 0 1000 + ≥

Notice that we show only the first quadrant of the

coordinate plane because dollar amounts should be

positive. Also, only the points with integer coordinates

are possible solutions.

60 130x y+

60 130 1000x y+ ≥

60 130 1000x y+ ≥



Personal Cakes Cupcakes Total

Amount 2 1 22

Time 2 4 40

Profit $14.99 $16.99

1.3 I can represent real-world situations as a linear programming problem and demonstrate an

understanding of how to find reasonable solutions.

A Linear Programming Problem

James is trying to expand his pastry business to include personal cakes and cupcakes. He has

40 hours available to decorate the new items and can use no more than 22 pounds of cake mix.

Each personal cake requires 2 pounds of cake mix and 2 hours to decorate. Each cupcake order

requires one pound of cake mix and 4 hours to decorate. If he can sell each personal cake for

$14.99 and each cupcake order for $16.99, how many personal cakes and cupcake orders

should James make to make the most profit?

Step 1: Organize the given information in a chart or table.

Step 2: Write the objective function for which you are trying to find the maximum or minimum value.

James wants to maximize his revenue. First we need to assign variables. Let’s use p for the number of personal

cake orders and c for the number of cupcake orders.

Total Profit

=

amount of money

earned from personal

cake orders

+

amount of money

earned from

cupcake orders

Profit = 14.99p + 16.99c

Items or products that are the

focus of the problem.

Items that are

the constraints

(the restrictions).

Goal – what you are trying to

maximize or minimize.



Step 3: Write the constraints as a system of inequalities.

James has two constraints that limit the number of personal cakes and cupcakes that can be made.

First is the constraint of the available amount of cake mix. It takes 2 pounds of cake mix to make a personal cake

and 1 pound of cake mix to make a cupcake. He can only use 22 pounds of cake mix.

Amount of mix for

personal cakes +

Amount of mix

for cupcakes ≤ 22

2p + 1c ≤ 22

Second is the constraint of available time to decorate. It takes 2 hours to decorate a personal cake and 4 hours to

decorate the cupcake. He only has 40 hours available.

Time to decorate

personal cakes +

Time to decorate

cupcakes ≤ 40

2p + 4c ≤ 40

Because this is a real-world situation, a third set of constraints occurs. The number of personal cakes and the

number of cupcakes must not be negative.

p > 0

c > 0

So the system of inequalities would be:

2p + 1c ≤ 22

2p + 4c ≤ 40

p ≥ 0

c ≥ 0

Step 4: Graph the system of inequalities to find the feasible region.

Graphing a system of inequalities requires you to graph two or more linear inequalities on the same coordinate

plane. The inequalities are graphed separately on the same graph and the solution for the system of inequalities

is the common shaded region between all the inequalities in the system. The common shaded region of the

system of inequalities is called the feasible region.

Graph the first inequality: 2p + 1c ≤ 22

-8 -4 4 8 12 16 20 24

-8

-4

4

8

12

16

20

24

p

c



On the same coordinate plane, graph the second inequality:

2p + 4c < 40.

The final 2 inequalities: p > 0 and c > 0 limit the graph to the

first quadrant.

The feasible region is the common shaded region. All of the points in the feasible region are solutions to all four

inequalities.

Number of Personal Cakes

Num

ber

of

Cupca

kes

A B

C

D



Step 5: Find the vertices of the feasible region.

• Vertex A is (0,0)

• Vertex B is the p-intercept of 2p + 1c = 22 which is (11, 0)

• Vertex C is the c-intercept of 2p + 4c = 40 which is (0, 10)

• Vertex D is the intersection of 2p + 1c = 22 and 2p + 4c = 40

This can be found by solving by graphing, by elimination, or by substitution.

To solve by elimination:

Substitute into an original equation to

find p

Vertex D is (8, 6)

To solve by substitution:

Isolate a variable in one equation.

2 1 22

1 22 2

p c

c p

+ =

= −

Substitute 22 2 p− into the other

equation for c and solve to find p.

2 4(22 2 ) 40

2 88 8 40

6 88 40

6 48

8

p p

p p

p

p

p

+ − =

+ − =

− + =

− = −

=

Substitute into an original equation

to find c

2(8) 1 22

1 6

c

c

+ =

=

Vertex D is (8, 6)

The feasible region has four vertices: {(0, 0), (11, 0), (0, 10), (8, 6)}.

The optimization of the objective function will most often occur at one of these vertices.

Step 6: Test the vertices in the objective function to find the maximum or minimum.

14.99p + 16.99c = Profit

Substitute each ordered pair to determine which makes the most money.

Vertices 14.99p + 16.99c Profit

(0, 0) 14.99(0) + 16.99(0) $0

(0, 10) 14.99(0) + 16.99(10) $169.90

(11, 0) 14.99(11) + 16.99(0) $164.89

(8, 6) 14.99(8) + 16.99(6) $221.93

Solution: To make the most profit, James should make 8 personal cakes and 6 cupcake orders.

( )2 1 22

2 4 40

p c

p c

− + =

+ =

2 1 22

2 4 40

p c

p c

− − = −

+ =

3 18c =

6c =

( )2 1 6 22

2 16

8

p

p

p

+ =

=

=



Solving Linear Programming Problems

Step 1: Organize the given information in a chart or table.

Step 2: Write the objective function to find the maximum or minimum value.

Step 3: Write the constraints as a system of inequalities.

Step 4: Graph the system of inequalities to find the feasible region.

Step 5: Find the vertices of the feasible region.

Step 6: Test the vertices in the objective function to find the maximum or minimum.

Vocabulary

• A system of inequalities is two or more linear inequalities.

• Linear programming is the mathematical process of analyzing a system of inequalities to

make the best decisions given the constraints of the situation.

• Constraints are the particular restrictions of a situation due to time, money, or

materials.

• In an optimization problem, the goal is to locate the feasible region of the system and use

it to answer a profitability, or optimization, question.

• An objective function is the function in an optimization problem that needs to be

maximized or minimized.

• The solution for the system of inequalities is the common shaded region between all the

inequalities in the system.

• The common shaded region of the system of inequalities is called the feasible region.

Video Resource:

• CK-12 Foundation: Linear Programming

Name ______________________________ Period __________

1.1A Introduction to Linear Inequalities

22 UNIT 1 – LINEAR INEQUALITIES AND SYSTEMS OF LINEAR INEQUALITIES (PRACTICE)

1) For Molly’s reading assignment homework she receives 10 points for every nonfiction book she reads and 5

points for every fiction book she reads. Copy the table below onto your assignment sheet. Identify 5

combinations of nonfiction and fictions books that Molly can read to earn exactly 50 points.

a) Draw a coordinate grid. Number and label the graph, then graph the points you have identified above

along with all of the possible combinations of nonfiction and fiction books that she can read to get

exactly 50 points. Is this a linear relationship? Explain why or why not.

b) Describe, using words, the meaning of the rate of change. For example, how does the number of fiction

books read change as the number of non-fiction books read changes?

c) Molly will earn a free homework pass for the next reading assignment if she earns more than 50 points.

Copy the table below onto your assignment sheet. Identify 5 possible combinations of nonfiction and

fiction books that would give her more than 50 points. On the coordinate grid, graph these 5 possible

combinations along with all possible combinations that represent situations where she would earn more

than 50 points.

d) What differences are there between the graph of a linear equation (creating exactly 50 points) and the

graph of a linear inequality (creating more than 50 points)?

# of nonfiction

books

# of fiction

books

Record your thinking of how you know this combination

satisfies the conditional goal

# of nonfiction

books

# of fiction

books

Record your thinking of how you know this combination

satisfies the conditional goal for part (b).

Name ______________________________ Period __________

1.1A Introduction to Linear Inequalities

1.1 I can demonstrate understanding of how to represent a region on a graph with an inequality. 23

2) Members of the Anoka High School Ski Club went on a ski-trip where members can rent skis for $16 per day

and snowboards for $20 per day. The members of the club brought a total of $240 with them on the trip.

a) Copy the table below onto your assignment sheet. Identify four possible combinations of ski rental and

snowboard rental that would allow the Ski Club to spend exactly $240. Then graph them on the

coordinate grid. (Remember to number and label your graph appropriately.)

b) Identify the x- and y-intercepts and explain the meaning of these two ordered pairs.

x-intercept _________ meaning: ______________________________________________________

y-intercept _________ meaning: ______________________________________________________

c) Describe what the graph would look like if you graphed all possible combinations of renting skis and

snowboards that the club would be able to rent with the $240 they brought with.

#3 – 4: Determine whether each of the given points is a solution to the given linear inequality.

3) 52 ≥+− yx 4) 43 −<− yx

a) (2, 9) b) (0, 2) a) (–1, 1) b) (0, 5)

#5 – 9: Without graphing, determine if each point is in the shaded region for each inequality.

5) ( )2,1 and 2 5x y+ >

6) ( )1, 3− and 2 4 10x y− ≤ −

7) ( )5, 1− − and 2 8y x> − +

8) ( )6, 2 and 2 3 2x y+ ≥ −

9) ( )5, 6− and 2 3 3y x< +

# of ski rentals

# of snowboard

rentals

Record your thinking of how you know this

combination satisfies the conditional goal

Name ______________________________ Period __________

1.1B Graphing Linear Inequalities


#1 – 2: Copy the inequality, the graph, and the table of values onto your assignment sheet. For each

inequality and graph given, complete the table of values for the boundary line then pick a point

and use it to determine which half-plane should be shaded. Shade the correct half-plane.

1) 1≥y 2) 3x < −

Looking at the table of values for these two problems, what patterns exist, and how does it relate to the given

inequality?

#3 – 4: Determine whether each of the given points is a solution to the given linear inequality.

3) 2 5y x≥ − 4) 4 1y x< − +

a) (0, 0) b) (1, –3) a) (–2, 9) b) (3, 3)

5) Copy the table of values onto your assignment sheet. Complete the table of values for the boundary

line. Write the inequality for each of the following graphs.

a) __________________ b) __________________

Points on the

boundary line Points on the

boundary line

x y x y

Points on the


boundary line

x y x y

Name ______________________________ Period __________



6) Copy the table of values onto your assignment sheet. Complete the table of values for the boundary

line. Write the inequality for each of the following graphs.

a) __________________ b) __________________

#7 – 10: For each inequality and graph, pick a point and use it to determine which half-plane should be

shaded. Copy the graph onto your assignment sheet. Then shade the correct half-plane.

7) 3 5y x≥ − 8) 3y x< −

9) 5

13

y x≤ − − 10) 2 3 2x y< − +

Points on the


boundary line

x y x y

Name ______________________________ Period __________



#11 – 12: Fill in the blank with the appropriate inequality sign.

11) 2 5 15x y− − 12) 1

4

y x −

13) Write the inequality for each of the following graphs.

a) ________________________ b) ________________________

Points on the

boundary line

Points on the

boundary line

x y x y

Name ______________________________ Period __________

1.1C Graphing Linear Inequalities in Standard Form and Slope-Intercept Form


#1 – 4: Copy the table of values onto your assignment sheet. Complete the table of values for the boundary

line. Test a point not on the boundary line and graph each inequality.

1) 3 1y x≥ − 2) 3

14

y x< − +

x y

0

3) 2 3 12x y+ ≥ 4) 3 6x y− ≤

x y

0

#5 – 8: For each inequality and graph, pick a point and use it to determine which inequality symbol should

be used.

5) 2 3y x + 6) 3

2 4

x y− −

7) 2 2x y + 8) 5 0x y+

x y

0

4

x y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

Name ______________________________ Period __________

1.1C Graphing Linear Inequalities in Standard Form and Slope-Intercept Form


Number of DVD players (x)

5 10 15 20

5

10

15

20

0

Nu

mb

er o

f te

lep

hon

es

(y)

9) An electronics store sells DVD players and telephones. The store makes a $24 profit on the sale of each DVD

player (x) and a $16 profit on the sale of each telephone (y). The store wants to make a profit of at least $240

from its sales of DVD players and telephones.

The following inequality describes this situation: 24 16 240x y+ ≥ .

Points on the boundary line can be identified by finding solutions to the related equation

24 16 240x y+ ≥ .

a) Find the x-intercept (x, 0). Interpret this point in

context of the problem.

b) Find the y-intercept (0, y). Interpret this point in

context of the problem.

c) On your assignment sheet, create an axis structure

similar to the one to the right. Plot the

x- and y–intercepts on the grid and connect them with

the appropriate style boundary line.

d) Should the boundary line be dashed or solid for the

inequality: 24 16 240x y+ ≥ ? Explain why.

e) Locate two or more points on the graph with at least

one point from each side of the boundary line. Test to

determine which half-plane is the solution to the inequality and shade the solution region appropriately.

Name ______________________________ Period __________

1.1D Graphing Linear Inequalities in Any Form using Graphing Technology


#1 – 3: Graph the following inequalities on a graphing utility. Then draw the graph onto your assignment

sheet.

1) 1

42

y x≤ − + 2) 7 2y x≥ − 3) 9

45

y x> − −

#4 – 5: For each of the following linear inequalities:

a) Rewrite the inequality in slope-intercept form.

b) Graph the inequality using a graphing utility and draw the graph onto your assignment sheet.

c) Identify a test point that lies in the shaded region of the graph on the graphing utility. Using the

original inequality, test to determine if it is a solution to verify. Record your thinking.

4) 627 −≤+ yx 5) 2−>− yx

6) The Coon Rapids girls’ lacrosse team is having a bake sale to raise money for a team camp. They are

selling cakes (x) for $4 each and a bag of cookies (y) for $2. Their goal is to sell $50 of cakes and cookies.

Kassie and Krissy are both players on the team.

Kassie said, “I made a table (to the right) and noticed that we need to

sell 2 fewer bags of cookies for every cake that we sell so the slope

equals –2. I also saw that if we don’t sell any cakes we need to

sell 25 bags of cookies so the y-intercept is 25. So the inequality

should be 2 25y x≥ − + .”

Krissy said, “I noticed that if x represents the number of cakes,

then 4x would be the amount of money that we made with from cakes.

If y represents the bags of cookies, then 2y would be the amount of money

that we made from cookies. We want to make at least $50 so the

inequality should be 4 2 50x y+ ≥ .”

Which player is correct, or are they both correct? Justify your answer.

Cakes Cookies

0 25

1 23

2 21

3 19

Name ______________________________ Period __________

1.2 Writing Equations and Inequalities


1) A library is trying to decrease the number of overdue books by increasing the fines. They plan to charge a flat

fee of $2.25 for an overdue book and $0.10 for each day a book is overdue.

a) Which equation shows the amount of a fine (F) for a book that is overdue for d days?

[A] 2.25 0.10F d= − +

[B] 0.10 2.25F d= − +

[C] 2.25 0.10F d= +

[D] 0.10 2.25F d= +

b) Justify which equation you chose and explain why the others are incorrect.

2) The number of people (n) who will attend a dance depends on the admission price (p), in dollars. This

relationship is represented by the equation shown below.

800 50n p= −

a) Which of these is a correct interpretation of this equation?

[A] The number of people attending the dance will increase by 50 for every dollar the admission price

increases.

[B] The number of people attending the dance will decrease by 50 for every dollar the admission price

increases.

[C] The minimum number of people attending the dance will be 800.

[D] The maximum number of tickets that can be sold is 50.

b) Justify which statement you chose and explain why the others are incorrect.

3) The senior class is producing a play to raise money for graduation activities. They would like to earn at least

$500. They intend to charge $2 for each student ticket, and $5 for each adult ticket. Define the variables and

write an inequality that represents this situation.

4) Several times a week a student runs part of the way and walks part of the way on a trail that is less than 4

miles long. The student’s running speed is 6 miles per hour and her walking speed is 4 miles per hour. Define

the variables and write an inequality that represents this situation.

5) Satchi found a used bookstore that sells pre-owned videos and CDs. Videos cost $9 each, and CDs cost $7

each. Satchi can spend no more than $35.

a) Define the variables and write an inequality that represents this situation.

b) Does Satchi have enough money to buy 2 videos and 3 CDs?

6) The perimeter of a rectangular lot is less than 800 feet. Define the variables and write an inequality that

represents the amount of fencing that will enclose the lot.

Name ______________________________ Period __________

1.2 Writing Equations and Inequalities

1.2 I can demonstrate understanding of real-world situations that can be modeled as linear 31

equations or linear inequalities.

7) At a grocery store the price of a kiwi fruit is $0.50 and the price of a lime is $0.25. Define the variables and

write an inequality to model the relationship between the number of kiwis and the number of limes that can be

purchased for less than $5.00.

8) You have $4000 to buy stock and have decided on The Clothes Store (TCS) and United Computers (UC).

TCS sells for $20 per share and UC sells for $15 per share. Define the variables and write an inequality

which restricts the purchase of x shares of TCS and y shares of UC.

9) Tickets for the school play cost $5 per student and $7 per adult. The school wants to earn at least $5400 on

each performance.


b) If 500 adult tickets are sold, what is the minimum number of student tickets that must be sold?

10) A fast food restaurant charges $11.00 for a Pizza and $4.00 for wings. The basketball team has no more than

$100 to spend on pizza and wings for their post-game celebration.


b) Determine if the team can purchase 8 pizzas and 6 orders of wings. Show your thinking.

11) A moving van has an interior height of 7 feet (84 inches). You have boxes in 12 inch and 15 inch heights, and

want to stack them as high as possible to fit. Define the variables and write an inequality that represents this

situation.

12) The grocery store has grapes that sell for $2.25 a pound and oranges that sell for $1.90 a pound. Define the

variables and write an inequality that represents how much of each type of fruit can be bought with no more

than $20.

13) A wholesaler has $80,000 to spend on certain models of mattress sets and bed frames. If the mattress sets may

be obtained at $200 each and the bed frames at $100 each, define the variables and write an inequality that

restricts the purchase on x mattress sets and y bed frames.

Name ______________________________ Period __________

1.3A Modeling Real-World Situations with Equations and Inequalities


#1 – 8: Address each statement presented or answer each question that is asked.

1) Fuel x costs $2 per gallon and fuel y costs $3 per gallon. You have at most $18 to spend on fuel.

a) Identify the variables for this situation.

b) Write an inequality to represent the amount of money you can spend on fuel x and fuel y.

2) A salad contains ham and chicken. There are at most 6 pounds of ham and chicken in the salad.


b) Write an inequality to represent the amount of ham and chicken in the salad.

3) Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only allowed to

work 13 hours per week. She wants to make at least $65.


b) Write an inequality that shows the amount of time that Mary wants to spend working.

c) Write an inequality that shows the amount of money that Mary wants to earn from working.

4) You can work a total of no more than 41 hours each week at your two jobs. Housecleaning pays $5 per hour

and your sales job pays $8 per hour. You need to earn at least $254 each week to pay your bills.


b) Write an inequality that shows the amount of time that you want to spend at work.

c) Write an inequality that shows the amount of money that that you want to earn from work.

5) You manufacture cell phone accessories: cases and clips. You need to produce at least 40 cases each day. You

have to produce at least 20 clips each day. You can produce no more than 80 items total in a day. You make

a profit of $6 for each case sold and $4 for each clip sold.

Let c = the number of cases produced and p = the number of clips produced

a) Write an equation for the anticipated profit.

b) Write an inequality that shows how many cases you have to produce each day.

c) Write an inequality that shows how many clips you have to produce each day.

d) Write an inequality that shows how many total clips and cases can be produced each day.

6) The automotive plant in Rockaway makes the Topaz and the Mustang. The plant has a maximum production

capacity of 1200 cars per week. During the spring, a dealer orders up to 600 Topaz cars and 800 Mustangs

each week. The profit on a Topaz is $500 and on a Mustang it is $800.

Let t = the number of Topaz produced and m = the number of Mustangs produced.

a) Write an equation for the anticipated profit.

b) Write an inequality that shows how many total cars are produced per week.

c) Write an inequality that shows how many Topaz are ordered per week.

d) Write an inequality that shows how many Mustangs are ordered per week

Name ______________________________ Period __________

1.3A Modeling Real-World Situations with Equations and Inequalities

1.3 I can represent real-world situations as a linear programming problem and demonstrate 33


#1 – 8 (continued): Address each statement presented or answer each question that is asked.

7) A wii manufacturer is making wii remotes and wii nunchucks. For each wii remote sold they make $20 profit.

For each wii nunchuck sold they make $12 profit. The wii remotes take 4 hours to prepare the parts and 1

hour to assemble. The wii nunchuck takes 2.5 hours to prepare the parts and 2.5 hours to assemble. The

maximum preparation time available is 16 hours. The maximum assembly time available is 10 hours.

a) Organize the given information in a chart or a table.

Remotes Nunchucks Total

Prep Time

Assembly Time

Profit

b) Identify and define the variables.

____ ⇒ _____________________________________

____ ⇒ ______________________________________

c) Determine the objective function used to maximize the profit.

d) Write the constraints as a system of inequalities.

Include inequalities for prep time, for assembly time and ones that indicate that neither the

number of remotes nor the number of nunchucks can be negative.

8) A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and

on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60

minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of

ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket. Find

the maximum profit and the amount of pants and jackets to maximize the profit.

a) Organize the given information in a chart or a table.

b) Identify and define the variables.

____ ⇒ _____________________________________

____ ⇒ ______________________________________

c) Determine the objective function used to maximize the profit.

d) Write the constraints as a system of inequalities.

Include inequalities for sewing time, cutting time and ones that

indicate that neither the number of ski pants nor ski jackets

can be negative.

Name ______________________________ Period __________

1.3B Graphing Systems of Linear Inequalities to Find a Feasible Region


1) How do we know if a point is part of the feasible region?

2) Look at the graph to the right. Determine whether each of

following points is a solution to the system and explain why or

why not.

a) ( )3, 2−

b) ( )4, 2−

c) ( )5,0

d) ( )0,3

#3 – 8: Match each system of inequalities with its graph.

3) 3

2

y x

y

x

≥ −

≥ −

≤

4) 3

2

y x

y

x

≤ −

≥ −

≤

5) 2

2 3 1

x y

x y

+ ≥

− >

6) 2

2 3 1

x y

x y

+ ≥

− <

7)

2 3

13

2

2

y x

y x

x

≤ − +

≥ −

≥ −

8)

2 3

13

2

2

y x

y x

x

≥ − +

≥ −

≥ −

A. B. C.

D. E. F.

Name ______________________________ Period __________




9) Tina and Boyang both graphed the following system of equations to find the feasible region. Did either of

them do the problem correctly? If not, explain to them what they did wrong and show the correct solution.

4

2 1

y x

y x

< +

≥ − +

Tina Boyang

#10 – 13: Graph the system of inequalities to find the feasible region. Use an appropriate method to graph

the boundary line of each inequality. A table of values may be useful to organize and record your thinking.

10)

3

2

5

y

x

x y

> −

<

+ ≤

11) 6

2 3 17

x y

y x

− < −

≥ +

12) 5 5

2 10

x y

y x

− ≥

− ≥ − 13)

5 2 24

3 2 16

6 12

x y

x y

x y

+ ≤ −

− ≤

− ≥

#14 – 15: Write the system of inequalities that would create each graph.

14) 15)

Name ______________________________ Period __________



16) A local street vendor sells hotdogs and pretzels. To make a profit, the street vendor must sell at least 30

hotdogs but cannot prepare more than 70. The street vendor must also sell at least 10 pretzels but cannot

prepare more than 40. The street vendor cannot prepare more than a total of 90 hotdogs and pretzels

altogether. The profit is $0.48 on a hotdog and $0.25 on a pretzel.

30 70

10 40

90

x x

y y

x y

≥ ≤

≥ ≤

+ ≤

a) Jane forgot to define her variables. What do x and y represent? Label the axes.

b) Within each “callout box”, label each line with its inequality.

c) Shade the feasible region.

d) Is it possible for the vendor to sell 5 pretzels and 60 hotdogs? Record your thinking.

e) To make a profit can the vendor sell 30 pretzels and 40 hotdogs? Record your thinking.

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

0

Name ______________________________ Period __________

1.3C Linear Programming – Finding Vertices Graphically



1) Wayne and Bubba both play football. The team trainer has put them each on their own special healthy diet.

Wayne currently weighs 120 lbs. The trainer puts him on a special bulk-up plan to gain 10 lbs per month.

Bubba currently weighs 210 lbs. The trainer puts him on a special get-lean plan to lose 8 lbs per month.

a) Make a table of values that represents Wayne’s weight and Bubba’s weight for 6 months.

b) Make a graph that represents Wayne’s weight for the 6 month time period.

c) Make a graph that represents Bubba’s weight for the 6 month time period.

d) Find the point where the lines intersect. List the coordinates of these points.

e) Explain the meaning of the point of intersection.

#2 – 5: Solve the systems of equations by graphing. Check your solution by showing, when using

substitution, that the ordered pair creates a true statement for both equations.

2)

71

2

15

2

y x

y x

= +

= −

solution: ( , ) 3)

17

6

51

6

y x

y x

= +

= − +

solution: ( , )

4)

25

7

21

7

y x

y x

= − −

= − −

solution: ____________ 5) 2 4

3 2 4

x y

x y

− = −

+ = − solution:_____________

6) Graph the system of inequalities. Find the vertices of the feasible region and label them on your graph.

�� ≥ −� − 2� ≥ 2� + 1� ≤ 12� + 7

7) Graph the system of inequalities. Find the vertices of the feasible region and label them on the graph.

2 10

1

2

7

x y

x

y

y x

+ ≥

≥

≥ ≥ − +

Name ______________________________ Period __________

1.3C Linear Programming – Finding Vertices Graphically


8) A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres.

However, he has only $1200 to spend and each acre of wheat costs $200 to

plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get

the planting done in 12 hours and it takes an hour to plant an acre of wheat and

2 hours to plant an acre of rye. The profit is $500 per acre of wheat and $300

per acre of rye.

a) Identify the variables and label the axes.

b) Determine the objective function used to maximize the profit.

c) Write the constraints as a system of inequalities.

Acres:

Cost:

Time:

d) Scale and number the axes to allow the x- and y-intercepts of the

inequalities to be plotted. What are the minimum and maximum

values that are necessary to include the critical values?

e) Graph the constraints on the coordinate grid. Find (and list) the vertices of the feasible region and

label them on the graph.

9) A carpenter makes tables and chairs. Each table can be sold for a profit of $30 and each chair for a profit of

$10. The carpenter can afford to spend up to 42 hours per week working and it takes six hours to make a table

and three hours to make a chair. The carpenter has a small shop and has limited room for storage. He has

only 40 cubic feet available for storage. Chairs take 5 cubic feet of storage; fortunately the tables are

collapsible and only take 4 cubic feet of storage.




Minimum number of tables:

Minimum number of chairs:

Time available:

Storage available:

d) Graph the constraints on the grid. Find (and list) the vertices of the

feasible region and label them on the graph.

Name ______________________________ Period __________

1.3D Linear Programming – Finding Vertices Algebraically



#1 – 4: Solve the systems of equations by elimination and check that it makes true statements for both

equations.

1) 3 8 18

7 8 10

x y

x y

+ = −

− − = solution:__________ 2)

10 3 25

5 9 25

x y

x y

− = −

− = solution:__________

3) 9 4 13

3 8 11

x y

x y

− + = −

− = solution:__________ 4)

8 6 16

3 17

x y

x y

+ =

− = solution:__________

5) Jack and Jill went to the Taco Shack for lunch. Jack ordered three tacos and three burritos for lunch and his

bill totaled $11.25. Jill paid $6.25 for one taco and two burritos.

a) Define your variables.

b) Write an equation to represent Jack’s lunch order.

c) Write an equation to represent Jill’s lunch order.

d) Use the elimination method to find the amount that each taco cost and each burrito cost.

e) How do you know your answer is correct?

#6 – 9: Use elimination to find each vertex of the feasible region graphed below. Label each vertex.

6) 7)

8) 9)

x + y ≥ 5

0.5x + y ≤ 8

–x + y ≥ –1

x ≥ 0

x – y ≤ 0

3x + y ≤ 12

2x + y ≥ –10

–x + y ≤ 8

–x + y ≥ 2

–3x + y ≥ 4

13x + 3y ≤ 93

x + 2y ≤ 16

4x + 9y ≥ –36

–9x + 7y ≤ 81

Name ______________________________ Period __________



10) A shoe company makes basketball and soccer shoes using

two machines (A and B). Each pair of basketball shoes that are produced requires

50 minutes processing time on machine A and 30 minutes processing time on

machine B. Each pair of soccer shoes that are produced requires 24 minutes

processing time on machine A and 33 minutes processing time on machine B.

Available processing time on machine A is forecast to be 40 hours (2400 minutes) and on machine B is

forecast to be 33 hours (1980 minutes).

The profit for a pair of basketball shoes in the current week is $75 and for a pair of soccer shoes is $95.

Company policy is to maximize the profits. The shoe company is trying to decide how many basketball shoes

and soccer shoes it should make in order to maximize profits.




Machine A time:

Machine B time:

# of pairs of basketball shoes:

# of pairs of soccer shoes:

d) Graph the constraints on a coordinate graph. Find and list the vertices of the feasible region (you may

need to use elimination to find one or more of the vertices). Label on the graph all vertices.

11) A potter is making cups and plates. It takes him 6 minutes to make a cup and 3

minutes to make a plate. Each cup uses 3/4 lb. of clay and each plate uses one lb. of

clay. He has 20 hours available for making the cups and plates and has 250 lbs. of

clay on hand. He makes a profit of $2 on each cup and $1.50 on each plate. The

potter is trying to determine how many cups and plates he should make in order to

maximize profits.


b) Determine the objective function.


Time:

Clay:

Cups:

Plates:


need to use elimination to find one or more of the vertices). Label on the graph all vertices.

Name ______________________________ Period __________




#12 – 15: Solve the systems of equations by substitution and check that it makes true statements for both

equations.

12) 7 1

6 2

y x

y x

= − −

= − − solution: ____________ 13)

3 5

7 5

y x

y x

= −

= − − solution: ____________

14) 7

3 5 19

y x

x y

= −

+ = − solution: ____________ 15)

3 6 9

2 3

x y

y x

− + = −

= + solution: ____________

16) An amusement park charges admission plus a fee for each ride. Admission plus two rides costs $10.

Admission plus five rides cost $16. What is the charge for admission and the cost of a ride?

#17 – 20: Use substitution to find each vertex of the feasible region graphed below. Label each vertex.

17) 18)

19) 20)

y ≥ 2

y ≥ x – 3 y ≥ 2x + 2

x – 2y ≤ –4

y ≤ x + 4

Name ______________________________ Period __________



21) The Northern Wisconsin Paper Mill can convert wood pulp to either notebook paper or newsprint. The mill

can produce at most 200 units of paper a day. At least 10 units of notebook paper and 80 units of newsprint

are required daily by regular customers. The profit on a unit of notebook paper is $500 and the profit on a

unit of newsprint is $350. The company manager is trying to determine how many units of notebook paper

and newsprint the mill should manufacture in order to maximize profits.




Minimum notebook:

Minimum newsprint:

Production limit:


need to use substitution to find one or more of the vertices). Label on the graph all vertices.

22) TeeVee Electronics, Inc., makes console and widescreen televisions. The equipment in the factory allows for

making at most 450 console televisions and 200 widescreen televisions in one month. It costs $600 per unit to

make a console television and $900 per unit to make a widescreen television. During the month of November,

the company can spend $360,000 to make these televisions. TeeVee makes $125 profit on each console

television and $200 on each widescreen television. The production manager is trying to determine how many

console and widescreen televisions the company should manufacture in order to maximize profits.




Minimum console televisions:

Minimum widescreen televisions:

Maximum console televisions:

Maximum widescreen televisions:

Costs:



Name ______________________________ Period __________




23) A farmer has a 320 acre farm on which she plants two crops: corn and

soybeans. For each acre of corn planted, her expenses are $50 and for each

acre of soybeans planted, her expenses are $100. Each acre of corn requires

100 bushels of storage and yields a profit of $60; each acre of soybeans

requires 40 bushels of storage and yields a profit of $90. The total amount of

storage space available is 19,200 bushels and the farmer has only $20,000 on

hand. The farmer is trying to determine how many acres of corn and

soybeans that she should plant in order to maximize profits.




Minimum corn:

Minimum soybeans:

Acres available:

Expenses:

Storage:



Name ______________________________ Period __________

1.3E Solving Linear Programming Problems


1) Given the graph of the constraints and the objective function, determine the vertices of the feasible region of

the graph given, the values of the objective function and the maximum and minimum values.

a) 2 3C x y= +

i) Find and list the vertices of the feasible region.

ii) Copy and complete the table. Test each of the vertices to

determine the maximum and minimum values of the

objective function.

Ordered

pair Calculations Value

iii) Summarize your findings by identifying the maximum and minimum values of the objective function

and the ordered pair that created each significant value.

b) 3 6C x y= +

i) Find and list the vertices of the feasible region.

ii) Copy and complete the table. Test each of the

vertices to determine the maximum and minimum

values of the objective function.

Ordered

pair Calculations Value



2) Given the following objective function and the vertices of the feasible region, determine the maximum and

minimum values.

Objective function: 3C x y= +

Vertices: (3, 0), (4, 5), (–1, 6), (–7, 5)

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

-3 -2 -1 1 2 3 4 5 6 7 8

-3

-2

-1

1

2

3

4

5

6

7

8

x

y

Name ______________________________ Period __________




3) Given the constraints and the objective function:

i) Graph the constraints on a coordinate graph.

ii) Find the vertices of the feasible region.



a)

5

4

2

x

y

x y

≤

≤

+ ≥

b)

2x + y ≤ 18

x + 2y ≤ 21

x ≥ 1

y ≥ 4

Objective function: 3 2C x y= − Objective function: 3 6C x y= +

4) In section 1.3B we looked at the selling patterns of a local street vendor sells hotdogs and pretzels. To make a

profit, the street vendor must sell at least 30 hotdogs but cannot prepare more than 70. The street vendor must

also sell at least 10 pretzels but cannot prepare more than 40. The street vendor cannot prepare more than a

total of 90 hotdogs and pretzels altogether.

a) Find the vertices of the feasible region and label them on the graph.

b) The profit is $0.48 on a hotdog and $0.25 on a pretzel. Write an objective function.

c) What combination of hotdogs and

pretzels would maximize her profits?

d) She realized that pretzels were better

sellers than hotdogs and wanted to

increase her profits. She changed the

profit of a hotdog to $0.25 and the profit

of a pretzel is now $0.48. Write the

revised objective function. What

combination of hot dogs and pretzel

sales gives the best profit now? Show

the calculations and write a summary

statement.

e) She decided that it would be easier if the

sale of every product created a profit of

$0.40. Write an objective function for

this situation. What combination gives

the best profit now? Show the

calculations and write a summary statement.

f) Taking into consideration all factors, what profit scenario would you recommend that the street vendor

function with?

Nu

mb

er o

f P

retz

els

Number of Hot Dogs

Name ______________________________ Period __________



5) Use Linear Programming to solve the problem.

A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread they make $35

profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to

bake. The muffins take 2 hours to prepare and 2 hours to bake. The maximum preparation time available is

16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should

be made to maximize profits?

a) Define the variables x =________________________ y = _________________________________

b) Write the constraints as a system of inequalities. (Hint: There are 4 restrictions)

c) Graph the system of inequalities on a coordinate graph.

d) What are the vertices of the feasible region?

e) What is the profit for each of these combinations?

f) Summarize your findings to answer the question presented.

6) Larry is starting a yard care business this summer featuring lawn

mowing services. He charges $25 per lawn mowed and $40 per

yard for yard cleanup (trimming/raking) services. He is on the

summer baseball team so he can only work 24 hours a week. It

takes him 1½ hours to mow a lawn and 3 hours to trim and rake a

yard. Larry is borrowing the equipment from his uncle who is

charging him $2 every time he uses the lawnmower and $1 every

time he uses the weed wacker for trimming. He doesn’t want to

spend more than $20 each week on equipment.

Larry wants to make as much money as possible on his new business. He has talked to his neighbors and is

pretty confident that he will have plenty of customers. Using a linear programming strategy for problem

solving, what combination of mowing lawns and yard trimming will give him the most profit? Keep in mind

that profit = income – expenses.

7) The manager of a travel agency is printing brochures and fliers to advertise special discounts on vacation

spots during the winter months. Each brochure costs $0.08 to print, and each flier costs $0.04 to print. A

brochure requires 3 pages, and a flier requires 2 pages. The manager does not want to use more than 600

pages, and she needs at least 50 brochures and 150 fliers. Using a linear programming strategy for problem

solving, how many of each should she print to minimize the cost?

Name ______________________________ Period __________

1.3F Using Graphing Technology for Linear Programming



#1 – 4: Use a graphing utility to help you solve the following linear programming problems.

1) The Plexus Dance Theatre Company will appear at the University of Georgia. According to school policy, no

more than 2000 general admission tickets can be sold and no more than 4000 student tickets can be sold. It

costs $0.50 per ticket to advertise the dance company to the students and $1 per ticket to advertise to the

general public. The dance company has an advertising budget of $3000 for this show.

Define the variables ___ =_______________________ ___ = ________________________________

Write the constraints as a system of inequalities.

Graph the system of inequalities on a coordinate graph. (Copy the graph from the graphing utility.)

What are the vertices of the feasible region?

Find the maximum profit the company can make if it charges $4 for a student ticket and $7 for a general

admission ticket. Keep in mind that profit = income – expenses.

Write a summary statement and identify the number of student tickets should they sell.

2) Funtime Airways flies from Palau to Nauru weekly if at least 12 first class tickets and at least 16 tourist class

tickets are sold. The plane cannot carry more than 50 passengers. Funtime Airways makes $26 profit for each

tourist class seat sold and $24 profit for each first class seat sold





In order for Funtime Airways to maximize its profits, how many of each type of seat should they sell?

Write a summary statement identifying the maximum profit and the number of each type of seat they

need to sell to obtain that profit amount.

3) Marcus is creating a low-fat pie crust recipe for his pie shop. Butter has six grams of saturated fat and one

gram of polyunsaturated fat per tablespoon. Vegetable shortening has one gram of saturated fat and four

grams of polyunsaturated fat per tablespoon. In the recipe, the butter and vegetable shortening will not be

more than 25 tablespoons. The butter and vegetable shortening combine for at least 34 grams of saturated fat

and at least 44 grams of polyunsaturated fat. Minimize the number of calories in the recipe if butter has 100

calories per tablespoon and vegetable shortening has 115 calories per tablespoon.





Record the calculations used to determine the minimum number of calories.

Write a summary statement identifying the minimum number of calories in the recipe and the number

of tablespoons of butter and vegetable shortening Marcus should use.

Name ______________________________ Period __________

1.3F Using Graphing Technology for Linear Programming


#1 – 4 (continued): Use your graphing utility to help you solve the following linear programming problems.

4) Reynaldo Electronica manufactures radios and cd players. The manufacturing plant has the capacity to

manufacture at most 600 radios and 500 cd players. It costs the company $10 to make a radio and $12 to

make a cd player. The company can spend $8400 to make these products. Reynaldo Electronica makes a

profit of $19 on each radio and $12 on each cd player.





Record the calculations used to determine the maximum the profit.

Write a summary statement identifying the maximum profit and the product they should produce.

Name ______________________________ Period __________

Unit 1 Review Material

Review Material 49

-5 5

-5

5

x

y1) Antonio believes (0, –1) is a solution to the inequality graphed

here. Mark believes it is not. Determine which student is

correct. Support your answer graphically and algebraically.

2) For each inequality graphed, determine the correct inequality symbol to correctly represent the graph. Show

how you determined which inequality symbol to use.

a) ________________ b) ________________

3) On a coordinate graph, graph each of the following inequalities.

a) Graph 8 4x y> − b) Graph 3

24

y x≤ − +

Points on the


boundary line

x y x y

-5 5

-5

5

2

13

y x +

-5 5

-5

5

3 4 x y+

Name ______________________________ Period __________



-5 5

-5

5

x

yGraph B

4) Explain where to find the solutions on a graph for each inequality graphed.

a) Graph A:

b) Graph B:

5) Write the inequality for Graph A.

6) Write the inequality for Graph B.

7) Prove algebraically that (5, 6) is a solution to the inequality for Graph A.

8) Prove algebraically that (4, 2) is not a solution to the inequality for Graph B.

9) Raymond must sell $500 in magazines and newspaper subscriptions in order to meet his fundraiser goal. A

magazine costs $15.00 and a newspaper subscription costs $22.00.

a) Define the variables you will use. ____ � _________________ and ____ � _________________

b) Write an inequality to describe the fundraising situation for Raymond to meet his goal.

c) Number and label the axes of a coordinate graph appropriately. Graph the inequality.

d) Identify one solution. Explain what your solution means in the context of this real-world situation.

e) Identify one non-solution. Explain what your non-solution means in the context of this real-world

situation.

-5 5

-5

5

x

yGraph A

Name ______________________________ Period __________


Review Material 51

10) During our basketball game, teams score points by shooting baskets (2 points) or by making free throws

(1 point). We do not make any 3-point shots. The opposing team scores 44 points.

a) Define the variables you will use. ____ � _________________ and ____ � _________________ \

b) Write an inequality to describe how many regular baskets and how many free throws our team must make

to win the game.

c) Number and label the axes of a coordinate graph appropriately. Graph the inequality.

d) Would making 15 baskets (2 point shot) and 4 free throws (1 point shot) be a solution? Explain in terms

of the context of the problem.

11) A plant makes aluminum and copper wire. Each piece of aluminum wire

requires 5 kilowatt hours (kwh) of electricity and 1/4 hour of labor. Each

pound of copper wire requires 6 kwh of electricity and 1/5 hour of labor.

Electricity is limited to 450 kwh per day and labor to 20 hours per day. If the

profit from aluminum wire is $0.25 per pound and the profit from copper is

$0.40 per pound, how much of each should be produced to maximize profit

and what is the maximum profit?

a) Define the variables you will use. ____ � _________________ and ____ � _________________ \

b) Organize the information and write the constraints.

c) Write the objective function.

d) Scale and number the axes to allow the x- and y-intercepts of the inequalities to be plotted. What are the

minimum and maximum values that are necessary to include the critical values?

e) Graph the constraints and write their inequality near each boundary line.

f) Name the vertices and write each on the graph.

g) What is the maximum profit? What is the minimum profit?

h) Write a summary statement answering the question that was presented.

12) A farmer wants to customize the fertilizer he uses for his current crop. He can buy plant food mix A and plant

food mix B. Each cubic yard of food A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen and 5

pounds of potash. Each cubic yard of food B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen

and 10 pounds of potash. He requires a minimum of 460 pounds of phosphoric acid, 960 pounds of nitrogen

and 220 pounds of potash. If food A costs $30 per cubic yard and food B costs $35 per cubic yard, use a

linear programming problem solving strategy to determine how many cubic yards of each food should the

farmer blend to meet the minimum chemical requirements at a minimal cost. What is this cost?

52 UNIT 2 – FUNCTIONS

FUNCTIONS

The concept of function may be the single most important concept spanning

all branches of mathematics. We will learn how to recognize when a

relationship is a function and to evaluate a function for a given input. We will

work with functions represented in various ways. We will recognize and

interpret key features of functions represented with graphs, tables, and

equations. We will relate these features to what is occurring in mathematical

and real world situations. The key features include: intercepts; intervals

where the function is increasing or decreasing; when the value of the function

is positive or negative; relative maximums and minimums; symmetries; domain (input) and range

(output); inverse relationships; and rate of change.

Practice Problems

53 Unit 2 Reference and Resource Material

2.1 I can demonstrate understanding of the

definition of a function and can determine

when relations are functions given a graph,

table or real-world situation.

71 2.1A Functions and Situations

72 2.1B Classifying Functions

2.2 I understand the meaning of function

notation and can evaluate a function for a

given input.

74 2.2A Evaluating Functions Using Function

Notation

77 2.2B Graphing Functions


significant features of a function

represented by a graph, a table, or an

equation and the relationship these features

have to real-world situations.

81 2.3A Significant Features of Functions – Part I

84 2.3B Significant Features of Functions – Part II

87 2.3C Significant Features of Functions – Part III

88 2.3D Rate of Change

89 2.3E Relations as Inverses

92 2.3F Using Graphing Technology


UNIT

2


2.1 I can demonstrate understanding of the definition of a function and can determine when

relations are functions given a graph, table or real-world situation.

2.2 I understand the meaning of function notation and can evaluate a function for a given

input.

2.3 I can demonstrate understanding of the significant features of a function represented by a

graph, a table, or an equation and the relationship these features have to real-world

situations.



2.1 I can demonstrate understanding of the definition of a function and can

determine when relations are functions given a graph, table or real-world

situation.

Input/Output Tables

The following diagrams show a map of the relationship between the input values and

output values.

Function Not a Function

This is a function because each input

value is mapped to exactly one

output value.

This is NOT a function because the

input value 3 is mapped to two

different output values, 6 and 7.

Input and output values could also be organized in a table.

Function

Not a Function

x y

–1 3

–2 5

–3 3

–5 –3

x y

1 –1

5 –2

1 2

–3 –5

This table represents a function. None of the

independent values (x) are repeated and each

has only one corresponding dependent value (y).

This table does NOT represent a function.

The x column has two values that are 1 and

they correspond to two different values for y.

Ordered Pairs

Functions can also be represented by sets of ordered pairs of x and y values, inputs and outputs. As with

other methods of representing relations, we can check the characteristics of a set of ordered pairs to

determine if it is a function. Since the first value in each pair is the input and the second is the output, we

can scan the set to see if each input is associated with a single, consistent output. If it is, the set of ordered

pairs is a function.

For example, {(–1, 3), (–2, 5), (–3, 3), (–5, –3)} is a function because each x-value (input) has

one y-value (output).

However, {(1, 5), (2, –1), (3, 6), (3, 7)} is NOT a function because the x-value, 3, is repeated

with two different outputs, 6 and 7.

Unit 2 Resources



Real-World Situation

Consider the relation “students in your school” as inputs and their “birth dates” as outputs. This is a

function, because each student can only have one birthday.

Reverse the relation using “birthdates” as the input and “students in your school” as the outputs. The

relation is no longer a function because multiple students (output) may share a common birthdate (input).

Vocabulary

• The x-intercept is the location where the graph crosses the x-axis, which is the value of x where ( ) 0f x =

• The y-intercept is the location where the graph crosses the y-axis, which is the value of ( )0f .

• A relation is a relationship between variables that change together.

• A function is a relation where there is exactly one output for every input.

• Independent variables represent the input of a function.

• Dependent variables represent the output of a function.

Video Resources:

• http://www.montereyinstitute.org/courses/Algebra1/U03L2T1_RESOURCE/index.html

• CK-12 Foundation: Relations and Functions

• Khan Academy: Relations and Functions

• Khan Academy Functions as Graphs

2.2 I understand the meaning of function notation and can evaluate a function for a given input.

Writing Equations in Function Notation

Previously you have written and used equations such as 3 7y x= − + . Many equations represent

functions. In the equation 3 7y x= − + , if we input a value for x, such as 10, we get a corresponding

output for y of –23. We can represent this result as the ordered pair (10, –23). A function is a set of

ordered pairs in which the first coordinate, usually x, matches with exactly one second coordinate, y.

Equations that follow this definition can be written in function notation. The y-coordinate represents the

dependent variable (output), meaning the values of this variable depend upon what is substituted for the

input variable.

The equation 3 7y x= − + is written in function notation as ( ) 3 7f x x= − + .

A function rule replaces the variable y with its function name, usually ( )f x . f represents the function

name and the symbol (x) represents the independent variable (input). The parentheses, in this case, do

not indicate multiplication. The parentheses separate the function name from the variable.

( )f x is read “the function f of x” or simply “f of x”

( ) 3 1h x x= − is read “h of x equals 3 times x minus 1”



2.2 Example 1:

Rewrite the following equations in function notation.

a) y = 7x – 3

b) d = 65t

c) 9

325

F C= +

Solution:

a) ( ) 7 3f x x= − because x is the input, this is the variable in the ( ).

b) ( ) 65d t t= because t is the input, this is the variable in the ( ).

c) ( )9

325

F C C= + because C is the input, this is the variable in the ( ).

2.2 Example 2:

Evaluate the following functions.

a) ( )21

2f x x= − … determine the value of ( )4f

b) ( ) 14 3g x x= − … determine the value of ( )0.5g

c) ( ) 4h x x= + … determine the value of ( )12h

Solution:

a)

( )

( ) ( )

( ) ( )

( )

2

2

1substitute 4 for

2

14 4 simplify the expression

2

14 16 continue to simplify

2

4 8 input: 4 output: 8

f x x x

f

f

f

= −

= −

= −

= − −

b)

( )

( ) ( )

( )

14 3 substitute 0.5 for

0.5 14 0.5 3 simplify the expression

0.5 4 input:: 0.5 output: 4

g x x x

g

g

= −

= −

=

c)

( )

( )

( )

( )

4 substitute 12 for

12 12 4 simplify the expression

12 16

12 4 input:: 12 output: 4

h x x x

h

h

h

= +

= +

=

=



f (5) = 0

f (–3) =10

f (3) = 8

f (1) = 2

2.2 Example 3:

The value V of a digital camera t years after it was bought is represented by the function V(t) = 875 – 50t.

a) Determine the value of V(4) and explain what the solution means to this problem.

b) Determine the value of t when V(t) = 525 and explain what this represents.

c) Determine the value of V(0) and explain what the solution means to this problem.

Solution:

a) V(4) = 675 The value of the camera after 4 years is $675.00

b) 525 = 875 – 50t solve for t… –350 = –50t t = 7

After 7 years, the value of the camera will be $525.00

c) V(0) = 875 – 50(0) V(0) = 875

When time is zero (starting amount), the value of the camera is $875.00

Function as a table of values

You can also represent a function as a table of values.

a) ( )3 10f − =

b) ( ) 0f x = when x is 5

Function as a graph

You can also represent a function as a graph.

a) ( )3 8f =

… the x-value is 3

… the function value when x = 3 is 8

b) ( ) 2f x = when 1x =

… need to think… where does the

function value equal 2?

… locate the point and determine the x-

value that creates a function value of 2.

This creates the ordered pair of (1, 2)

x f(x)

–3 10

0 4

3 6

5 0



Function as a Graph

A function can be represented by a graph on a coordinate grid. The independent value is plotted on the

x-axis and the dependent value is plotted on the y-axis. The fact that each input value has exactly one

output value means graphs of functions have certain characteristics. For each input (x-coordinate) on the

graph, there will be exactly one output (y-coordinate). Use the vertical line test to demonstrate this.

Function Not a Function

The graph of the semicircle represents a function.

For each x-coordinate (input) there is exactly one

y-coordinate (output). The graph touches the

vertical line at only one location.

The graph of the circle does not represent a

function. There are many x-coordinates (inputs)

that have more than one y-coordinate (output).

The graph touches the vertical line at two or more

locations.

Vocabulary

• The vertical line test is a test to determine if a graph of an equation is a function. It involves drawing

several vertical lines over the graph. If the graph touches any vertical line more than once, it is not a

function.

Video Resources:

• CK-12 Basic Algebra: Linear Function Graphs



2.3 I can demonstrate understanding of the significant features of a function represented by a graph, a

table, or an equation and the relationship these features have to real-world situations.

Significant Feature: Domain and Range

There is another pair of components we must consider when talking about relations, called domain and range. The

domain of a function or relation is the set of all possible independent values the relation can take. It is the collection

of all possible inputs. The range of a function or relation is the set of all possible dependent values the relation can

produce from the domain values. It is the collection of all possible outputs. By putting all the inputs and all the

outputs into separate groups, domain and range allows us to find and explore patterns in each type of variable.

Here’s a series of figures, each made of groups of squares.

We can make a function out of this series by using the number

of the figure as the input, and the number of squares that make

up the figure as its output. An input of 1 has an output of 1,

since figure 1 has just 1 square. An input of 2 has an output of

5, since figure 2 has 5 squares. An input of 3 has an output of

9, since figure 3 has 9 squares. This function’s domain is the

counting numbers 1, 2, 3 that identify which figure is used. The

inputs to this function are discrete values, or values that

change in increments and not continuously. There are only the

three figures, so the only possible inputs are 1, 2, and 3.

Thus the domain of this function is 1, 2, and 3.

Domain: {1, 2, 3}

The range is the number of squares in each figure. The figures have only 1, 5, or 9 squares, so that’s the range. There

is no figure that has 2 or 3.5 or any other number of squares. Like the domain, the range is made of a set of discrete

values.

Range: {1, 5, 9}

If the pattern goes on indefinitely we would add a series of ellipses to the end of each set of values, to indicate that

the sequence continues, like this:

Domain: {1, 2, 3, …}

Range: {1, 5, 9, …}

Relations can also be shown as tables or as sets of ordered pairs. When a mathematical relationship is given in a

table, the independent values, generally listed in the left-hand column, are the domain, and the dependent values,

usually found in the right-hand column, make up the range.

FIGURE 1 FIGURE 2 FIGURE 3



Domain: {–1, 2, 5, 9}.

Range: {–3, 4, 6, 7}.

When it comes to sets of ordered pairs, we simply need to split the pairs apart into x-coordinates and

y-coordinates. The x-coordinates make up the domain and the y-coordinates are the range.

In the set of ordered pairs {(–2, 0), (0, 6), (2, 12), (4, 18)},

The domain is the set of x-coordinates: {–2, 0, 2, 4}.

The range is the set of the y-coordinates: {0, 6, 12, 18}.

On a graph, the independent variable of a function is usually

graphed on the horizontal (x) axis. That means the

x-coordinates of the points are the domain. Since the dependent

variable is usually graphed on the vertical (y) axis, the

y-coordinates make up the range.

First, examine this graph of discrete data. The only values that we

know to satisfy the function are the marked points. We simply

read off the x-coordinates, and place them in the set of domain

values. Then we read the y-coordinates, and put them into the

range. For this graph:

The domain is {–2, 2, 6}.

The range is {0, 5, 9}.

This function has a domain that is discrete and finite. We can count how many elements are in the domain. You

can have a domain that is discrete and infinite such as the set of all integers.

This function has a continuous domain because the domain is the

set of all real numbers. The domain here is also infinite since you

cannot count the elements in the domain.

For this function, there are no restrictions to the domain and range.

Any real number can be an input or an output. That means all

whole numbers, integers, fractions and other rational numbers, and

even irrational numbers, are all part of the domain and part of the

range. Since we cannot write down all those possibilities, we

simply say:

Both the domain and the range are all real numbers.

Independent

Value

Dependent

Value

–1 7

2 –3

5 6

9 4



x-intercept (3, 0) x-intercept:

y-intercept (0, –3)

In some situations only the domain or the range is unrestricted, but not

both.

In this function, the line extends indefinitely in both directions along the

x-axis. However, the function is always positive; the range values never

get less than zero.

The domain is all real numbers.

The range is all real numbers y ≥ 0.

The domain and range of a function are often limited by the nature of the

relationship.

For example, consider the function of time and height that occurs when you punt a football into the air and

someone catches it. Time is the input, height is the output.

The domain (input) is time. The values included in the domain begin the instant the football leaves the foot

of the punter (when t = 0) and end the instant the football is caught. Time before you kick it and after you

catch it are irrelevant, since the function only applies for the duration of the punt. Say the football was in the

air for 4 seconds prior to being caught – in that case, the domain is 0 – 4 seconds. Because time runs

continuously during this interval, we cannot write down every possible input, only the starting and stopping

values.

The range (output) is the height of the football. It includes all heights of the football from the time it contacts

the punter’s foot to the highest point the football reaches and until the ball is caught. If the punter’s foot was

2 feet above the ground when contact was made, it reached its highest point at 100 feet off the ground and was

caught when 4 feet above the ground, then the range is 2 – 100 feet. Because height changes continuously

during this interval, we cannot write down every possible output, only the least and greatest values.

Significant Feature: Intercepts

Intercepts are the locations where the graph of a function crosses an axis. The x-intercept is the location where the

graph crosses the x-axis, which is the value of x where y = 0. The y-intercept is the location where the graph crosses

the y-axis, which is the value when x = 0.

To interpret an intercept in a real world situation it is important to remember that the x-intercept is the x-value

where ( ) 0f x = and the y-intercept is the value of ( )0f .



Significant Feature: Extrema (minima and maxima)

The point where the function changes from increasing to decreasing is called a relative maximum. The point

where the function changes from decreasing to increasing is called a relative minimum. Relative maximums and

relative minimums are also called relative extrema.

In this function the extrema that exist are:

Absolute maximum: ( )3, 2−

Relative minimum: ( )2.5, 7.6−

Relative maximum: ( )8,1

These extrema points identify the boundary

point between where the function is increasing

and where the function is decreasing.

This function has no maximum or minimum as it is

always decreasing.

This function has extrema at:

( )3, 2− an absolute maximum as this

is the absolute largest value of the

function.

( )2.5, 7.6− a relative minimum as this

is not the smallest value of the

entire function, but it is the

smallest value in this relative area.

( )8,1 a relative maximum as it is the

largest function value in this

relative area.

Absolute maximum

(–3, 2) Relative maximum

(8, 1)

Relative minimum

(2.5, –7.6)



Significant Feature: Increasing and Decreasing Intervals

One way to describe functions is whether they are increasing or decreasing when you look at them from left to

right. An increasing function is one with a graph which goes up from left to right. A decreasing function is one

with a graph which goes down from left to right.

A line like the graph shown is always decreasing.

Sometimes a function might change from increasing to decreasing or from decreasing to increasing. We use

intervals to describe which parts of the function are increasing and which are decreasing. An interval is a specific

and limited part of a function. The interval can be described with an inequality using x.

In this function the extrema that exist are:

Absolute maximum: ( )3, 2−

Relative minimum: ( )2.5, 7.6−

Relative maximum: ( )8,1

These extrema points identify the breaking

point between where the function is increasing

and where the function is decreasing.

This function is:

Increasing over the intervals:

3 and 2.5 8x x< − < <

Decreasing over the intervals:

3 2.5 and 8x x− < < >

Increase Decrease Increase Decrease

Absolute maximum

(–3, 2)

Relative maximum

(8, 1)

Relative minimum

(2.5, –7.6)



Positive function values

Negative function values

Significant Feature: Positive and Negative Intervals

Another way to describe graphs is to identify whether the value of the function, ( )f x , is positive or negative. The

function is described as positive when ( ) 0f x > and the function is described as negative when ( ) 0f x < . The

x-axis is the dividing line between positive function values and negative function values.

This function is:

positive when x < 5 and

negative when x > 5.

This function is:

negative when x < 3.1 and

positive when x > 3.1.

Significant Feature: Rate of Change

2.3 Example 1:

A gym had 25 members at the end of its first week of operation, 5 of whom had preregistered. Two weeks

after it opened it had 45 members, and 3 weeks after it opened it had 65 members. How would you

calculate the rate of change in the number of gym members? How is this different than the slope? If this

rate continues, how long will it take for the gym to have 300 members?

Solution:

Organize the information.

Seeing this, it can be concluded that there is an increase of 20 members per 1

week, remembering that upon opening, there were 5 members. Continuing the

table or completing calculations, after 14 weeks, there would be 285 members,

and after 15 weeks they will have achieved their goal of having 300 members

(305 actual members). It will take 15 weeks (15 weeks x 20 members per

week + 5 that preregistered) for the gym to reach their goal of 300 members.

Week # of members

1 25

2 45

3 65

4 85

Positive function values

Negative function values



Significant Feature: Inverses

By now, you are probably familiar with the term “inverse”. Multiplication and division are inverses of each other.

More examples are addition and subtraction and the square and square root. We are going to extend this idea to

functions. An inverse relation maps the output values to the input values to create another relation. In other words,

we switch the x and y values. The domain of the original relation becomes the range of the inverse relation and the

range of the original relation becomes the domain of the inverse relation.

2.3 Example 2:

Find the inverse mapping of ( ) ( ) ( ) ( ) ( ){ }6, 1 , 2, 5 , 3, 4 , 0,3 , 2, 2S = − − − − .

Solution:

Here, we will find the inverse of this relation by mapping it over the line y x= . As was stated above in

the definition, the inverse relation switched the domain and range of the original function. So, the inverse

of this relation, S , is 1S − (said “ S inverse”) and will flip all the x and y values.

( ) ( ) ( ) ( ) ( ){ }1 1, 6 , 5, 2 , 4, 3 , 3, 0 , 2, 2S −

= − − − −

If we plot the two relations on the x, y plane, we have:

The dot points are all the points in and

the “××××” points are all the points in .

Notice that the points in are a

reflection of the points in over the line,

. All inverses have this property.

If we were to fold the graph on , each

inverse point should lie on the original

point from . The point lies on this

line, so it has no reflection. Any value on

this line will remain the same.

Domain of :

Range of :

Domain of :

Range of :

By looking at the domains and ranges of

and , we see that they are both functions (no x-values repeat). When the inverse of a function is also

a function, we say that the original function is a one-to-one function. Each value maps one unique value

onto another unique value.

S

1S −

1S −

S

y x=

y x=

1S −

S ( )2, 2

S { }3, 2, 0, 2, 6− −

S { }5, 1, 2, 3, 4− −

1S −{ }5, 1, 2, 3, 4− −

1S −{ }3, 2, 0, 2, 6− −

S

1S −

×

×

×

×

×



2.3 Example 3:

Find the inverse of ( )2

13

f x x= − .

Solution: This is a linear function. Let’s solve by

doing a little investigation. First, draw the

line along with on the same set of

axes.

Notice the points on the function f

(Line 1). Map these points over by

switching their x and y values. You could

also fold the graph along y x= and trace

the reflection.

Line 2, in the graph to the right is the

inverse of ( )2

13

f x x= − . Using slope

triangles between ( )1, 0− and ( )1, 3 , we

see that the slope is 3

2. Use ( )1, 0− to

find the y-intercept.

( )

( )

( )

1

1

3

2

30 1

2

3

2

3 3

2 2

f x x b

b

b

f x x

−

−

= +

= − +

=

= +

The equation of the inverse, read “f inverse” is ( )1 3 3

2 2f x x−

= + .

You may have noticed that the slopes of f and 1f − are reciprocals of each other. This will always be

the case for linear functions. Also, the x-intercept of f becomes the y-intercept of 1f − and vice versa.

y x=

y x=

(Line 1)

(Line 1)

(Line 2)



2.3 Example 3 (continued):

Find the inverse of ( )2

13

f x x= − .

Solution - Alternate Method: There is also an algebraic approach to finding the inverse of any function.

Let’s repeat this example using algebra.

1. Change ( )f x to y. 2

13

y x= −

2. Solve for x (isolate the independent variable from the initial equation).

The output value (y) from the initial equation will become input (x) of the

inverse equation. When writing the inverse equation, be sure to notate the

difference by using 1y − to represent the equation of the inverse.

Vocabulary

• The domain of a function or relation is the set of all possible independent values the relation can take. It is

the collection of all possible inputs.

• The range of a function or relation is the set of all possible dependent values the relation can produce

from the domain values. It is the collection of all possible outputs.

• An intercept is a location where the graph of a function crosses an axis.

• The x-intercept is the location where the graph crosses the x-axis which is the value of x where ( ) 0f x = .

• The y-intercept is the location where the graph crosses the y-axis which is the value of ( )0f .

• An increasing function is one with a graph which goes up from left to right.

• A decreasing function is one with a graph which goes down from left to right.

• An interval is a specific and limited part of a function that can be described with an inequality using x.

• The point where the function changes from increasing to decreasing is called a relative maximum.

• The point where the function changes from decreasing to increasing is called a relative minimum.

• Relative maximums and relative minimums are called relative extrema.

• The highest point on a graph is called the absolute maximum. The lowest point on a graph is called the

absolute minimum.

• Functions are positive where f(x) > 0. Functions are negative where f(x) < 0.

• For the ordered pairs ( )( )1, 1x f x and ( )( )2, 2x f x , the Rate of Change = ( ) ( )2 1

2 1

f x f x

x x

−

−

Video Resources:

• - James Sousa: Determine Where a Function is Increasing and Decreasing

21

3

21

3

3 3 2

3 3

2 2

y x

y x

y x

y x

= −

+ =

+ =

+ =

1

1

3 3

2 2

3 3

2 2

3 3

2 2

y x

x y

y x

−

−

+ =

+ =

= +



Graphing Features on a TI-84+

Graph a

Function

Select

Enter the function

Use to enter x-value

Select

Select option 6: ZStandard

To see points on the graph:

Select

Use

Use a Table

Select TBLSET by pressing

Enter a starting value

for the table (TblStart)

Enter the amount x

should be incremented

in the table (∆Tbl)

Select TABLE by pressing

Use



Finding the

y-intercept

Select CALC by pressing

Choose option 1: value

At the prompt x = ?,

enter 0

Press

The y‐intercept is shown:

(0, 2)

Finding the

x-intercept


Choose option 2: zero

At the prompt…Left Bound?

Use the L/R arrows to

move the cursor left of

the x‐intercept.

Press

At the prompt…Right Bound?


move the cursor right of

the x‐intercept.

Press

At the prompt…Guess?

Press

The x-intercept is shown:

(–0.67, 0)



Finding

the

Minimum

or

Maximum

Graph 2 4y x x= −


Choose option

3:minimum

(or 4: maximum)

At prompt… Left Bound?


move the cursor left

of the minimum.

Press

At prompt… Right Bound?

Use the L/R arrows to move

the cursor right of the

minimum.

Press

At the Guess?

Press

The minimum is shown:

(2, –4)



Change a

Viewing

Window

Sometimes functions are entered that do not show a

good view of the graph in the standard window.

To determine a better

viewing window, look at

the table. Notice the range

of y values.

Use the arrow keys to

scroll up and down to see

the patterns of the

y-values.

Use that information to

enter more appropriate

values for

Xmin

Xmax

Ymin

Ymax

Select

Name ______________________________ Period __________

2.1A Functions and Situations

2.1 I can demonstrate understanding of the definition of a function and can determine when 71


Time Time Time Time

Am

ou

nt

of

Lem

on

ade

Am

ou

nt

of

Lem

on

ade

Am

ou

nt

of

Lem

on

ade

Am

ou

nt

of

Lem

on

ade

#1 – 2: Choose a graph that matches the statement. Give reasons for your answer.

1) A bicycle tire valve’s distance from the ground as a boy rides at a constant speed.

[A] [B] [C] [D]

2) A child swings on a swing, as a parent watches from the front of the swing.

[A] [B] [C] [D]

3) Ian and his friends were sitting on a deck and drinking lemonade. Each person had a glass with the same

amount of lemonade. The graphs below show the amount of lemonade remaining in each person’s glass over

a period of time.

Write sentences that describe what may have happened for each person (person a, b, c, and d).

a) b) c) d)

4) Sketch a graph to model each of the following situations. Remember to label the axes of each graph.

a) Candle: Each hour a candle burns down the same amount.

x = the number of hours that have elapsed. y = the height of the candle in inches.

b) Letter: When sending a letter, you pay quite a lot for letters weighing up to an ounce. You then pay a

smaller, fixed amount for each additional ounce (or part of an ounce).

x = the weight of the letter in ounces. y = the cost of sending the letter in cents.

c) Bus: A group of people rent a bus for a day. The total cost of the bus is shared equally among the

passengers.

x = the number of passengers. y = the cost for each passenger in dollars.

d) Car value: My car loses about half of its value each year.

x = the time that has elapsed in years. y = the value of my car in dollars.

Name ______________________________ Period __________

2.1B Classifying Functions

72 UNIT 2 – FUNCTIONS (PRACTICE)

1) How are input / output similar to an independent variable/a dependent variable?

2) In your own words, define a function.

3) Give three real-life examples of relations that are functions.

4) Is it mandatory for a function to have both an input and an output? Explain.

Give an example.

Give a non-example.

5) Can a statement be a function if there is only one input and one output? Explain.

#6 – 13: For each relation below:

a. Tell whether it is a function or not a function

b. If it is not a function, change it so that it is

6) Input Output 7) Input Output

8) 9) 10)

x y x y x y

1 4 1 –2 2 –1

4 7 2 –2 2 –2

3 10 3 –2 2 –3

4 13 4 –2 2 –5

11) 12) 13)

#14 – 27: Tell whether the situation represents a function or not. Explain your reasoning.

14) Input: Age

Output: Names of Students who are that age

Name ______________________________ Period __________

2.1B Classifying Functions

2.1 I can demonstrate understanding of the definition of a function and can determine when 73


#14 – 27 (continued): Tell whether the situation represents a function or not. Explain your reasoning.

15) Input: Name of Student

Output: Age of the student named

16) ( ) ( ) ( ) ( ) ( ) ( ){ }2, 4 , 4, 6 , 6,8 , 3, 4 , 5, 7 , 8, 2

17) ( ) ( ) ( ) ( ) ( ){ }1, 6 , 0, 4 , 4, 0 , 1, 6 , 3, 8− − − − − −

18) ( ) ( ) ( ) ( ) ( ){ }Jim, Alice , Joe, Alice , Brian, Betty , Jim, Kitty , Ken, Anissa

19) ( ) ( ) ( ) ( ) ( ){ }Jim, Kitty , Jim, Betty , Brian, Alice , Jesus, Anissa , Ken, Kelli

20) Your age and your weight on your birthday each year.

21) The name of a course and the number of students enrolled in that course.

22) The diameter of a cookie and the number of chocolate chips in it.

23) At a Prom dance, each boy pins a corsage on his date.

24) Later, at the same dance, Cory shows up with two dates, does this change the answer? Explain why or why

not.

25) {(0, 0), (1, 1), (1, –1), (2, 2), (2, –2)}

26) {(4, 2), (5, 1), (6, 0), (7, –1), (8, –2)}

27) {(–2, 2), (–1, 1), (0, 0), (1, 1), (2, 2)}

28) Create your own example of a relation that is a function. Represent this relation in all three forms

(Input/Output Columns (Mapping), Table, and Ordered Pairs).

29) Create your own example of a relation that is NOT a function. Represent this relation in all three forms

(Input/Output Columns (Mapping), Table, and Ordered Pairs).

Name ______________________________ Period __________

2.2A Evaluating Functions Using Function Notation


#1– 2: Use the table to find the following values to find the values in parts (a) through (d).

1) x f(x) 2) x g(x)

–1 0 –5 –1

1 5 2 –2

3 3 3 2

5 –3 10 –5

a) f(–1) = _______ a) g(–5) = _______

b) f(5) = _______ b) g(2) = _______

c) f(x) = 5 x = _______ c) g(x) = 2 x = _______

d) f(x) = 0 x = _______ d) g(x) = –2 x = _______

3) Fran collected data on the number of feet she could walk each second and wrote the following rule,

( ) 4d t t= , to model the distance she travels ( )d t in t seconds.

a) What is Fran looking for if she writes “d(12) = __”?

b) In this situation what does d(t) = 100 tell you?

c) How can the function rule be used to indicate that a time of 16 seconds was walked?

d) How can the function rule be used to indicate that a distance of 200 feet was walked?

4) Mr. Multibank has developed a population growth model for the rodents in the field by his house. He

believes that starting each summer the population can be modeled with the function ( ) ( )8 2t

p t = with t

representing the number of weeks since the beginning of summer.

a) Find p(4).

b) Find p(10).

c) Find t, if p(t) = 512.

d) Find the number of weeks it will take for the population to be over 20,000.

e) In a year with 16 weeks of summer, how many rodents would he expect by the end of the summer?

What real-world factors might deter or interrupt actual population numbers?

5) How are “y =” and “f(x) =” similar? How are they different?

6) Which is the independent variable, x or ( )f x ? Which is the dependent variable? Why?

#7 – 10: Use the function to find the following values. Record your thinking.

7) 32)( += xxf 8) 13)( +−= xxg

a) f(0) a) g(0)

input: _______________________ input: _______________________

output: ______________________ output: ______________________

b) f(–2) b) g(3)

c) f(x) = 7 c) g(x) = –5

input: _______________________ input: _______________________

output: ______________________ output: ______________________

d) f(x) = 13 d) g(x) = 7

Name ______________________________ Period __________


2.2 I understand the meaning of function notation and can evaluate a function for a given input. 75

#7 – 10 (continued): Use the function to find the following values. Record your thinking.

9) 2)( tth = 10) ( ) 2 xT x =

a) h(1) a) T(1)

b) h(5) b) T(2)

c) h(t) = 0 c) T(x) = 16

d) h(t) = 16 d) T(x) = 64

#11 – 12: Find the value of x when given the following information.

11) ( ) ( )6 3 for 15g x x g x= − = .

12) ( ) ( )2

5 for 33

h x x h x= − + = − .

13) One human year is equivalent to seven dog years. Write a function to translate a Dog's Human Age into its

comparative Dog Age.

#14 – 15: For each problem below, copy the 4-representations to your assignment sheet. Given one of the

following (a table, a graph, an equation or a story), create the rest so each is representative of

the other.

14) Table:

x y

Graph:

Equation:

5 50y x= − +

Story:

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

0x

y

Name ______________________________ Period __________



#14 – 15 (continued): For each problem below, copy the 4-representations to your assignment sheet. Given

one of the following (a table, a graph, an equation or a story), create the rest so each is

representative of the other

15) Table:

x y

Graph:

Equation:

Story:

A car travels down a highway at 50 mph. Let x represent the time

and let y represent the total distance travelled in miles.

16) Ms. Callahan works hard to budget and predict her costs for each month. She is currently attempting to

determine how much her cell phone company will likely charge her for the month. She is paying a flat fee of

$80 a month for a plan that allows for unlimited calling and texting, but costs her an additional twenty cents

per 100KB per month for data usage over 1GB.

a) Write a function for Ms. Callahan’s current cell phone plan that will calculate the cost for the

month, ( )c k , based on the number of 100KB blocks of data, k, over 1GB of data that she uses.

b) Find c(45) and explain the meaning of this situation in the context of the problem.

c) Find k, if c(k) = 90 and explain the meaning of this situation in the context of the problem.

d) When would unlimited data, at an additional $20 a month, be cheaper than her current plan?

1 2 3 4 50

25

50

75

100

125

150

175

200

225

250

275

300

0x

y

Name ______________________________ Period __________

2.2B Graphing Functions


#1 – 6: For each graph, categorize each of the following as a function or not a function. If it is not a

function, copy the graph to your assignment sheet and mark at least two points on the graph that

demonstrate it is not a function and list the ordered pairs marked.

1) 2) 3)

4) 5) 6)

7) One of the following graphs represents a function and one is not a function. Identify the function and

explain the difference between the two graphs. Why does one represent a function and the other does not?

Name ______________________________ Period __________



#8 – 11: Use the graph to find the following values.

8) 9)

a) ( )4 _____f =

a) ( ) 2 _____h x x= ≈

b) ( )3 _____f − ≈ b) ( ) 0 _____h x x= ≈

c) ( ) 4 _____f x x= = c) ( )0 _____h =

d) ( ) 5 _____f x x= ≈

d) ( )1 _____h =

10)

11)

a) ( )1 _____g − =

a) ( )1 _____d − =

b) ( )3 _____g − =

b) ( ) 4 _____d x x= =

c) ( ) 4 _____g x x= − =

c) ( )3 _____d ≈

d) ( ) 1 _____g x x= − =

d) ( ) 0 _____d x x= =

Name ______________________________ Period __________



12) You are going on vacation and you are wondering if a specific day of the week is better than others to

purchase gas before beginning your travels.

Included below is a graph of the average gas prices throughout the state of Minnesota. In questions a)

through e) below, round prices to the nearest cent.

a) Copy the graph to left onto your assignment

sheet. Referencing the graph above, what is the

average price for a gallon of gas on Saturday,

June 14th? Plot this point on the graph to the left.

(Round prices to the nearest cent.)

b) Referencing the graph above, what is the average

price for a gallon of gas on Sunday, June 15th?

Plot this point on the graph to the left.

c) Continue to transfer all the data from the graph

above to the graph on the left. You will plot the

gas price for each day of the week over the

course of the entire month onto the graph. For

example, there are 4 Sundays in the month and

you will place a data point for each of these days

on the Sunday value on the x-axis. Repeat this

for the remaining days of the week.

d) Is the graph above a function? Explain.

e) Is the graph to the left a function? Explain. Day of the Week

z Su Mo Tu We Th Fr Sa

Aver

age

Gas

Pri

ce i

n M

innes

ota

($ p

er g

allo

n)

3.60

http://www.gasbuddy.com/gb_retail_price_chart.aspx

Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa

3.62

3.61

3.60

3.59

3.58

3.57

3.56

3.55

3.54

3.53

3.52

3.62

3.61

3.60

3.59

3.58

3.57

3.56

3.55

3.54

3.53

3.52 6/1

3

6/1

4

6/1

5

6/1

6

6/1

7

6/1

8

6/1

9

6/2

0

6/2

1

6/2

2

6/2

3

6/2

4

6/2

5

6/2

6

6/2

7

6/2

8

6/2

9

6/3

0

7/0

1

7/0

2

7/0

3

7/0

4

7/0

5

7/0

6

7/0

7

7/0

8

7/0

9

7/1

0

7/1

1

7/1

2

Name ______________________________ Period __________



13) You throw a ball in the air and its height (meters) at t seconds is given by the following equation

( )24.9 30 2h t t t= − + + .

a) Copy the table and graph below onto your assignment sheet. Complete the table (investigate at least five

input values) and sketch the graph onto your assignment sheet.

b) Choose 3 ordered pairs and explain the meaning of the numeric values, in the context of the problem.

Time

(seconds)

(t)

Height of ball

(meters)

h(t)

Time (in seconds) (t)

Hei

ght

of

bal

l (i

n m

eter

s) h

(t)

0

Name ______________________________ Period __________

2.3A Significant Features of Functions – Part I

2.3 I can demonstrate understanding of the significant features of a function represented by a graph, 81

a table, or an equation and the relationship these features have to real-world situations.

#1 – 3: Use the graphs to identify significant features of each. Copy the organizational chart onto your

assignment sheet and identify the significant feature for each graph.

Graph 1 Graph 2 Graph 3

Graph 1 Graph 2 Graph 3

Domain

Range

Discrete or continuous?

x-intercept(s)

y-intercept

Function? yes or no

#4 – 5: Determine if each set of ordered pairs is a function or not. State the domain and range.

4) ( ) ( ) ( ) ( ) ( ){ }7, 2 , 3,5 , 8, 4 , 6,5 , 2,3− − − 5) ( ) ( ) ( ) ( ) ( ) ( ){ }1, 2 , 2,3 , 3, 4 , 4,5 , 5, 6 , 6, 7

a) Function: Yes / No a) Function: Yes / No

b) Domain: b) Domain:

c) Range: c) Range:

#6 – 11: Give the domain and range of the following functions.

6)

7)

-1 1 2 3 4 5 6

-3

-2

-1

1

2

3

-1 1 2 3 4 5 6 7

-8

-7

-6

-5

-4

-3

-2

-1

1

2

Name ______________________________ Period __________



#6 – 11 (continued): Give the domain and range of the following functions.

8)

9)

10)

11)

#12 – 13: Describe key features for each of the following situations.

12) Marcus bought a $900 couch on an interest free payment plan. He makes $100 payments to the loan each

week. Consider the amount that remains as a balance on his loan.

Summarize the information. Possible organizational tools include, but are not limited to: a table, a graph,

an equation, etc.

Considering the information you have organized for the scenario described, identify the following

significant features.

answer explain what this means in the context of the situation

a) domain:

b) range:

c) x-intercept:

d) y-intercept:

e) discrete/continuous:

f) function?

-6 -4 -2 2 4 6

-2

2

4

6

8

Name ______________________________ Period __________




#12 – 13 (continued): Describe key features for each of the following situations.

13) An empty 15 gallon tank is being filled with gasoline at a rate of 2 gallons per minute.

Summarize the information. Possible organizational tools include, but are not limited to: a table, a graph,

an equation, etc.

Considering the information you have organized for the scenario described, identify the following

significant features.

answer explain what this means in the context of the situation

a) domain:

b) range:

c) x-intercept:

d) y-intercept:

e) discrete/continuous:

f) function?

#14 – 15: Copy the graph onto your assignment sheet. Draw a function with the following features.

14) x-intercepts at (–2, 0), (4, 0) 15) Domain: –4 ≤ x < 2

y-intercept at (0, 5) Range: –1 ≤ y ≤ 4

x-intercept at (–3, 0)

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Name ______________________________ Period __________

2.3B Significant Features of Functions – Part II


1) Fred Baker has a new pickup. The graph below shows how much gas his pickup is using.

a) Find the y-intercept and explain its

meaning.

b) Given the equation of the graph,

25 375x y+ = , find the x-intercept and

explain its meaning.

c) Is this function increasing or decreasing?

Explain why.

d) Find the domain and explain its meaning.

e) Find the range and explain its meaning.

2) The following graph shows the height of a rock launched by a catapult.

a) Find the y-intercept and explain its

meaning.

b) Find the x-intercept and explain its

meaning.

c) Find the maximum and explain its

meaning.

d) Find the domain and explain its

meaning.

e) Find the range and explain its

meaning.

Hei

gh

t o

f th

e ro

ck (

feet

)

Number of SECONDS since launch

Gasoline Usage

Miles Driven

Gal

lons

in T

ank

40 80 120 160 200 240 280 320 360 4000

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

0x

y

Name ______________________________ Period __________




#3 - 4: Given the graph, identify the interval(s) that the function is increasing, the interval(s) that the

function is decreasing, and the extrema that occur (relative and/or absolute minimums or

maximums).

3) 4)

Increasing interval(s): __________________________ Increasing interval(s): __________________________

Decreasing interval(s): __________________________ Decreasing interval(s): _________________________

Absolute maximum: ___________________________ Relative maximum: ___________________________

Relative minimum: ____________________________

#5 – 8: Copy the graph onto your assignment sheet. Draw a function with the following features.

5) Increasing: x < 1 6) Relative minimum at (2, 1) and (–4, –3)

Decreasing: x > 1 Relative maximum at (–1, 4)

x-intercept at (–2.5, 0)

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

x

y

Name ______________________________ Period __________



#5 – 8 (continued): Copy the graph onto your assignment sheet. Draw a function with the following features.

7) Relative minimum at (–3, –2) 8) Increasing: –4 < x < –2 and x > 3

Relative maximum at (2, 5) Decreasing: x < –4 and –2 < x < 3

9) The following graph shows the hours of daylight each month in Minneapolis, MN.

a) Find the x-intercept(s) and

y-intercept(s) and explain

their meaning.

b) Identify the intervals where

the function is increasing

and the intervals where it is

decreasing and explain the

meaning.

c) Find the relative minimum

and maximum and explain

their meaning.

d) Find the domain and range

and explain their meaning.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Name ______________________________ Period __________

2.3C Significant Features of Functions – Part III



#1 – 4: Create a coordinate grid. Sketch a graph of a function with the following features.

1) Relative minimum at (2, –1) 2) x-intercepts at –3, 1 and 4

Relative maximum at (4, 5) Decreasing: x < 0 and x > 2

y-intercept at 5 Domain: –5 ≤ x ≤ 5

Range: –4 ≤ y ≤ 5

3) Relative minimum at (–3, 2) 4) Decreasing: x < –2 and 0 < x < 2

Positive: x < 1 and x > 4 Negative: –4 < x < 4

Increasing: –3 < x < –1 and x > 3 y-intercept at –2

Decreasing: x < –3 and –1 < x < 3

5) Use the graph to the right to answer the following

questions.

a) Identify the x-intercept(s) and y-intercept(s) and

explain their meaning for this situation.

b) Estimate the relative extrema and explain their

meaning for this situation.

c) Identify the increasing and decreasing interval(s) and

explain their meaning for this situation.

d) Why is there no x-intercept? Use the context of the

situation in your answer.

e) Are there any negative intervals? Why or why not? Use the context of the situation in your answer.

f) Identify the domain and range and explain their meaning for this situation.

Name ______________________________ Period __________

2.3D Rate of Change


#1 – 7: Answer the questions below using the following situation.

Examine the following graph of a function. It represents a journey

made by a large delivery truck on a particular day. The truck left

the distribution center and made two deliveries. Each delivery

took one hour. The driver also took a one-hour break for lunch,

and then returned to the distribution center at the end of the day.

1) What is the rate of change over the interval 0 < t < 2? What does

the rate of change tell you about the delivery truck’s journey

over this interval?


the rate of change tell you about the delivery truck’s journey

over this interval?


the rate of change tell you about the delivery truck’s journey over this interval?

4) What is the rate of change over the interval 4 < t < 6? What does the rate of change tell you about the

delivery truck’s journey over this interval?

5) What is the rate of change over the interval 6 < t < 8? What does the rate of change tell you about the

delivery truck’s journey over this interval?

6) When is the rate of change negative? Why is it negative?

7) Over what interval(s) is/are the rate of change value(s) the least? Explain your reasoning.

#8 – 12: Answer the questions below using the following situation.

The following table contains data for stopping distances at various speeds. This information is based

on full braking, on dry, level asphalt, with an average perception-reaction time of 1.5 seconds.

Speed (x)

(mph) 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Braking Distance (y)

(feet) 5 11 19 30 43 59 76 97 119 144 172 202 234 268 305 345 386

Find the average rate of change of braking distance between the given speeds:

8) 10 MPH and 15 MPH 9) 20 MPH and 30 MPH 10) 40 MPH and 80 MPH

11) Is there a linear relationship between the speed of a

vehicle and the braking distance? Use the context

of the situation in your answer.

12) Is this a proportional relationship? (i.e. If the speed

is doubled, is the braking distance also doubled?)

13) Use the graph to estimate the average rate of change for the

first 2 decreasing intervals. Compare the two rates and explain

the meaning for this situation.

Time (hours)

Dis

tance

(m

iles)

Name ______________________________ Period __________

2.3E Relations as Inverses



#1 – 2: Find the ordered pairs of each inverse relation.

1) x 1 4 1 0 1 2) x 1 –2 4 2 –3

y 3 –1 6 –3 9 ( )f x 0 3 –2 2 1

x x

1y − ( )1f x−

Copy the graph below. Be sure to include the oblique line y = x. On the graph you create, with one color (or

symbol), graph the original relation. With a different color (or symbol), graph the inverse relation.

#3 – 5: Match the graph with its inverse. Consider the table of values that could be created for each.

3) 4) 5)

A B C

Ma

tch

es w

/ gra

ph

____

Ma

tch

es w

/ gra

ph

____

Ma

tch

es w

/ gra

ph

____

Name ______________________________ Period __________



6) a) Copy and complete the table of values for the following function.

( )1

23

f x x= −

x ( )f x Copy and create a table of values for the

inverse of the previously given function. x ( )

1f x−

–6

–3

0

3

6

b) On a coordinate grid, draw the graph of each. Label the first line ( )f x and the second line ( )1f x− .

c) Draw the line for the equation y x= .

d) What do you notice about ( )f x and ( )1f x− in relation to the line y x= ?

e) Working with the function ( )1

23

f x x= − , solve this equation for x (get x by itself).

You should have created the equation ( )3 6x f x= ⋅ + .

initial input = 3 · initial output + 6

knowing that the initial output becomes the ‘new input’ of the inverse relation…

and the ‘new output’ is the same as the initial input…we can rewrite this…

inverse output = 3 · inverse input + 6

( )1 3 6f x x−

= ⋅ +

f) Copy and complete the table of values for the function ( )1 3 6f x x−

= + .

x ( )1f x−

–4

–3

–2

–1

0

g) What do you notice when comparing this table to the second table of values from 6a) from this practice

section?

Name ______________________________ Period __________




7) Considering the following function, complete each of the following. ( )2

33

f x x= −

a) Copy and complete the table of values for the initial function. Use that table to create the table for the

inverse of the function.

x ( )f x . x ( )1f x−

–6

–3

0

3

6

b) Create a coordinate graph and graph each table of values on that graph.

c) Verify that they are inverses of each other by drawing the line y x= and checking for the reflection.

d) Rewrite the function in terms of x (get x by itself)

e) Write this inverse relation in function notation ( )( )1 expressionf x−

= . Verify the second table of

values and the graph for the second table accurately represent this inverse function.

#8 – 9: Find an equation for the inverse relation. On a coordinate grid, graph both linear functions.

Verify the two functions are inverses of one another by drawing the line y = x and verifying the

reflection.

8) ( ) 2 5f x x= − + 9) ( )1

32

f x x= +

#10 – 15: Find an equation for the inverse relation. Graph them on a graphing calculator to verify that the

equations are inverses.

10) ( )4

115

f x x= − + 11) ( ) 11 5f x x= − 12) ( ) 12 7f x x= − +

13) ( )1 13

8 8f x x= + 14) ( )

3 5

7 7f x x= − + 15) ( )

26

3f x x= − +

Name ______________________________ Period __________

2.3F Using Graphing Technology


#1 – 4: Use a graphing utility to make a table and sketch a graph for the following. Copy the answers from

your calculator.

1) 2 3y x= + 2) 3 22 3y x x= + −

x y

–2

–1

0

1

2

x y

–2

–1

0

1

2

3) 3xy = 4) 1

yx

=

x y

–2

–1

0

1

2

x y

–2

–1

–½

0

½

1

2

#5 – 6: Enter the following equations into the table setting of a graphing utility. Make a table to match the

one shown and complete the missing values.

5) ( )21

20 1002

y x= − − 6) 30y x= +

x y x y

60 –32

80 –30

100 –28

120 –26

140 –24

160 –22

#7 – 9: Graph the following functions using a graphing utility. Copy the graph onto your assignment

sheet. Identify the x-intercept(s) and the y-intercept(s) of each graph.

x-intercept(s) y-intercept(s)

7) 3 7y x= − +

8) 214 4

2y x x= − + +

9) 2 5xy = −

Name ______________________________ Period __________

2.3F Using Graphing Technology



#10 – 12: Graph the following functions on your graphing calculator. Identify the relative minimum(s)

and relative maximum(s).

relative minimum(s) relative maximum(s)

10) 22 6 5y x x= − − +

11) 4 5y x= +

12) 3 3 5y x x= − +

13) What significant point(s) will you need to identify to be able to find the interval(s) of increasing and

decreasing function values?

14) What significant point(s) will you need to identify to be able to find the interval(s) of positive and negative

function values?

#15 – 16: Use your table function to find an appropriate viewing window for the following functions and

find the indicated features of the graph.

15) 2 32 240y x x= − +

Give an appropriate viewing window: Draw a viewing window similar to the one below.

Sketch the graph displayed in the window. Identify and

label significant points that appear on the graph.

Xmin = ______________

Xmax = _____________

Xscl = ______________

Ymin = ______________

Ymax = _____________

Yscl = ______________

Find the:

x-intercept(s): ______________ y-intercept: __________

interval(s) where the function is positive: ________________________

interval(s) where the function is negative: ________________________

16) ( ) ( ) ( )2 1 5 8y x x x= − + + +

Give an appropriate viewing window: Draw a viewing window similar to the one below.

Sketch the graph displayed in the window. Identify and

label significant points that appear on the graph.

Xmin = ______________

Xmax = _____________

Xscl = ______________

Ymin = ______________

Ymax = _____________

Yscl = ______________

Find the:

relative maximum(s): _______________ relative minimum(s): ____________________

increasing interval(s): _______________ decreasing interval(s): __________________

Name ______________________________ Period __________



1) TABLE A TABLE B

x y x y

input output input output

domain range domain range

1 2 1 2

1 3 2 3

2 3 3 3

3 5 4 5

4 5 5 5

4 6 6 6

5 7 7 7

Which is a function? Why?

2) Set A: ( ) ( ) ( ) ( ) ( ) ( ){ }0,1 , 0, 1 , 1, 2 , 2,1 , 3,1 , 4,1−

Set B: ( ) ( ) ( ) ( ) ( ) ( ){ }0,1 , 1, 1 , 2, 2 , 2,1 , 3, 2 , 4,1− −

Which is a function? Why?

State the domain and range of the set that is a function.

3) Which graph (or graphs) represents a function?

[A] [B] [C] [D]

4) TABLE A TABLE B

Cell phone use by

country Million Users Million Users

Cell phone use by

country

United States 219 219 United States

China 438 438 China

United Kingdom 61 61 United Kingdom

Mexico 47 47 Mexico

Germany 72 72 Germany

Italy 72 72 Italy

France 48 48 France

Pakistan 48 48 Pakistan

Japan 95 95 Japan

Russia 120 120 Russia

Which is a function (Table A or Table B)? Why?

x

y

x

y

x

y

Name ______________________________ Period __________


Review Material 95

5) Identify whether each relation below is or is not a function and state your reason for classifying it that way.

a) Names and social security numbers

b) Addresses and names

c) ( ) ( ) ( ){ }2, 4 , 2, 5 , 3, 7−

d) ( ) ( ) ( ){ }4,1 , 4, 3 , 5, 6

e) ( ) ( ) ( ){ }2, 5 , 3, 5 , 4, 5

6) Vehicle Car Bike Motorcycle Boat Tricycle

Number of Wheels 4 2 2 0 3

Given the table above, which relation is a function? Why?

a) ( )Vehicle, Number of Wheels

b) ( )Number of Wheels, Vehicle

7) Every morning Tom walks along a straight road from his home to a bus stop, a distance of 160 meters. The

graph shows his journey on one particular day.

Describe what is happening in this graph in context.

8) Describe what Tom is doing in each of these graphs.

a)

b)

c)

Dis

tan

ce f

rom

ho

me

Time

Dis

tan

ce f

rom

ho

me

Time Time

Dis

tan

ce f

rom

ho

me

20 40 60 80 100 120

20

40

60

80

100

120

140

160

180

200

x

y

Dis

tan

ce f

rom

hom

e in

met

ers

Time in seconds

Name ______________________________ Period __________



9) You are able to make $5 an hour babysitting and already have $22. The function ( ) 5 22m h h= + describes

the money (m) you have after babysitting h hours.

a) Find m(8), explain what it means.

b) Find m(h) = 37, explain what it means.

10) Describe what is happening in this graph. Answer each

question below and explain it in context of the graph.

a) What are the x- and y-intercepts?

b) What is the domain? What is the range?

c) Over what interval(s) is the graph decreasing?

d) Over what interval(s) is the graph increasing?

e) What is the relative minimum?

f) Identify ( )2f .

g) Determine the value of x when ( ) 116f x = .

h) At what rate is the person losing weight?

i) At what rate is the person gaining weight?

11) What are the significant features of the graph of the

function?

a) Domain:

b) Range:

c) Relative and/or absolute max:

d) Relative and/or absolute min:

e) Interval(s) with increasing function values:

f) Interval(s) with decreasing function values:

g) y-intercept:

h) x-intercept(s):

i) Interval(s) with positive function values:

j) Interval(s) with negative function values:

1 2 3 4 5 6 7 8 9 10 11 120108

110

112

114

116

118

120

122

124

126

128

130

y

Months (Month 1 = January)

wei

ght

-4 -2 2 4

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

2

4

x

y

Name ______________________________ Period __________


Review Material 97

12) In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers

increased by 75% per year after 1985.

a) Identify and explain the y-intercept.

b) Identify and explain the x-intercept.

c) State the domain and the range.

d) Find ( )9f and explain what it

means in context of the problem.

e) Find ( ) 8000f x = and explain what

it means in the context of the

problem.

f) When did the number of cell phone

subscribers increase at a rate of

about 2000 per year?

Years since 1985

# o

f ce

ll p

hone

subsc

riber

s

Name ______________________________ Period __________



13) Gas prices flucuate with supply and demand. Below is a graph that shows the average price of a gallon of

gasoline in the United States during the years 1995 through 2014. The x-axis represents the year and the

y-axis represents the cost of gallon of gasoline in US dollars. The US Department of Transportation (DOT)

monitors the prices of gasoline. They have modelled the price using the function g(t).

a) What is the range of g(t) in this time interval? Explain the meaning in the context of the problem.

b) Estimate the year was the gas price the greatest?

c) Estimate g(2010) from the graph. Explain the meaning in the context of the problem.

d) Estimate when g(t) = 1. Explain the meaning in the context of the problem.

e) Find the average rate of change in the gas price over the time interval from t = 1995 until t = 2005.

Give the proper units for this rate. Explain the meaning in the context of the problem.

99

EXPONENTIAL FUNCTIONS

Exponential functions occur frequently in real world situations. They are used to

model the growth of human and animal populations, chemical processes such as

radioactive decay, and financial applications such as the compound growth of the

value of an investment. Students learn about exponential functions by comparing the

situations and equations for exponential functions to those for linear functions.

Students recognize, use, and create tables, graphs and situations modeling

exponential growth and decay. Students are able to evaluate exponential functions

with rational number inputs and to relate the meaning of a rational exponent to the

context of the situation. Students use tables and graphs to solve exponential equations

and translate between representations.

Practice Problems

100 Unit 3 Reference and Resource Material

3.1 I can demonstrate understanding about

exponential functions and compare

situations and equations for exponential

functions to those for linear functions.

111 3.1 Compare and Contrast Exponential and

Linear Functions

3.2 I can use tables and graphs to solve

exponential equations including

real-world situations and translate between

representations.

114 3.2A Explore – Exponential Growth/Decay:

Tables, Graphs & Real-World Situations

118 3.2B Solve – Exponential Growth/Decay: Tables,

Graphs & Real-World Situations

3.3 I can evaluate exponential functions in the

form y=abx and relate the meaning to the

context of a real-world situation.

120 3.3 Evaluate Exponential Functions


significant features of a graph of an

exponential function and their relationship

to real-world situations.

122 3.4 Significant Features of an Exponential

Function Graph



3.1 I can demonstrate understanding about exponential functions and compare situations and equations

for exponential functions to those for linear functions.

3.2 I can use tables and graphs to solve exponential equations including real-world situations and

translate between representations.

3.3 I can evaluate exponential functions in the form y = a·bx and relate the meaning to the context of a

real-world situation.

3.4 I can demonstrate understanding of the significant features of a graph of an exponential function and

their relationship to real-world situations.

UNIT

3


100 UNIT 3—EXPONENTIAL FUNCTIONS

3.1 I can demonstrate understanding about exponential functions and compare

situations and equations for exponential functions to those for linear functions.

Exponential functions can be used to model situations of rapid growth or decay, such

as population growth and radioactive decay. A linear function has a constant rate of

change (common difference) while an exponential function has a common ratio. The

rate of change in an exponential function is growing or shrinking rapidly as the x-value

gets larger.

Linear Functions versus Exponential Functions

Linear Function Example

For each increase of one in the input value, the output value

increases at a constant rate.

Exponential Function Example – Exponential Growth

For each increase of one in the input value, the output value

changes by a common multiplier greater than one.

Common ratio = 3

This function is written

( ) 4 3xf x = ⋅ .

*Note that an exponential function

has x in the exponent.

Exponential Function Example – Exponential Decay

For each increase of one in the input value, the output vale

changes by a common multiplier between zero and one.

Common ratio = 1

3

This function is written

( )1

273

x

f x

= ⋅

.

*Note that an exponential

function has x in the exponent.

Unit 3 Resources

x y

0 4

1 12

2 36

3 108

4 324

• 3

• 3

• 3

• 3

x y

0 27

1 9

2 3

3 1

4 1/3

• 1/3 (or ÷ 3)

• 1/3 (or ÷ 3)

• 1/3 (or ÷ 3)

• 1/3 (or ÷ 3)

Common difference: 2

1

rise

run=

This function is written as ( ) 2 1f x x= +

x y

–2 –3

–1 –1

0 1

1 3

2 5 + 2

+ 2

+ 2

+ 2



g (x)

f (x)

Linear vs. Exponential Functions Example

Compare the graphs of the linear and exponential

functions shown here.

f (x) g (x)

Function Type Linear Exponential

Evaluate for x = 0, 1, 2, 3 (0,1), (1, 3) (2, 5) (3, 7) (0,1), (1, 2) (2, 4) (3, 8)

Rate of change

from x = 0 to x = 3

Constant rate of 2

Changing rate:

x = 0 to x = 1 Rate = 1

x = 1 to x = 2 Rate = 2

x = 2 to x = 3 Rate = 4

Which function is greater? (use graph to approximate interval)

( )f x is greater over the interval

0 2.6x< <

( )g x is greater over the intervals

0 and 2.6x x< >

How to recognize the type of graph from a table

To recognize if a function is linear, exponential or neither of these without an equation or graph, look at the

differences of the y-values between successive integral x-values. If the difference is constant, the graph is linear.

If the differences follow a pattern similar to the y-values, the graph is exponential or if the ratio of y-values is

determined, the ratio between terms is the same if the relation is exponential. See the examples below for clarity.

3.1 Example 1: Based on the table, identify the shape of the graph.

x –3 –2 –1 0 1 2 3

y –3.5 –2 –0.5 1 2.5 4 5.5

Solution:

x –3 –2 –1 0 1 2 3

y –3.5 –2 –0.5 1 2.5 4 5.5

Linear Exponential

x y x y

0 1 0 1

1 3 1 2

2 5 2 4

3 7 3 8

–2 – (–3.5) = 1.5

–0.5 – (–2) = 1.5

1 – (–0.5) = 1.5 , etc.

⇐ investigate differences 1.5 1.5 1.5 1.5 1.5 1.5

The difference in y-values is always

1.5, a constant. The function graphed

is linear and it is verified in the graph.




x –3 –2 –1 0 1 2 3

y 9 4 1 0 1 4 9

Solution:

x –3 –2 –1 0 1 2 3

y 9 4 1 0 1 4 9


x –3 –2 –1 0 1 2 3

y 1

8

1

4

1

2 1 2 4 8

Solution:

x –3 –2 –1 0 1 2 3

y 1

8

1

4

1

2 1 2 4 8

4 ÷ 9 = =

1 ÷ 4 =0.25 = , etc.

⇐ investigate differences –5 –3 –1 1 3 5

⇐ investigate ratios

The difference in y-values is not constant. There is a pattern, but the differences themselves are not constant.

We’ll check ratios to determine if the function values demonstrate an exponential pattern. The ratios must be

EQUAL (common) over equal sized intervals to be able to say the function is exponential.

The differences are not common so this relation is not

linear. The ratios between successive values are not

common. This relation is neither linear nor exponential.

, etc.

⇐ investigate differences

1 2 4

⇐ investigate ratios 2 2

The difference in y-values is not constant. There is a pattern,

but the differences themselves are not constant. This created

an interesting pattern – it ‘matches’ the function values! We’ll

investigate the ratios to determine if it is exponential.

There is a common ratio (2).

Therefore, we can say this table of

values DOES represent an

exponential function.

This is verified by plotting the

values.

x

y2 …



Vocabulary

• Linear function: a polynomial function of degree zero or one

A linear function has a constant rate of change. A line is created when this function is

graphed.

• Exponential function: a function whose value is a constant raised to the power, where the power

includes a variable

� An exponential function represents rapid growth or decay and has a common growth or

decay ratio. Exponential functions have an input variable in the exponent.

� An exponential growth function has a common ratio greater than one.

� An exponential decay function has a common ratio between zero and one.

Video Resources

• http://youtube.com/M7dACw8gyHU

3.2 I can use tables and graphs to solve exponential equations including real-world situations and

translate between representations.

3.2 Example 1: The typical car loses 15% to 20% of its value each year. The graph below shows

the value of a car that is depreciating 20% each year.

Use the graph to answer the following

questions.

� What was the value of the car when

it was new?

� When did the car lose the most

value?

� How does the depreciation compare

for each interval of five years?

Solution:

� What was the value of the car when it

was new? $30,000

� When did the car lose the most value?

The first year it lost about $6,000 in value.

� How does the depreciation compare for each interval of five years?

o From 0-5 years it lost $20,000 in value

o From 5-10 years it lost about $6,000 in value.

o From 10-15 years it lost about $2,000 in value.



-8 -6 -4 -2 2 4 6

-2

2

4

6

8

10

x

y

-2 -1 1 2

-1

1

2

3

4

x

y

3.2 Example 2:

Use a graphing utility to solve the equation: 3 12 7x−=

Solution:

Graph the function 3 12 xy −= and find the point where the

graph intersects the horizontal line y = 7.

The solution is x ≈ 1.27.

3.2 Example 3:

Use a graphing utility to solve the equation: 4 8 56 2x x− −=

Solution:

Graph the functions 46 xy −= and 8 52 xy −

= and find their

intersection point.

The solution is x ≈ 0.27.

3.3 I can evaluate exponential functions in the form y = abx and relate the meaning to the context of a

real-world situation.

The equation of an exponential function is different from other function families because the exponent is the

variable and the base is a constant. The formula for an exponential function takes the form

f(x) = abx

where a is the initial amount and b is the growth or decay factor. Note: a and b are constants, and

a ≠ 0, b > 0 and b ≠ 1. This kind of function can be used to model real situations, such as population growth,

compound interest, or the decay of radioactive materials.

Evaluating Exponential Functions:

Consider the function ( ) 2 xf x = .

In this example a = 1 and b = 2. ( xy a b= ⋅ 1 2 xy = ⋅ )

When we input a value for x, we find the function value by raising 2 to the exponent of x.

For example,

if x = 3, we have ( ) ( )33 2 , so 3 8f f= = .

If we choose larger values of x, we will get larger function values, as the function values will be larger

powers of 2.

For example, ( ) ( )1010 2 , so 10 1024f f= = .



Exponential Growth Model xy a b= ⋅= ⋅= ⋅= ⋅

Exponential functions where b > 1 are called exponential growth functions. They are called this because larger

values of x give larger function values.

Investigating an exponential growth model: Create a table of values for the equation 5 3 xy = ⋅

x 5 3 xy = ⋅ (x, y)

–2 25 3 5 3 0.556xy y y−= ⋅ = ⋅ = (–2, 0.556)

–1 15 3 5 3 1.667xy y y−= ⋅ = ⋅ = (–1, 1.667)

0 05 3 5 3 5xy y y= ⋅ = ⋅ = (0, 5)

1 15 3 5 3 15xy y y= ⋅ = ⋅ = (1, 15)

2 25 3 5 3 45xy y y= ⋅ = ⋅ = (2, 45)

3 35 3 5 3 135xy y y= ⋅ = ⋅ = (3, 135)

3.3 Example 1: Suppose that 1000 people visited an online auction website during its first month in existence and that

the total number of visitors (a cumulative total since the site opened) to the auction site is tripling

every month.

� Create a table of values to record the total number of online visits during the first five months.

The situation can be modeled with the function rule ( ) ( )1

1000 3x

f x−

= .

� Looking at the function rule, explain the meaning of the 1000 in the context of the problem.

� Explain the meaning of the 3 in the context of the problem.

� How many total visitors will the auction site have after two years?

Solution:

� Because we know the number of visitors for the first month and that the number of visitors is triple

from the previous month, the table can be created from those two known facts.

# of months since the

website ‘opened’ (x) 1 2 3 4 5

# of visitors (y) 1000 3000 9000 27,000 81,000

� Because there are 1000 people that visit during the first month, this is an important fact that will need

to be considered in the function rule for the situation.

� Because the number of visitors to the site is tripling each month, the 3 is the common multiplier

(common ratio) in this problem.

� Because the function rule uses x to represent the number of months of since they opened, 24 will be

substituted for x in the function rule. ( ) ( )1

1000 3x

f x−

=

( ) ( )

( ) ( )

24 1

14

24 1000 3

24 281, 000, 000, 000, 000 or 2.81 10

f

f

−

=

= ×

There will have been a total of 281 trillion visitors by the time they are operational for two years.



Exponential Decay Model xy a b= ⋅= ⋅= ⋅= ⋅

Exponential functions where 0 < b < 1 are called exponential decay functions. They are called this because

larger values of x give smaller function values.

Investigating an exponential growth model: Create a table of values for the equation 1

53

x

y

= ⋅

x 1

53

x

y

= ⋅

(x, y)

–2 2

1 15 5 45

3 3

x

y y y

−

= ⋅ = ⋅ =

(–2, 45)

–1 1

1 15 5 15

3 3

x

y y y

−

= ⋅ = ⋅ =

(–1, 15)

0 0

1 15 5 5

3 3

x

y y y

= ⋅ = ⋅ =

(0, 5)

1 1

1 15 5 1.667

3 3

x

y y y

= ⋅ = ⋅ =

(1, 1.667)

2 2

1 15 5 0.556

3 3

x

y y y

= ⋅ = ⋅ =

(2, 0.556)

3 3

1 15 5 0.185

3 3

x

y y y

= ⋅ = ⋅ =

(3, 0.185)

3.3 Example 2:

For her fifth birthday, Nadia’s grandmother gave her a full bag of candy. Nadia counted her candy and

found out that there were 160 pieces in the bag. Nadia loves candy, so she ate half the candy on the

first day. Her mother told her that if she continues to eat at that rate, it will be gone within a week and

she will not have any more until her next birthday.

� Create a table of values to record the number pieces of candy she has at the beginning of each

day (her birthday is considered day zero) for seven days.

The situation can be modeled with the function rule ( )1

1602

x

f x

=

.

� Looking at the function rule, explain the meaning of the 160 in the context of the problem.

� Explain the meaning of the1

2in the context of the problem.

� Will the candy be gone in a week (seven days)?

Solution: �

� The value, 160, represents the starting amount of candy that Nadia had.

� The value, ½, represents that ½ of the candy remained at the beginning of the following day.

� By using the table, we see that there would be 1.25 pieces of candy left (essentially 1 piece).

Days since her birthday 0 1 2 3 4 5 6 7

# of pieces of candy (y) 160 80 40 20 10 5 2.5 1.25



-4 -3 -2 -1 1 2 3 4

-5

5

10

15

20

25

x

y

Vocabulary

• An exponential function is a function for which the input variable x is in the exponent of some base b.

It is written in the form f(x) = abx

where a is the initial amount when x = 0 and b is the amount by which

the total is multiplied every time. Note: a and b are constants where a ≠ 0, b > 0 and b ≠ 1.

• Exponential functions where b > 1 are called exponential growth functions. They are called this

because larger values of x give larger function values.

• Exponential functions where 0 < b < 1 are called exponential decay functions. They are called this

because larger values of x give smaller function values.

• When a quantity increases or decreases by a percent each time period, the model becomes:

o Growth: f(x) = a(1 + r)t

o Decay: f(x) = a(1 – r)t

(a is the initial amount, r is the growth or decay rate in decimal form, and t is the time)

Video Resources:

• Writing Exponential Growth Functions - Overview

• CK-12 Basic Algebra: Exponential Growth Functions

• Writing Exponential Decay Functions - Overview

• CK-12 Basic Algebra: Exponential Decay Functions

3.4 I can demonstrate understanding of the significant features of a graph of an exponential function and

their relationship to real-world situations.

3.4 Example 1: Changing the value of “a” to a LARGER number

Compare the graphs of 2 xy = and 3 2 xy = ⋅ . First make tables for each function.

Solution:

x 2 xy = (x, y) x 3 2 xy = ⋅ (x, y)

–3 3

3

1 12

82

−= =

13,

8

−

–3 3

3

1 1 33 2 3 3

8 82

−⋅ = ⋅ = ⋅ =

33,

8

−

–2 2

2

1 12

42

−= =

12,

4

−

–2 2

2

1 1 33 2 3 3

4 42

−⋅ = ⋅ = ⋅ =

32,

4

−

–1 11

1 12

22

−= =

11,

2

−

–1 11

1 1 33 2 3 3

2 22

−⋅ = ⋅ = ⋅ =

31,

2

−

0 02 1= ( )0,1 0 03 2 3 1 3⋅ = ⋅ = ( )0, 3

1 12 2= ( )1, 2 1 13 2 3 2 6⋅ = ⋅ = ( )1, 6

2 22 4= ( )2, 4 2 23 2 3 4 12⋅ = ⋅ = ( )2,12

3 32 8= ( )3,8 3 33 2 3 8 24⋅ = ⋅ = ( )3, 24

� We can see that the function 3 2xy = ⋅ is bigger than the function 2 xy = .

� In both functions, the value of y doubles every time x increases by one.

However, 3 2xy = ⋅ has a y-intercept of 3, while 2 xy = has a y-intercept of

1. Note that the y-intercept is the same value as a. It would make sense that

3 2xy = ⋅ would be bigger as its value of y keeps getting doubled.

� The domain of both graphs is all real numbers and the range is all positive

numbers. The x-axis is an asymptote. Both functions approach but never

equal 0.

y=3(2)x

y=(2)x



3.4 Example 2: Changing the value of “a” to a SMALLER (still positive) number

Compare the graphs of 2 xy = and 1

23

xy = ⋅ . First make tables for each function.

Solution:

x 2 xy = (x, y) x 1

23

xy = ⋅ (x, y)

–3 3

3

1 12

82

−= =

13,

8

−

–3 3

3

1 1 1 1 1 12

3 3 3 8 242

−⋅ = ⋅ = ⋅ =

13,

24

−

–2 22

1 12

42

−= =

12,

4

−

–2 22

1 1 1 1 1 12

3 3 3 4 122

−⋅ = ⋅ = ⋅ =

12,

12

−

–1 11

1 12

22

−= =

11,

2

−

–1 11

1 1 1 1 1 12

3 3 3 2 62

−⋅ = ⋅ = ⋅ =

11,

6

−

0 02 1= ( )0,1 0 0

0

1 1 1 1 1 12

3 3 3 1 32⋅ = ⋅ = ⋅ =

10,

3

1 12 2= ( )1, 2 1 11 1 22 2

3 3 3⋅ = ⋅ =

21,

3

2 22 4= ( )2, 4 2 21 1 42 4

3 3 3⋅ = ⋅ =

42,

3

3 32 8= ( )3,8 3 31 1 82 8

3 3 3⋅ = ⋅ =

83,

3

� Here we can see that the function 1

23

xy = ⋅ is smaller than the

function 2 xy = and has a y-intercept of 1

3 which is the value of a.

� Again, the domain of both graphs is all real numbers and the range

is all positive numbers. The x-axis is an asymptote. Both functions

approach but never equal 0.

( )1

23

xy =

2xy =



3.4 Example 3: Changing the value of “a” to a SMALLER (negative) number

Graph the exponential function 5 2xy = − ⋅ .

Solution:

� Since “a" is negative and keeps doubling

over time, it makes sense that the value of y

gets farther from zero, in a negative

direction.

� The y-intercept is the value of “a”,

which is –5.

� The domain again is all real numbers but

the range is all negative numbers with the

x-axis as an asymptote.

3.4 Example 4: Changing the value of “b” to a VARIETY OF POSITIVE numbers

Graph the exponential functions 2 3 4x x xy y y= = = .

Solution:

x

2 xy = 3xy = 4 xy =

–3 33

1 12

82

−= = 3

3

1 13

273

−= = 3

3

1 14

644

−= =

–2 2

2

1 12

42

−= = 2

2

1 13

93

−= = 2

2

1 14

164

−= =

–1 11

1 12

22

−= = 1

1

1 13

33

−= = 1

1

1 14

44

−= =

0 02 1= 03 1= 04 1=

1 12 2= 13 3= 14 4=

2 22 4= 23 9= 24 16=

3 32 8= 33 27= 34 64=

x 5 2xy = − ⋅ (x, y)

–3 3

3

1 1 55 2 5 5

8 82

−− ⋅ = − ⋅ = − ⋅ = −

53,

8

− −

–2 22

1 1 55 2 5 5

4 42

−− ⋅ = − ⋅ = − ⋅ = −

52,

4

− −

–1 11

1 1 55 2 5 5

2 22

−− ⋅ = − ⋅ = − ⋅ = −

51,

2

− −

0 05 2 5 1 5− ⋅ = − ⋅ = − ( )0, 5−

1 15 2 5 2 10− ⋅ = − ⋅ = − ( )1, 10−

2 25 2 5 4 20− ⋅ = − ⋅ = − ( )2, 20−

3 35 2 5 8 40− ⋅ = − ⋅ = − ( )3, 40−

2 xy =

� The graphs of the three functions have the same overall

shape, they are all increasing and they all have the same

y-intercept (0, 1).

� They all have the domain of all real numbers and a range of

all positive numbers.

� The x-axis is an asymptote for each function.

-3 -2 -1 1 2 3

-40

-35

-30

-25

-20

-15

-10

-5

5

10

x

y



3.4 Example 5: Changing the value of “b” to a SMALL number (between –1 and 0 or between 0 and 1)

Graph the exponential functions 1

52

x

y

=

.

Solution:

This graph looks very different than the graphs from the previous example.

� When the base b of an exponential function is between 0 and 1, the graph is decreasing instead of

increasing like we’ve seen in the previous examples.

� Graphs like this represent exponential decay instead of exponential growth.

� Exponential decay functions are used to describe quantities that decrease over a period of time.

When we raise a number greater than 1 to the power of x, it gets bigger as x gets bigger. But when we raise a

number smaller than 1 to the power of x, it gets smaller as x gets bigger—as you can see from the table of

values above. This makes sense because multiplying any number by a quantity less than 1 always makes it

smaller.

Video Resources:

• CK-12 Foundation: Exponential Functions

• http://vimeo.com/47092096

x 1

52

x

y

= −

(x, y)

–3 ( )

331

5 5 2 5 8 402

−

= = ⋅ =

(–3, 40)

–2 ( )

221

5 5 2 5 4 202

−

= = ⋅ =

(–2, 20)

–1 ( )

111

5 5 2 5 2 102

−

= = ⋅ =

(–1, 10)

0

01

5 5 1 52

= ⋅ =

(0, 5)

1

11 1 5

5 52 2 2

= ⋅ =

51,

2

2

21 1 5

5 52 4 4

= ⋅ =

52,

4

3

31 1 5

5 52 8 8

= ⋅ =

53,

8

Name ______________________________ Period __________

3.1 Compare and Contrast Exponential and Linear Functions

3.1 I can demonstrate understanding about exponential functions and compare situations and 111

equations for exponential functions to those for linear functions.

#1 – 3: Identify the type of change as linear or exponential. Then, describe the pattern of change and how

you found it.

1) Using the figure structure below, what is the pattern of change in the perimeter of the figures from one step

to the next?

Step 1 Step 2 Step 3 Step 4

Description of the pattern and how I found it:

Type of pattern of change:

2) Using the table to the right, what are the patterns that exist in the number set?



3) What is the distance between a person and their home if they are traveling west at a constant rate of 60 mph?



#4 – 17: For each problem below, state if it models linear or exponential change. Explain your decision.

4)

5) Badminton Tournament

Rounds

Number of

players left

1 64

2 32

3 16

4 8

5 4

6 2

x f(x)

–30 –37

–25 –27

–20 –17

–15 –7

–10 3

–5 13

0 23

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Name ______________________________ Period __________


112 UNIT 3—EXPONENTIAL FUNCTIONS (PRACTICE)

#4 – 17 (continued): For each problem below, state if it models linear or exponential change. Explain

your decision.

6) The number of rabbits doubles every month.

7) You withdraw $100 from your savings account

every month and do not make any deposits.

8)

9) 1, 5, 25, 125,…

10) ( ) 3 5f x x= + 11) Each term in a sequence is exactly 5 more than the

previous term.

12) Each term in the sequence is exactly 1/3 of the

previous term. 13)

x y

–2 13

0 8

2 3

4 –2

6 –7

14) The car loses 7% of its value every year. 15) –3.6, –5.4, –8.1, –12.15,…

16) 1, 4, 7, 10, 13, 16, 19, 22, …

17) 5 xy =

18) Describe in your own words how you know if a function is modeling linear or exponential change.

Name ______________________________ Period __________


3.1 I can demonstrate understanding about exponential functions and compare situations and 113

equations for exponential functions to those for linear functions.

19) Use the graph to the right to answer the

following questions.

a) Examine the graph at the right from

x = 0 to x = 1.

What is the average rate of change

for ( )r x over this interval?

What is the average rate of change

for ( )s x over this interval?

b) Which graph is changing faster on

this interval?

c) Now look at the functions ( )r x and

( )s x over the interval from x = 2 to x

= 3.

Which graph is growing faster on this interval? Explain your answer using rate of change.

d) Now look at the graph from x = 3 to x = 4.

Which graph is growing faster in this interval? Explain your answer using rate of change.

20) Malcolm says that exponential functions grow faster than linear functions. Marietta says that linear functions

grow faster than exponential functions. Who is correct? Explain your answer. (Hint: look at your results

from problem #19).

r(x)

s(x)

Name ______________________________ Period __________

3.2A Explore – Exponential Growth/Decay: Tables, Graphs and Real-World

Situations


1) The half-life of DDT is 15 years. This means that if you started with 100 grams of DDT, there would be 50

grams left after 15 years. This pesticide DDT (used until 1972) is toxic to a wide range of animals and aquatic

life and is also suspected to cause cancer in humans. Copy the graph and label the axes appropriately.

a) How many grams were there in the

beginning sample?

b) How many grams are left after 20 years?

c) How long would it take for there to be

less than 10 grams of DDT?

d) If scientists believe an area with less

than 5 grams of DDT is safe, how many

years would it take for this area to be

safe?

2) This is a graph of the population in Coleman, Texas from 2000 to 2100. Copy the graph and label the axes the

axes appropriately.

a) Estimate the population in 2010.

b) What is the y-intercept and what does

it represent in the context of the

problem?

c) If x was –5, what would that mean in

the context of the problem?

d) The population of Coleman, Texas

grows at a 2% rate annually. The

population in 2000 was 5981 people;

use the graph to find the expected

population in 2050.

Name ______________________________ Period __________


Situations

3.2 I can use tables and graphs to solve exponential equations including real-world situations 115


3) Each year there is a regional tennis tournament. They start with 128 athletes. During each round, half the

players are eliminated.

a) Create a table that models this situation. Create a coordinate grid. Label and number the axes

appropriately and graph the data from your table.

b) How many athletes will be left after 5 rounds?

c) Would it make sense to ask how many athletes were left after 6.5 rounds? Explain your reasoning

d) What is the smallest number of athletes remaining? Explain your thinking.

4) The number of bacteria in a sample doubles every hour. Copy and complete the table below to record values

to aid in organizing information to help answer the questions below. Create a coordinate grid. Label and

number the axes appropriately and graph the data from your table.

Hours passed 0 1 2 3 4

Number of

bacteria

a) If there were 64 bacteria in a sample after 3 hours, how many bacteria were in the original sample?

b) How many bacteria will be in the sample after 6 hours?

c) Number and label the axes of the graph to the right. Graph the values from table you created above.

d) Continue the pattern you have created in either the table or graph (or both) to determine when there will

be over 500 bacteria?

Name ______________________________ Period __________


Situations


5) Donovan took a $32,000 loan out to buy a new car. The car will lose 15% of its value every year. His

monthly payment on the loan is $300 (consider what he would pay for 1 full year in this setting). Use the

graph to approximate the values for the table and answer the questions below.

Years since taking out

the loan 0 1 2 3 4 5 6

Amount owed toward

the loan

Value of the car at the

end of given year

a) Which graph represents

the amount owed toward

the loan? (circle one)

Linear (the line)

- - or - -

Exponential (the curve)

b) Which graph represents

the value of the car?

(circle one)

Linear (the line)

- - or - -

Exponential (the curve)

c) Would you recommend

that he sell the car after

2 years? Why?

d) When is the earliest he

should sell the car?

Why? How can you

determine that from the

table of values? …from

the graph?

2 4 6 8 10 12 14 16 18 20

4000

8000

12000

16000

20000

24000

28000

32000

0

Years since taking out the loan

Doll

ar a

mount

Name ______________________________ Period __________


Situations



6) Use the function model �� = �2�� to answer the following:

a) Copy and complete the table of values and graph the function on a coordinate grid.

x 0 1 2 3 4 5 6

( )f x

b) Copy and complete the tables below.

x 0 1 2 3 4 5 6

( )f x

Difference between

function values

x 0 1 2 3 4 5 6

( )f x

Ratio between

function values

Using the information above, identify the following characteristics of the function.

� Function values (increasing or decreasing):

� Common difference, common ratio or neither:

� Linear, exponential growth, exponential decay or other:

Name ______________________________ Period __________

3.2B Solve – Exponential Growth/Decay: Tables, Graphs and Real-World Situations


1) How can you use graphing technology to solve an equation?

2) Use the graph to find the solution to the following equation.

a) 1

162

x

=

b) 5 25x=

c) 6

12

2

xx

−

=

d) ( )643 5 0.5

xx −−− + =

3) You are investing $10,000 at 6% interest, compounded annually. Use the function rule

( ) ( )10000 1.06x

f x = to determine how long it will take for there to be $25,000 in the account. Round your

answer to the nearest year.

-5 5

5

10

15

20

x

y

-5 5

4

8

12

16

20

24

28

x

y

-4 -2 2 4 6 8

4

8

12

16

20

24

28

x

y

-2 2 4 6 8

-2

2

4

6

8

x

y

Name ______________________________ Period __________

3.2B Solve – Exponential Growth/Decay: Tables, Graphs and Real-World Situations



4) An initial population of 750 endangered turtles triples each year. Use the function rule ( ) 750(3)xf x = to

determine how many years it will take for the population to reach 60,750.

5) Suppose that you are given a choice of investing $10,000 at a rate of 7% ( 10000 1.07 xy = ⋅ ) or $5,000 at a

rate of 12% ( 5000 1.12 xy = ⋅ ). x is the number of years since the money was invested. Either choice will be

compounded annually.

a) When will the investments be worth the same amount?

b) If the money will be yours after 50 years, which investment plan should you choose?

Record your thinking.

6) Use a graphing utility to solve the following problems.

a) 2 13 27x x+= b) 2 1 3 27 7x x+ −

=

c) 5 7x= d) 2 13 14x +

=

Name ______________________________ Period __________

3.3 Evaluate Exponential Functions


1) The function rule ( ) ( )15000 0.65t

f t = models the value of a boat after t years.

a) How much is the boat worth when you bought it? In other words, find ( )0f . Show the calculations.

b) How much is the boat worth after 5 years? In other words, find ( )5f . Show the calculations.

c) Is this an example of exponential growth or decay? How do you know?

d) Find ( )7f and explain what it means in the context of the problem.

e) Find ( )3f − . What does this mean in the context of the problem? Does this make sense?

2) The artwork of an aging artist is appreciating at a rate of 7% a year. A new painting costs you $2000.

The function rule ( ) ( )2000 1.07t

V t = models the value of the painting after t years.

a) Using the table function of a graphing calculator, how much is the painting worth after 0 (zero) years?

b) Using the table function of a graphing calculator, how much is the painting worth after 5 years?

c) Find ( )10V and explain the meaning of this value in the context of the problem.

d) Is this an example of exponential growth or decay? How do you know?

3) The percentage of light visible at d meters is given by the function rule ( ) 0.70dV d = .

a) Find the percent of light visible at 0 meters.

b) Find the percent of light visible at 65 meters.

c) Find ( )10V and explain the meaning in the context of the problem.

4) The function ( ) ( )20, 000 1 0.06t

A t = + models the amount of money in Mama Bigbuck’s account after t

years.

a) How much money is in the account to begin with?

b) Is this a model of exponential growth or decay? How do you know?

c) How much money will be in the account after 20 years? Record your thinking/calculations.

d) How much interest will she have earned after 25 years? Record your thinking/calculations.

e) She needs $1,500,000 to retire comfortably. How long will she need to save at this rate to retire? Record

your thinking/calculations.

f) Explain the real-world restrictions on the domain and range.

Name ______________________________ Period __________

3.3 Evaluate Exponential Functions

3.3 I can evaluate exponential functions in the form xy a b= ⋅ and relate the meaning to 121

the context of a real-world situation.

5) Your dream car costs $42,000. You find that depreciation is 16% per year.

a) Copy and complete the table using the formula ( )42000 1 0.16x

y = − to find the value of the car in

future years.

Years (x) 0 2 4 6 8 10

Value (y)

b) If you could only afford a $20,000 car, how old would the car be when you can afford it? Explain how

you determined the car’s age.

6) Joanne has a bacterial infection. When she goes to the doctor, the population of bacteria is 2 million. The

doctor prescribes an antibiotic that reduces the bacteria population to 1

4 of its size each day. The function

rule ( )1

2, 000, 0004

d

P d

=

models the number of bacteria in the population after d days.

a) Find the size of the population at day 0 (zero). Use function notation to express your answer.

b) Find the size of the population after 10 days. Use function notation to express your answer.

c) Is this a growth or decay model? How do you know?

d) Will the infection be gone after 20 days? Explain your thinking.

e) When is the first time there are fewer than 500 bacteria? How did you find your answer?

7) ( ) 5xf x = ( ) 3 4tg t = ⋅ ( )1

62

k

p k

= ⋅

Use the function rules above to find the following values.

a) ( )3g b) ( )2f − c) ( )5p d) ( )4f e) ( )2p

8) Using the function rule ( ) 4 2xH x = ⋅ , Jenna and Joua evaluated ( )2H − and got the following answers.

Who is correct (if either)? Explain what they did wrong and state the correct answer if necessary.

Jenna Joua

( )

( )

( 2)2 4 2

4 4

16

H −− = ⋅

= ⋅ −

= −

( )( 2)

( 2)

2 4 2

8

1

64

H −

−

− = ⋅

=

=

9) A super-ball has a 75% rebound ratio. The maximum height (in feet) the ball reaches after each bounce is

modeled by the function ( ) ( )20 0.75n

H n = .

a) What was the height of the ball at the time it was dropped? Show the calculations.

b) How high will the ball bounce after it strikes the ground for the 3rd

time? Record your thinking.

Name ______________________________ Period __________

3.4 Significant Features of an Exponential Function Graph


1) Tell whether the function represents exponential growth, exponential decay, or neither. How do you know?

a) ( ) 3xf x = b) ( )1

2

x

f x

=

c) ( ) ( )2000 0.97x

f x =

d) ( ) ( )20 1 0.02x

f x = ⋅ + e) ( ) ( )400 1.06x

f x = f) ( ) ( )365 1 0.15x

f x = −

g) ( ) 3 7f x x= + h) ( )2 9f x x= − i) ( )

2

1f x

x=

2) Tell whether the function represents exponential growth, exponential decay, or neither.

a)

b)

c)

d)

e)

f)

3) Investigate the rate of change over the following intervals.

a)

b) Is the absolute value of the rate of change increasing or decreasing?

c) Describe the pattern of the function values.

d) Is the function increasing or decreasing? Explain.

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Interval x = 0

to

x = 1

x = 1

to

x = 2

x = 2

to

x = 3

x = 3

to

x = 4

Rate of change

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Name ______________________________ Period __________


3.4 I can demonstrate understanding of the significant features of a graph of an exponential 123

function and their relationship to real-world situations.

4) Investigate the rate of change over the following intervals.

a)

b) Is the absolute value of the rate of change increasing or

decreasing?

c) Describe the pattern of the function values.

d) Is the function increasing or decreasing? Explain.

5) For each of the functions, create a table of values, create a coordinate grid and number the axes appropriately.

Graph the functions, draw and label the asymptote(s), and give the indicated information.

a) ( ) 2 3xf x = ⋅ Domain

Range

Asymptote

y-intercept

b)

( )1

32

x

f x

= −

Domain

Range

Asymptote

y-intercept

6) For each of the functions, create a table of values, create a coordinate grid and number the axes

appropriately. Graph the functions, draw and label the asymptote(s), and give the indicated information.

a)

( )1

4 34

x

f x

= − +

Domain

Range

Asymptote

y-intercept

b) ( )

( 2)4 xf x −= − Domain

Range

Asymptote

y-intercept

Interval x = 0

to

x = 1

x = 1

to

x = 2

x = 2

to

x = 3

x = 3

to

x = 4

Rate of change

Name ______________________________ Period __________



7) The graph represents the value, ( )v t , of an insurance policy t years after it was purchased. Copy the graph

onto your assignment sheet and label the axes.

a) What was the original value of the

policy?

b) What aspect of the graph shows the

maximum value that the policy is

approaching? What is that maximum

value?

c) What does ( )3v − represent? Does it

make sense in the context of the

situation?

8) The graph represents the value, v(t), of a painting t years after it was purchased. Copy the graph onto your

assignment sheet and label the axes.

a) What was the value of the painting

when it was purchased?

b) What aspect of the graph shows the

maximum value that the painting is

approaching? What is that maximum

value?

c) What does ( )3v − represent? Does it

make sense in the context of the

situation?

Name ______________________________ Period __________


3.4 I can demonstrate understanding of the significant features of a graph of an exponential 125

function and their relationship to real-world situations.

9) Imagine that 2000 people catch a cold, all at the same time. Each day, 38% of those who are sick get well.

This situation can be modeled by the function �� = 2000�1 − 0.38��, where x is the number of days that

have passed.

a) What is f (0)? What point is this on the graph? What does this point represent in the context of the

problem?

b) How many people will be well on day 7? Find the value and then write your answer using function

notation.

c) How long will it take for the cold to affect only 10 people or less? Explain how you found your answer.

d) What are the domain and range of the function? Explain the meaning of the domain and range

restrictions in the context of the problem.

10) �� = 15,000�1.17�� describes the growth in students attending Happy College starting in 2010.

a) What is the annual percent increase?

b) What is the attendance in 2010? What significant feature on the graph does this point represent?

c) Find f (–2). What does this represent in the context of the problem and does it make sense? Explain

your thinking.

d) Find the number of students attending Happy College in 2020. Write your answer using function

notation.

Name ______________________________ Period __________



#1 – 2: Copy the table and complete the table to create the type of function described. Create a coordinate

graph and number the axes appropriately. Create a real-life situation that could be represented

by the table of values and graph.

1) What does it mean to be a linear function?

x ( )f x

–2

–1

0 1

1 4

2

3

4

Create a coordinate grid and graph the function.

Create a Real-Life Situation.

2) What does it mean to be an exponential function?

x ( )f x

–2

–1

0 1

1 4

2

3

4

Create a coordinate grid and graph the function.

Create a Real-Life Situation.

3) Given ( )1

3 26

x

f x

= ⋅ +

, evaluate the following.

a) ( )0f b) ( )2f c) ( )1f −

4) Given ( )11

24

xxh x −

= ⋅

, evaluate the following.

a) ( )1h − b) ( )4h c) ( )1h

5) In a laboratory, one strain of bacteria will double each half hour. Suppose a culture starts with 60 cells.

a) How long would it take for 6,000 bacteria cells to grow?

Time period

(half-hour periods) 0 1 2

# of bacteria

b) What is the y-intercept? Explain what it means in the context of the problem.

c) Find the exact number of bacteria there would be in 2 hours (4 half hour time periods).

d) How many bacteria would there be if x were –2?

Name ______________________________ Period __________


Review Material 127

6) Use the function model ( ) ( )0.5x

f x = to answer the following:

a) Copy and complete a table of values and graph the function on a coordinate grid.

x –2 –1 0 1 2 3 4

( )f x

b) Identify the pattern of the function values (both common difference and common ratio).

c) Using the information above, identify the following characteristics of the function.




7) Use the function model ( ) ( )1.5x

f x = to answer the following:


x –2 –1 0 1 2 3 4

( )f x






8) Use the function model ( ) ( )1.5

f x x= to answer the following:


x –2 –1 0 1 2 3 4

( )f x






Name ______________________________ Period __________



-10 -5 5 10

-10

-5

x

y

9) Mr. Smith bought a new truck for $40,000. The value of the truck depreciates 6% annually. Use the model

( ) 40,000(0.94)tf t = to answer the questions below.

a) Find ��0�.

b) Explain the meaning of ��0� in the context of the problem.

c) Find ��6�.

d) Explain the meaning of ��6� in the context of the problem.

e) What is the domain and range of the function? Explain their meaning in the context of the problem.

Domain:

Range:

10) Graph the equation � = 2� − 3. Use a table of values to make the graph and find the missing information.

Growth or Decay? ______________

Domain _______________________

Range ________________________

Asymptote_____________________

y-intercept____________________

11) Use the equation and graph to determine the answer to each item below.

a) Find f (1).

b) Find f (0).

c) Find f (–1).

d) Find f (–10).

e) Find f (–100).

f) Does the function above ever cross the line y = –4? If yes, at what x-value does this occur? If no,

explain why not.

g) Identify the domain of the function above.

h) Identify the range of the function above.

Equation: ( ) ( )3 2 4x

f x = − −

Asymptote: y = – 4

Name ______________________________ Period __________


Review Material 129

-5 5 10

-5

5

x

y

12) Use the equation and graph to determine the answer to each item below.

a) Find f (–1).

b) Find f (0).

c) Find f (1).

d) Find f (10).

e) Find f (100).

f) Does the function above

ever cross the line y = 3?

If yes, at what x-value

does this occur? If no,

explain why not.

g) Identify the domain of the function above.

h) Identify the range of the function above.

13) Solve the exponential equation using any method.

a) 27 = 3� b) 100 = 10� c) 4�� = 15

d) 9�� + 3 = 81 e) 5 ∙ 18�� = 26 f) 2�� = 3��

14) A mother and daughter decided to register for a social media

group. The number of messages received by the mother on

day n is given by the equation M = 4n. The graph shows the

number of messages received by the daughter on day n

(where n is the number of days since joining the group.)

a) Create a table and create a table of values using the

mother’s equation M = 4n and graph the information on

the daughter’s graph.

b) What type of function would best model each of these

situations?

Mother:

Daughter:

c) What is the domain and range for each person?

Mother:

Daughter:

d) Who is getting more messages on day 3? Explain your

answer.

e) On what day(s) do both Mom and daughter receive the

same number of messages? Explain your answer.

Equation: ( ) ( )2 4 3x

f x−

= − +

Asymptote: y = 3

Number of days since joining the group

Nu

mb

er o

f m

essa

ges

Number of Messages

130 APPENDICES

Appendix A ~ Glossary ~ Units 1 – 3

Absolute maximum – p. 61 – resources, section 2.3

The highest point on a graph.

Absolute minimum – p. 61 – resources, section 2.3

The lowest point on a graph is called the.

Boundary line – p. 11 – resources, section 1.1

The linear function in a linear inequality forms the boundary line to the solutions which lie in a shaded

region.

• If the line is included as part of the solution it is a solid line.

• If the line is not included a part of the solution it is plotted as a dashed line.

Constraints – p. 18 – resources, section 1.3

The particular restrictions of a situation due to time, money, or materials.

Decreasing function – p. 62 – resources, section 2.3

A graph which goes down from left to right.

Dependent variables – p. 62 – resources, section 2.2

The output of a function.

Domain (of a function or relation) – p. 60 – resources, section 2.3

The set of all possible independent values the relation can take. It is the collection of all possible inputs.

Exponential decay function – p. 100 – resources, section 3.1

An exponential function with a common ratio between zero and one (0 < b < 1). Larger values of x give

smaller function values.

Exponential function – p. 100 – resources, section 3.1

A function for which the input variable x is in the exponent of some base b. It is written in the form

f(x) = abx

where a is the initial amount when x = 0 and b is the amount by which the total is multiplied every

time. Note: a and b are constants where a ≠ 0, b > 0 and b ≠ 1.

When a quantity increases or decreases by a percent each time period, the model becomes:

o Growth: f(x) = a(1 + r)t

o Decay: f(x) = a(1 – r)t

(a is the initial amount, r is the growth or decay rate in decimal form, and t is the time)

Exponential growth function – p. 100 – resources, section 3.1

An exponential function with a common ratio greater than one (b > 1). Larger values of x give larger function

values

Feasible region – p. 18 – resources, section 1.3

The common shaded region of a system of inequalities.

Function – p. 53 – resources, section 2.1

A relation where there is exactly one output for every input.

Appendix A ~ Glossary ~ Units 1 – 3 131

Increasing function – p. 62 – resources, section 2.3

A graph which goes up from left to right.

Independent variables – p. 54 – resources, section 2.1

The input of a function.

Inequality – p. 11 – resources, section 1.1

A mathematical sentence built from expressions using one or more of the symbols <, >, ≤, or ≥. Like an equation,

it is a relationship between two quantities that are not necessarily equal.

Intercept – p. 60 – resources, section 2.3

The location where the graph of a function crosses an axis.

Interval – p. 62 – resources, section 2.3

A specific and limited part of a function that can be described as an inequality using x.

Linear function – p. 100 – resources, section 3.1

A polynomial function of degree zero or one. A linear function has a constant rate of change. A line is

created when this function is graphed.

Linear inequality – p. 11 – resources, section 1.1

An inequality that involves a linear function.

Linear programming – p. 17 – resources, section 1.3

The mathematical process of analyzing a system of inequalities to make the best decisions given the

constraints of the situation.

Negative functions – p. 63 – resources, section 2.3

Functions are where f (x) < 0.

Objective function – p. 17 – resources, section 1.3

In an optimization problem, this is the function that needs to be maximized or minimized.

Optimization problem – p. 20 – resources, section 1.3

The goal is to locate the feasible region of the system and use it to answer a profitability, or optimization,

question.

Positive function – p. 63 – resources, section 2.3

Functions where f (x) > 0.

Range (of a function or relation) – p. 60 – resources, section 2.3

The set of all possible dependent values the relation can produce from the domain values. It is the collection

of all possible outputs.

Rate of change – p. 63 – resources, section 2.3

For the ordered pairs ( )( )1, 1x f x and ( )( )2, 2x f x , the Rate of Change = ( ) ( )2 1

2 1

f x f x

x x

−

−

Relation – p. 53 – resources, section 2.1

A set of ordered pairs. The first elements in the ordered pairs (the x-values), form the domain. The second

elements in the ordered pairs (the y-values), form the range.

132 APPENDICES

Relative extrema – p. 61 – resources, section 2.3

The general term for relative maximums and relative minimums.

Relative maximum – p. 61 – resources, section 2.3

The point where the function changes from increasing to decreasing.

Relative minimum – p. 61 – resources, section 2.3

The point where the function changes from decreasing to increasing.

Slope-intercept form – p. 11 – resources, section 1.1

A linear equation written in the form y mx b= + .

Solution for the system of inequalities – p. 11 – resources, section 1.1

The common shaded region between all the inequalities in the system.

Standard form – p. 12 – resources, section 1.1

A linear equation written in the form �� + �� = �.

System of inequalities in two variables – p. 18 – resources, section 1.3

Consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the

ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is the

graph of all solutions of the system.

Vertical line test – p. 57 – resources, section 2.2

A test to determine if the graph of an equation is a function. It involves drawing several vertical lines over the

graph. If the graph touches any vertical line more than once, it is not a function.

x-intercept – p. 12 – resources, section 1.1

The location where the graph crosses the x-axis, which is the value of x where ( ) 0f x =

y-intercept – p. 12 – resources, section 1.1

The location where the graph crosses the y-axis, which is the value of ( )0f .

Unit 0 – Practice Problem ANSWERS

Appendix B ~ Selected Answers ~ Unit 0 133

SKILL/CONCEPT (1)

1) slope: 1

y-int: 10

2) slope: –2

y-int: 7

3) slope: 1

4) slope: 1

3−

5) 4

15

y x= +

6) 2 3y x= −

7) 5

22

y x= − +

8) 0

05

m = = , horizontal

9) 1m = − , falls

10) 1

22m = , rises

11) 6

undefined0

m−

= = , vertical

12) Line 1: 5

12m = −

Line 2: 12m = −

Line 2 is steeper

13) Line 1: 3m =

Line 2: 3m =

The lines have the same slope

14) Line 1: 5m =

Line 2: 5m = −

Same steepness, but opposite

direction

15) 55.9

115.2

≈

16) –2° per hour, 65° at 9pm

SKILL/CONCEPT (2)

1) B

2) C

3) A

4) 2 8x y+ =

x-intercept: 8

y-intercept: 4

5) 3 4 10x y+ = −

x-intercept: 10

3−

y-intercept: 5

2−

6) 3 3x y− =

x-intercept: 1

y-intercept: 3−


134 APPENDICES

SKILL/CONCEPT (2)… continued

7) 5 6 30x y− = −

x-intercept: 6−

y-intercept: 5

8) 6y =

x-intercept: none

y-intercept: 6

9) 5x = −

x-intercept: 5−

y-intercept: none

10) 3 7y x= +

11) 8x = −

12) 2 7 14x y− =

13) 5

22

y x= −

14) 5 10 30x y+ =

15) 3

4y =



SKILL/CONCEPT (3)

Answers/Graphs will vary.

SKILL/CONCEPT (4)

1) 2 11y x= − −

2) 1 55

3 3y x= − +

3) 5

102

y x= −

4) 1

13

y x= − −

5) 5

53

y x= −

6) 4 7y x= −

7) 1 7

10 10y x= − +

8) 3 6y x= −

9) 5 1

6 3y x= +

10) 4 30

9 9y x= − +

11) 3 5

4 2y x= −

12) 4 16

9 9y x= −

13) 5y x< − +

14) 9

92

y x≤ − −

15) 9

94

y x< −

16) 1 1

4 3y x< −

17) 2 4

3 3y x≤ +

18) 2

25

y x> − −

SKILL/CONCEPT (5)

1) Yes

2) No

3) No Solution, Parallel Lines

4) Solution: ( )0, 5−

5) Same Line, Infinite Solutions

6) No Solution, Parallel Lines

7) Solution: ( )4, 6

8) Solution: ( )1, 1−

SKILL/CONCEPT (6)

1) 30 minutes inline skating,

10 minutes swimming

2) 2 hours bicycling,

3 hours inline skating

3) 20 bags of taffy,

15 bags of caramel

4) 3 hours

5) 8 nickels, 22 dimes

6) 5 trees, 20 bushes

7) Pat caught 6 bass and 12 trout

SKILL/CONCEPT (7)

1) ( )4, 1−

2) No Solution

3) ( )3, 3

4) ( )2, 2−

5) Infinite Solutions

6) ( )2, 6−

7) 4

0,5

8) ( )1, 1− −


10) No Solutions


136 APPENDICES

SKILL/CONCEPT (8)

1) ( )5, 2− −


3) ( )5, 0

4) 6 103

, 11 11

−

5) No Solution

6) 7

5,4

7) No Solution


9) ( )9, 6

10) a. Infinitely Many Solutions:

when both x and y terms are

eliminated and the constants

remaining are equal.

b, No Solutions: when both x

and y terms are eliminated

and the constants remaining

are not equal.

SKILL /CONCEPT (9)

1) a. The number of boys plus the

number of girls total 23.

b. There are 3 more girls than

boys.

c. 13 girls

2) She recycled 4 bottles and

17 cans.

3) The FF had 4 touchdowns and

9 field goals.

4) They sold 76 roses.

5) 120 single scoops,

130 double scoops

6) 15 1

,4 2

−

7) 371 student tickets,

566 non-student tickets

8) 20 cell phones at $67,

16 cell phones at $100

9) There are 35 two point

questions on the test

10) 28 corn dogs were sold

11) The gummy bears

cost $1per bag.

12) The car has traveled

210 miles.

.

SKILL/CONCEPT (10)

1) 4 is not a solution

2) 3 is a solution

3) 5x >

4) 11x ≤ −

5) 6x <

6) 15

4x >

7) 6x <

8) x < 0

9) 2 18x≤ ≤

10) 6 1n− ≤ ≤ −

11) 3 or 6x x≤ ≥

12) 1 1x− < <

13) a. The game can be played 13

times.

b. You will have spent $49.75,

so you

have $0.25 left.

14) She must get a score from 94 to

100 on the final exam

15) a. 31.15 89.15C− ° ≤ ≤ °

b. 242 362.3K° ≤ ≤ °

16) Answers will vary.



-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

SECTION 1.1A

1) Combinations of Non-Fiction

and Fiction books:

N 0 1 2 3 4 5

F 10 8 6 4 2 0

a. Yes, this is a linear

relationship. For every

value of non-fiction that

increases by 1 the fiction

decreases by 2.

b. Answers may vary

c. The linear inequality will be

combinations above the

linear points (0, 10), (1, 8),

(2, 6), (3, 4), (4, 2), (5, 0).

1) d. Solutions to a linear

equation are given on the

line. Solutions to the linear

inequality above 50 points

are any combination above

the line.

2) a. Answers include:

Skis 0 5 10 15

Snowboard 12 8 4 0

b. x-intercept: (15,0) 15 ski

rentals and 0 snowboard

rentals;

y-intercept (0, 12) 0 ski

rentals and 12 snowboard

rentals.

2) c. It would be the points at

(0, 12) and (15, 0) and all

integer combinations that lie

on that line.

3) a. Solution

b. Not a Solution

4) a. Not a Solution

b. Solution

5) No

6) Yes

7) No

8) Yes

9) Yes

SECTION 1.1B

1)

2)

3) a. Yes b. Yes

4) a. No b. No

5) a. � ≤ 4 b. � < −4

6) a. � > 1 b. � ≤ −1

7) Test point (–1, 2)

( )

3 5

2 3 1 5

2 8... TRUE

y x≥ −

≥ − −

≥ −

8) Test point (1, 2)

( )

3

2 3 1

2 3.... FALSE

y x< −

< −

< −

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y


138 APPENDICES

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

SECTION 1.1B (continued)

9) 5

13

y x≤ − − ; Test point (1, 1)

( )5

1 1 135

1 13

21 2 .... FALSE

3

≤ − −

≤ − −

≤ −

10) 2 3 2x y< − + ; Test point (3, 1)

( ) ( )2 3 2 1

2 9 3

2 6.... FALSE

< −3 +

< − +

< −

11) ≤

12) >

13) a. 2

35

y x≤ +

b. 2

33

y x> − +

SECTION 1.1C

1)

2)

3)

4)

5) ≤

6) <

7) >

8) >

9) a. (10, 0) Selling 10 DVD

players and 0 telephones will

produce a profit of $240.

b. (0, 15) Selling 0 DVD

players and 15 telephones

will produce a profit of $240.

c. Graph. Points of the

boundary line include (0, 15)

(2, 12) (4, 9) (8, 3) (10, 0).

Boundary line is solid.

d. The boundary line should be

solid because the inequality

symbol it can be equal to (≥).

e. Test a point below: (5, 2)

( ) ( )

?

24 5 16 2 240

152 240.... FALSE

+ ≥

≥

Test a point above: (11, 5)

( ) ( )

?

24 11 16 5 240

344 240.... TRUE

+ ≥

≥

Shading is above the line as

the point (11, 5) makes the

inequality true.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y



-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y

SECTION 1.1D

1)

2)

3)

4) a. 7

32

y x≤ − −

b.

c. Use test point (–3, 1)

( ) ( )

?

7 3 2 1 6

18 6... TRUE

− + ≤−

− ≤ −

Shade side with (–3, 1)

5) a. 2y x< +

b.

c. Use test point (0, 0)

( ) ( )

?

0 0 2

0 2... TRUE

− >−

> −

Shade side with (0, 0)

6) Both Kassie and Krissy’s

representation fits the situation

of cakes and cookies. Kassie’s

equation is in slope-intercept

form and Krissy’s equation is

in standard form. Both use the

correct symbol representing

wanting to sell at least $50.

SECTION 1.2

1) D. Explanations will vary. 2.25

is a fixed rate, therefore, it is

the y-intercept. $0.10 is the

unit rate and dependent on

the number of days.

2) B. 800 people will attend the

dance if you charge $0. As

the price increases, the

number of people attending

will decrease.

3) s= # of student tickets

a= #of adult tickets

2� + 5� ≥ 500

4) r = Time spent running (in hrs)

w = Time spent walking (hrs)

6 4 4r w+ <

5) a. v = # of videos

c = # of CD’s

9 7 35v c+ ≤

b. No, because

( ) ( )9 2 7 3 35+ ≤

6) l = length in feet

w = width in feet

2 2l w+ <800

7) k = # of kiwis

l = # of limes

0.5 0.25 5k l+ <

8) T = shares of TCS stock

U = shares of US stock

20 15 4000T U+ ≤

9) a. s = # of student tickets

a = # of adult tickets

5 7 5400s a+ ≥

b. 380 student tickets

10) a. p = # of pizzas

w = # of wings

11 4 100p w+ ≤

b. No, ( ) ( )11 8 4 6 100+ ≤

11) T = # of 12 inch boxes

F = # of 15 inch boxes

12 15 84T F+ ≤

12) G = # of pounds of grapes

R = # of pounds of oranges

2.25 1.90 20G R+ ≤

13) M = # of mattress sets

B = # of bed frames

200 100 80,000M B+ ≤

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

x

y


140 APPENDICES

-2 5

-4

7

x

y

-7 2-1

11

x

y

-2 8

-7

5

x

y

-8 14

-14

4

x

y

SECTION 1.3A

1) a. x = # of $2 gallons of fuel

y = # of $3 gallons of fuel

b. 2 3 18x y+ ≤

2) a. h = # of lbs of ham

c = # of lbs of chicken

b. 6h c+ ≤

3) a. B = # of hours babysitting

T = # of hours tutoring

b. 13B T+ ≤

c. 4 7 65B T+ ≥

4) a. H = # of hrs housecleaning

S = # of hours at sales job

b. 41H S+ ≤

c. 5 8 254H S+ ≥

5) a. Profit 6 4c p= +

b. 40c ≥

c. p ≥ 20

d. 80c p+ ≤

6) a. Profit 500 800t m= +

b. 1200 t m≥ +

c. 600 t≥

d. 800m≤

7) a.

rem

ote

s

nu

nch

uck

s

To

tal

Prep Time 4 2.5 16

Assembly Time 1 2.5 10

Profit 20 12

b. R = # of wii remotes sold

N = # of nunchucks sold

c. Profit 20 12R N= +

d.

8) a.

skip

ants

skij

acket

s

To

tal

Sewing 8 4 60

Cutting 4 8 48

Profit 2.00 1.50

b. p = # of pants

j = # of jackets

c. Profit 2 1.50p j= +

d. 8p + 4j ≤ 60

4p + 8j ≤ 48

p ≥ 0

j ≥ 0

SECTION 1.3B

1) You can substitute the ordered

pair into all inequalities of

the system and verify it

satisfies all of them.

2) a. yes, in the shaded region

b. yes, in the shaded region

c. yes, within the solution region

(on solid line)

d. no, not within the solution

region (not on dashed line)

3) A 4) F

5) B 6) D

7) C 8) E

9) Tina: Shading is wrong

Boyang: Shading is wrong,

dashed line should be solid

and solid line should be

dashed.

10)

11)

12)

13)

4 2.5 16

1 2.5 10

0

0

R N

R N

R

N

+ ≤

+ ≤

≥

≥




14) 1

33

y x≤ − +

7y x≤ − +

15) 2

13

y x≥ +

110

3y x< − +

16) a. x = # of hotdogs

b. y = # of pretzels

c. see the graph to the right

d. No, this situation is not in the shaded region and

it does not satisfy the restriction of selling at least

10 pretzels.

e. Yes, this situation is in the shaded region and

does satisfy all of the restrictions (constraints)

that were stated.

SECTION 1.3C

1) a.

Wayne’s weight Bubba’s Weight

# of

months

weight

# of

months

weight

0 120 0 210

1 130 1 202

2 140 2 194

3 150 3 186

4 160 4 178

5 170 5 170

6 180 6 162

1) b. Wayne is the dot

c. Bubba is the ×

d. (5, 170)

e. After 5 months, Wayne and

Bubba will weigh the same

amount, 170 pounds.

2) (–2, –6) (should be graphed)

3) (–6, 6) (should be graphed)

4) no solution (should be graphed)

5) (–2, 1) (should be graphed)

6) (–6,4), (4, 9), (–1, –1)

7) (1, 8), (3, 4), (5, 2)

8) a. W = # of acres of wheat

R = # of acres of rye

8) b. Profit 500 300W R= +

c. 7 10W R≤ + ≤

200 100 1200W R+ ≤

1 2 12W R+ ≤

d. Both x and y-axes from 0 to12

counting by 1’s.

e. vertices @ (5, 2),(2, 5),(4,4)

9) a. C = # of chairs

T = # of tables

b. Profit 10 30C T= +

c. 0 0T C≥ ≥

3 6 42C T+ ≤

5 4 40C T+ ≤

d. (C, T) (0, 0),(8, 0), (4, 5), (0, 7)

14

14

0

Chairs

Tables

PR

ET

ZE

LS

HOT DOGS

x ≤ 70 x ≥ 30

y ≤ 40

y ≥ 10

x + y ≤ 90

Number of Months

Wei

gh

t (i

n p

ou

nds)


142 APPENDICES

SECTION 1.3D

1) (2, –3) 2) (–4, –5)

3) (1, –1) 4) (5, –4)

5) a. T = price of tacos

B = price of burritos

b. 3 3 11.25T B+ =

c. 1 2 6.25T B+ =

d. Burrito $2.50; taco $1.25

e. Substitute the solution into

both equations to make sure

they are true.

6) (–6, 11), (6, 5), (3, 2)

7) (0, 12), (3, 3), (0, 0)

8) (–6, 2),(2, 10),(–1, 1),(–4, –2)

9) (–9, 0),( –2, 9),(6, 5),(9, –8)

10) a. x = # of pairs of basketball

shoes

y = # of pairs of soccer shoes

b. Profit 75 95x y= +

c. 50 24 2400x y+ ≤

30 33 1980x y+ ≤

0 0x y≥ ≥

d. (0, 60), (34, 29), (48, 0), (0, 0)

11) a. C = # of cups

P = # of plates

b. Profit = 2C + 1.5P

c. 6C + 3P < 1200

0.75C + P < 250

C > 0 P > 0

11) d. (0,0),(0,250),(120,160),(200,0)

12) (1, –8) 13) (0, –5)

14) (2, –5) 15) (–3, –3)

16) admission $6; ride is $2

17) (3, 2), (5, 2), (9, 6)

18) (–4, 0), (0, 2), (2, 6)

19) (–5, –0.5),(0, 2),(2, 1),(–2, –2)

20) (–1, –6), (1, –4), (2, 0), (1, 2)

21) a. x = number of notebooks

y = number of newspapers

b. Profit 500 350x y= +

c. Minimum nb: � ≥ 10

Minimum np: � ≥ 80

Prod. Limit: � + � ≤ 200

d. (10, 80), (10, 190), (120, 80)

22) a. C = # consoles each month

W = # widesc. each month

b. Profit = 125c + 200w

c. Min console: 0C ≥

Min widesc: 0W ≥

Max console: 450C ≤

Max widesc: 200W ≤

Cost:

600 900 360,000C W+ ≤

d. (0, 0), (0, 200), (300, 200),

(450, 100), (450, 0)

23) a. C = # of acres of corn

planted

B = # of acres of soybeans

planted

b. Profit 60 90C B= +

c. min. corn: 0C ≥

min. soybeans: 0B ≥

acres available: 320C B+ ≤

expenses: 50 100 20,000C B+ ≤

Storage: 100 40 19,200C B+ ≤

d. (0, 0),(0, 200),(140, 130),

(192, 0)

# of pairs of BB shoes

# o

f pai

rs o

f S

occ

er s

hoes

Number of cups

Nu

mb

er o

f P

late

s

50 100 150 200

25

50

75

100

125

150

175

200

0

# of notebooks

# o

f n

ewsp

aper

s

Number of console TV’s N

um

ber

of

Wid

escr

een T

V’s

# of acres of Corn

# o

f ac

res

of

So

yb

eans



SECTION 1.3E

1) a. i (–2, 0), (4, 3), (4, 0), (0, –2)

ii

Ordered

Pair Calculations Value

(–2, 0) � � 2�−2� + 3�0� –4

(0, –2) � � 2�0� + 3�−2� –6

(4, 0) � � 2�4� + 3�0� 8

(4, 3) � � 2�4� + 3�3� 17

iii max value of 17 at (4, 3) min at –6 at (0, –2)

b. i (2, 5), (3, 5), (5, 3), (4, 1), (2, 2)

ii

Ordered

Pair Calculations Value

(2, 2) � � 3�2� + 6�2� 18

(2, 5) � � 3�2� + 6�5� 36

(3, 5) � � 3�3� + 6�5� 39

(4, 1) � � 3�4� + 6�1� 18

(5, 3) � � 3�5� + 6�3� 33

iii max 39 at (3, 5), min 18 at (4, 1) and (2, 2)

and every point on the line segment joining

(4, 1) and (2, 2) on that boundary line.

2) max 17 at (4, 5); min –16 at (–7,5)

3) a. vertices: C(–2, 4) = –14, C(5, 4) = 7,

C(5, –3) = 21

max is 21 at (5, –3); min is –14 at (–2 , 4)

b. vertices: C(1, 10) = 63, C(1, 4) = 27,

C(5, 8) = 63, C(7, 4) = 45

min is 27 at (1 , 4); max is 63 at (1, 10) and

(5, 8) and every point on the line segment

joining (1, 10) and (5, 8).

4) a. (30, 10) (30, 40) (50, 40) (70, 20) (70, 10)

b. Let H = # of hot dogs

Let P = # of pretzels

Profit 0.48 0.25H P= +

c. maximum profit is $38.60 when 70 hotdogs and

20 pretzels were made and sold

4) d. Profit 0.25 0.48H P= +

maximum profit is $31.70 when 50 hotdogs and

40 pretzels were made and sold

e. Profit 0.40 0.40H P= +

maximum profit of $36.00 occurs at any point

of the line segment joining points (50, 40) and

(70, 20). Therefore several combinations of

making and selling hot dogs and pretzels will

achieve a maximum profit amount (50 hot dogs

and 40 pretzels… 70 hot dogs and 20

pretzels… etc.)

f. Scenario c) gives the maximum profit amount

($0.48 for a hot dog and $0.25 for a pretzel).

5) a. x = # of batches of whole wheat bread

y = # of batches of apple bran muffins

b. 0x ≥ 0y ≥

4 2 16x y+ ≤

2 10x y+ ≤

c. graph �

d. (0, 0), (0, 5),

(2, 4), (4, 0)

e. P(0, 0) = $0

P(0, 5) = $50

P(2, 4) = $110

P(4, 0) = $140

f. The bakery should bake 4 batches of bread and

no muffins for a profit of $140

Nu

mb

er o

f P

retz

els

Number of Hot Dogs

(30, 40) (50, 40)

(70, 20)

(70, 10) (30, 10)

# of batches of bread

# o

f b

atch

es o

f m

uff

ins


144 APPENDICES

SECTION 1.3E (continued)

6) m = # of lawns mowed

t = # of yards trimmed

0m ≥ 0t ≥

time: 1.5 3 24m t+ ≤

cost: 2 1 20m t+ ≤

income: 25 40m t+

expenses: 2 1m t+

Profit: 23 39m t+

vertices: P(0, 0) = $0, P(0, 8) = $312,

P(8, 4) =$340, P(10, 0) = $230

Mowing 8 lawns and cleaning up (trimming)

4 yards will maximize his profits at $340.

7) b = number of brochures printed

f = number of fliers printed

50b ≥

150f ≥

3 2 600b f+ ≤

Cost 0.08 0.04b f= +

Vertices: C(50, 150) = $10, C(50, 225) = $14,

C(100, 150) = $13

The manager will minimize printing costs if she

prints 50 brochures and 150 fliers for $10.

SECTION 1.3F

1) t = # of student admission tickets sold

g = # of general admission tickets sold

0 0t g≤ ≤

4000t ≤

2000g ≤

0.50 3000t g+ ≤

( ) ( )

Profit Income Expenses

Profit 4 7 0.50 1

Profit 3.50 6

t g t g

t g

= −

= + − +

= +

Vertices: P(0, 0) = $0,

P(0, 2000) = $12,000,

P(2000, 2000)$19,000,

P(4000, 1000) = $20,000,

P(4000, 0) = $14,000,

They will maximize their profit if they sell 4000

student admission tickets and 1000 general

admission tickets for a total of $20,000.

2) t = # of tourist class tickets sold

f = # of first class tickets sold

16t ≥

12f ≥

# of lawns mowed

# o

f la

wns

trim

med

# of brochures

# o

f fl

iers

# of student tickets

# o

f gen

eral

adm

issi

on t

icket

s



50t f+ ≤

SECTION 1.3F (continued)

2) (continued) Profit 26 24t f= +

Vertices: P(16, 12) = $704, P(38, 12) = $1276,

P(16, 34) = $1232

They will maximize their profits if they sell

12 first class tickets and 38 tourist class tickets for

$1276.

3) b = tablespoons of butter

v = tablespoons of vegetable shortening

0b ≥

0v ≥

6 1 34b v+ ≥

1 4 44b v+ ≥

25b v+ ≤

Calories = 100b + 115v

Vertices: C(4, 10) = 1550, C(1.8, 23.2) = 2848,

C(18.67, 6.33) = 2595

He can minimize the calories by using

4 tablespoons of butter and 10 tablespoons of

vegetable shortening. It will total 1550 calories.

4) R = number of radios made

C = number of CD players made

0R ≥

0C ≥

600R≤

500C ≤

10 12 8400R C+ ≤

Profit 19 12R C= +

Vertices: P(0, 0) = $0

P(0, 500) = $6,000

P(240, 500) = $10,560

P(600, 200) = $13,800

P(600, 0) =$11,400

They will maximize their profit at $13,800 if they

make 600 radios and 200 CD players.

# of tourist class tickets sold

# o

f fi

rst

clas

s ti

cket

s so

ld

# of tablespoons of butter # o

f ta

ble

spoo

ns

of

veg

etab

le s

ho

rten

ing

# of radios sold

# o

f C

D p

layer

s so

ld


146 APPENDICES

UNIT 1 REVIEW

1) Antonio is correct. (0, –1) is part

of the solution region. The point

is on the solid line, so it is a

solution. The inequality is

� ≥ 2� − 1, so test (0, –1)

−1 ≥ 2�0� − 1?

−1 ≥ −1, true

2) a. ≥ b. <

3) a.

b.

4) a. strictly above the boundary

line y = –2x – 1, in the

shaded region

b. below the boundary line and

on the boundary line

y = –2x – 1

5) 2 1y x> − −

6) 2 1y x≤ − −

7) 6 > –2(5) –1,

6 > –11, true, solution

8) 2 ≤ –2(4) – 1,

2 ≤ –9, false, not a solution

9) a. x = # of magazine

subscriptions

y = # of newspaper

subscriptions

9) b. 500 ≤ 15� + 22�

c.

d. Answers will vary, any point

on the line or to the above the

line, example: (22, 14)

If Raymond sells 22 magazines

and 14 newspaper

subscriptions, he makes $638

e. Answers will vary, any point

below the line, example (1, 2).

If Raymond sells 1 magazine

and 2 newspaper subscriptions,

he makes $59.

10) a. � = # of 1 point (free throws)

shots scored

y = # of 2 point shots scored

b. � + 2� ≥ 44

c.

d. No, (15, 4) would only give

you 34 point which is not

enough to win the game.

11) a. x = # of pounds of aluminum

wire

y = # of pounds of copper wire

b. .� ≥ 0, � ≥ 0

5� + 6� ≤ 450 1

4� +

1

5� ≤ 20

c. Profit = 0.25x + 0.40y

d. and e.

f. (0, 0), (0, 75), (60, 25), (80, 0)

g. Max profit $30, Min profit $0

h. To produce the maximum

profit of $30, 0 pounds of

aluminum should be produced

and 75 pounds of copper of

wire should be produced.

12) A = # of yards3 of plant food A,

B = # of yards3 of plant food B

20A + 10B ≥ 460, 30A + 30B ≥

960, 5A + 10B ≥ 220, A ≥0,

B ≥0,

� � 30� + 35� (0,46), (44,0), (20, 12), (14, 18)

Minimal cost: 20 cubic yards of

plant food A and 12 cubic yards

of plant food B for a cost of

$1020.

-2 10

-5

5

x

y

-5 5

-5

5

x

y

8 16 24 32

4

8

12

16

20

24

x

y

# of magazine subscriptions

# o

f n

ewsp

aper

su

bsc

rip

tion

s

# of 1 point shots scored

# o

f 2

po

int

shots

sco

red

# of pounds of

# o

f pounds

of

cop

per

# of yd3 plant food A

10 20 30 40 50

10

20

30

40

50

0

# o

f yd

3 p

lant

foo

d B



SECTION 2.1A

1) B. Answers will vary.

Example: Distance increases

until it reaches the top of the

tire then returns to its lowest

point, then repeats.

2) A. Answers will vary.

Example: As the child swings

away, the distance increases. It

decreases as the child

approaches the parent. Other

graphs indicate vertical aspects,

which are not part of this

problem.


a. The person does not drink

any lemonade.

b. The person started drinking

very fast but slowed down.

They finish their lemonade.

c. Drank about a third of the

glass at a constant rate, then

stopped for a bit before

finishing the glass at a

slower rate.

3) d. Didn’t start drinking right

away. Then drank about half

the lemonade at a constant

rate then stopped drinking.

4) a. Answers will vary.

b. Answers will vary.

4) c. Answers will vary.

d. Answers will vary.

SECTION 2.1B 1) Input is the independent

variable (x-variable). Output is

the dependent variable

(y-variable).

2) Answers will vary. Should

include that it is a relationship

with x-variables matched to y-

variables and there are no

repeated x-values with different

y-values. Functions can be

represented in tables, graphs,

equations, etc.

3) Answers will vary. For each

ride at the fair it will cost us

$1.50. Input: # of rides.

Output: cost

4) Yes

5) Yes, it is a function with a

limited domain.

6) No. A is a repeated input value

7) Function

8) No. Change the second 4 in

the input column to a 5 (for

example)

9) Function

10) No. Inputs must all be

different.

11) Function

12) No. The input value 7 is

repeated. Example: Change

the pairing (7, 0) to (8, 0).

13) Not a function; 5 cannot have

2 outputs. Change (5, –8) to

(6, –8), for example.

14) No. Multiple students can be

the same age.

15) function; each student has one

age

16) Yes. All x-values are different.

17) not a function; x = –1 has two

outputs, 6 and –6

18) No. Jim is a repeated input.

19) not a function; Jim has 2

partners, Kitty and Betty

20) Yes. The x-values are the

years, which will be different

on each birthday.

21) Yes, if it for only one term. Not

a function if it is different

numbers for different terms and

they are not combined.

22) Yes, if it is only one cookie.

No, if it is a batch of cookies

that may have equal diameters

but different numbers of chips.

23) Yes, if each boy only brings

one date. No, if multiple dates

are allowed.

24) Yes. Cory would represent a

repeated input.

25) No, there are multiple outputs

for the input 1.

26) Yes, no repeated inputs.

27) Yes, each input has no more

than 1 output.


148 APPENDICES




SECTION 2.2A

1) a. 0 b. –3

c. x = 1 d. x = –1

2) a. −1 b. −2 c. x = 3 d. x = 2

3) a. ��12� represents the

distance traveled after 12

seconds.

b. �� 100 represents the

time it takes to travel 100

feet.

c. ��16�

d. �� 200

4) a. 128 b. 8192

c. � � 6 or �6�

d. after approx. 12 weeks (with

a calc, 11.3) the population

will be over 20,000.

e. 524,288 rodents; This

number is unrealistic over

16 weeks. Animals do not

double this quickly for this

amount of time. Factors

could include weather

changes, animal health,

predator life, etc.

5) They both represent a relation

between inputs and outputs. but

f(x) represents a function.

6) !�� implies x is the

independent variable f(x) is the dependent variable

because it specifies that x is the

input.

7) a. input: 0 output: 3

b. input: −2 output: −1

c. input: 2 output: 7

d. input: 5 output: 13


b. input: 3 output: −8

c. input: 2 output: −5

d. input: −2 output: 7


b. input: 5 output: 25


d. input: +4, −4 output: 16


b. input: 2 output: 4


d. input: 6 output: 64

11) � = 3

12) � = 12

13) d(h) = 7h

14) May choose any x-values, but

y-values must match with the

equation.

# –1 0 1 2 3

$ 55 50 45 40 35

Equation: � = −5� + 50

Story: Answers will vary. A

rain barrel contains 50

gallons of water. It has a

slow leak in it and loses 5

gallons of water per day.

15) Possible values in table

include:

# 0 1 2 3 4 $ 0 50 100 150 200

Equation: � = 50�

Story: Given

16) a. �(%) = 0.20% + 80

b. �(45) = 89, so she will

pay $89 for 45 text

messages sent.

c. ( ) 90c k =

90 0.20 80

10 0.20

50

k

k

k

= +

=

=

She sent 50 texts during the

month for a cost of $90.

d. k = 100 Unlimited data

would be cheaper if she

used over 100 × 100Kb of

additional data use.

4 8 12

10

20

30

40

50

0

x

y

# of days

gal

lon

s of

wat

er i

n b

arre

l

Time (in hours) 1 2 3 4 5

25

50

75

100

125

150

175

200

225

250

275

300

0 x

y

Mil

es t

ravel

ed

1

2

3

4

2

4

{(1,2), (2,2),

(3,4), (4,4)}

x y

1 2

2 2

3 4

4 4

x y

1 2

2 2

3 4

4 4

1

2

3

4

2

4

{(2,1), (2,2),

(4,3), (4,4)}



SECTION 2.2B

1) Not a function. Answers will

vary.

(for example) (1, 1) and (1, –1)

2) Not a function.

Example answer (0, 2.3) and

(0, -2)

3) Function

4) Function

5) Function

6) Not a function. Example

answer (–1, 2) and (–1, –1)

7) The first graph is a function,

because each x-value has no

more than 1 y-value. The

second graph is not a function

because every whole number in

the domain has two y-values. If

the dots in the second graph

were connected at each x-value,

there would be 2 dots at each

input. SECTION 2.2B

8) a. 1 b. 6.3

c. � = 0 d. � = −1.3

9) a. � = 2 b. � = 1.2

c. −4 d. −1

10) a. −2 b. −8

c. � = −2 d. � = 0

11) a. 1 b. � = 5

c. 1.5 d. x = −6, 0, 1.8

12) a. $3.54 b. $3.54

c.

d. Yes. Each date had a

unique price.

e. No. Each day of the week

had multiple prices,

depending on the day of the

month.

13) a.

Time Height

0 2

1 27.1

2 42.4

3 47.9

4 43.6

5 29.5

6 5.6

7 –28.1

b. Answers will vary.

(0, 2) The ball is thrown from a

height of 2 meters

(3, 47.9) After 3 seconds the

ball is 47.9 meters in the

air

(5, 29.5) After 5 second the

ball is 29.5 meters in the

air

SECTION 2.3A

1) Domain: all real numbers, Range: y ≤ 4

continuous

x-intercept: (−3,0) and (1, 0)

y-intercept (0, 3)

function: yes

2) Domain:{–8, –3, –2, 0, 2, 3, 4, 6} Range: {–3, –1, 0, 1, 4, 6, 7}

Discrete

x-intercept = (6, 0)

y-intercept = (0, 1)

Not a function

3) Domain: x ≥ 1 Range: y ≥ –2

continuous

x-intercept: (5, 0) y-intercept: none

function: yes

4) a. Yes

b. Domain: {–7, –6, –2, 3, 8}

c. Range: {2, 3, 4, 5}

5) a. yes

b. x = {1, 2, 3, 4, 5, 6}

c. y = {2, 3, 4, 5, 6, 7}

6) Domain: � ≥ 1

Range: � ≥ 0

7) Domain: � ≤ 5

Range: 1y ≤ −

8) Domain: 1 < � ≤ 6

Range: −2 < � ≤ −1

9) Domain: – 8 ≤ � ≤ 10

Range: –3 ≤ � ≤ 5

10) Domain: � > −5

Range: � >(

)

11) Domain: all real numbers

Range: y > 0

x

y

S M T W T

F S

3.64

3.62

3.60

3.58

3.56

3.54

3.52

Time (in seconds)

1 2 3 4 5 6 7

5101520253035404550

0

Hei

ght

of

the

bal

l


150 APPENDICES

SECTION 2.3A (continued)

12) Many correct ways to approach

this, including using a table to

make a formula. Let x = the

number of weeks he makes a

payment and y = the remaining

amount owed.

a. Domain: all integers

between 0 and 9 inclusive.

Each week between 0 and 9

weeks he makes a payment.

b. Range: 0 ≤ � ≤ 900 in $100

increments. He pays $100

each week, dropping the

balance owed.

c. x-intercept = �9, 0�. It will

be paid off in 9 weeks.

d. y-intercept = 900. He starts

out owing $900.

e. Discrete. It drops in $100

each week, and the value of

what is owed is always a

whole number even in

between weeks.

f. It is a function. There is only

one y-value for each x-value.

The value of the couch will

never have more than one

value in a given week.

13) a. Domain: 0 ≤ x ≤ 7.5 It takes

7.5 minutes to fill the tank.

b. Range: 0 ≤ y ≤ 15

There was 15 gallons of gas

in the tank.

c. x-intercept: (7.5, 0)

It takes 7.5 minutes to fill the

tank.

d. y-intercept: (0, 7.5)

The tank is empty at the

beginning of the event.

e. continuous

You can have different

amounts of gas in the tank at

different fractional values of

a minute.

f. Yes, it is a function.

Each time value has one and

only one output value of

gallons in the tank. At any

given point in time, the

amount of water in the tank

will never be more than one

value.

14) Answers will vary. One

possible graph:


One possible graph. Segments

between points do not have to be

linear.

SECTION 2.3B

1) a. (0,15), He started with 15

gallons of gas

b. (375,0), He could drive

375 miles before running

out of fuel

c. decreasing, As the number

of miles increases, the

number of gallons of gas

decreases.

d. 0 ≤ x ≤ 375, the range of

miles that could be given

e. 0 ≤ y ≤ 15, He can have

anywhere from 0 to 15

gallons of gas in the tank.

2) a. y-intercept = (0, 7�. The

catapult releases the rock 7

feet off the ground.

b. x-intercept = �14, 0). The

rock hits the ground 14

seconds after launch.

c. (6, 16) – The rock will be

16 feet above the ground 6

seconds after the launch. It

will then start to decrease

in height.

d. The domain is 0 ≤ � ≤ 14

because the rock is

airborne between 0 and 14

seconds.

e. The range is 0 ≤ � ≤ 16

because the rock will be

anywhere between 0 and

16 feet off the ground.

3) Increasing: � < −2

Decreasing: � > −2

Absolute Max: (−2, 3)

4) Increasing: � < −1, � > 1.7

Decreasing: −1 < � < 1.7

Relative Max: �−1,1�

Relative Min: �1.7,−9.1�


possible answer is:

-5 5

-4

4

x

y





possible answer is:


possible answer is:


possible answer is:

9) a. There is no x-intercept

because there is never 0

hours of daylight. The y-intercept (0, 12) is

in mid-March. There are 12

hours of daylight and it is

the vernal equinox.

b. Decreasing mid-June to

mid-December;

Increasing mid-December

to mid-June

c. There are no more than 15

hours of daylight (June,

maximum) and no fewer

than 9 hours of daylight

(December, minimum)

d. Domain: months of 1 year;

Range: 9 ≤ � ≤ 15

SECTION 2.3C

1)

2)

3)

4)

5) a. no x-intercept; The jumper

never hits the ground.

y-intercept = �0, 98�; The

jumper starts at a height of

98 feet.

5) b. relative maxima at

approximately (0, 98),

(3.4, 50), (6.5, 42); These

are the heights the jumper

rebounds to after each

bounce. Relative minima at

approximately (1.8, 5), (5,

19), (8, 25); These are the

low points where the jumper

rebounded.

c. Increasing 1.8 < x < 3.5,

5 < x < 6.5; The jumper is

going up in these areas.

Decreasing 0 < x < 1.8,

3.5 < x < 5, 6.5 < x < 8. The

jumper is falling during

these times

d. The jumper does not touch

the ground.

e. No; a negative interval

would mean that the jumper

went below ground level.

f. Domain: 0 ≤ x ≤ 8 this is the

length of the jump in

seconds.

Range: 5 ≤ y ≤ 98, these are

the heights of the jumper

over this time period.


152 APPENDICES

SECTION 2.3D

1) 40; the driver drove 40 mph for 2 hours to the first

delivery

2) The rate of change = 0. The truck is not moving

during that hour. The driver may be unloading or

reloading the truck during that time.

3) 40; the driver drove 40 mph for 1 hour to the

second delivery

4) The truck is stopped again for two hours.

5) �60; the truck was driven at 60 mph for 2 hours

back to the distribution center.

6) The rate of change is negative between 6 and 8

hours. It is the return journey back to the

distribution center. As the truck moves towards

the center, the distance is decreasing (speed is

positive). This is the fastest rate of the journey.

7) 2 < t < 3 (region B) and 4 < t < 6 (region D). The

truck is not moving.

8) 1.2ft/mph 9) 2.4 ft/mph 10) 5.725 ft/mph 11) The relationship is not linear. As x increases by 5,

the y values increase differently each time. As the

speeds increase so do the stopping distances.

12) The relationship between speed and braking

distance is not proportional. If it was proportional,

values such as (10, 5) and (20, 10) would be in the

table.

13) Between (0, 98) and (1.8, 5) the average rate of

change is −57.1, so the jumper falls 57.1 ft/sec on

the first fall. Between (3.5, 50) and (5, 19) the

average rate of change is approximately –20.67

ft/sec. on the second bounce.

SECTION 2.3E

1)

• x 3 –1 6 –3 9

f –1

(x) 1 4 1 0 1

2)

▲ x 0 3 –2 2 1

*+,�#� 1 –2 4 2 –2

3) B

4) C

5) A

6) a.

x *�#� x *+,�#� –6 –4 –4 –6

–3 –3 –3 –3

0 –2 –2 0

3 –1 –1 3

6 0 0 6

b. and c.

d. They reflect each other over

the line y = x.

e. 3!�� 6 � �so!+(�� 3� 6

f. See part A.

g. In the table, the x and y

values switch.

7) a.

x f (x) x *+,�#� –6 –7 –7 –6

–3 –5 –5 –3

0 –3 –3 0

3 –1 –1 3

6 1 1 6

b. and c.

d. /) !�� 0) � �

e. !+(�� /) �

0)

( )1f x−

( )f x

y x=



SECTION 2.3E (continued)

8) � () !�� 5 � �

!+(�� () �

1)

9) 2!�� 6 � � !+(�� 2� � 6

!+(�� 2� � 6

10) !+(�� 54� 554

11) !+(�� (((�

1((

11) !+(�� (((�

1((

12) !+(�� (()�

2()

13) !+(�� 8� � 13

14) !+(�� 2/ �

1/

15) !+(�� /) � 9

SECTION 2.3F

1) � � �) 3

x y

�2 7�1 40 31 42 7

2) � � �/ 2�) � 3

x y

�2 �3�1 �20 �31 02 13

3) � � 33

x y

�2 0.11�1 0.330 11 32 9

4) � � (3

x y

�2 �0.5�1 �1�0.5 �20 error0.5 21 12 0.5

5)

6)

x y

–32 Does not exist

–30 0

–28 √2 5 1.4

–26 2

–24 √6 5 2.4

–22 √8 � 2√2 5 2.8

7) x-intercept: 62/ , 07

y-intercept: (0, 7)

8) x-intercept: (–0.889, 0),

(8.889, 0) y-intercept: (0, 4)

9) x-intercept: (2.32, 0)

y-intercept: (0, –4)

10) no minimum,

max: ��1.5, 9.5� 11) no relative max or min

12) min: (1, 3), max: (–1, 7)

13) relative maxima and minima

14) x-intercepts

15) Answers will vary. Example:

x-min= 0, x-max= 30, x-scl =5

y-min = –20, y-max = 40,

y-scl = 5

x-intercepts: (12, 0), (20, 0)

y-intercept: (0, 240)

intervals where function is

positive: x < 12, x > 20

interval where function is

negative: 12 < x < 20

16) Answers will vary. Example:

x-min=�10, x-max=1,

x-scl= 1

y-min = �30, y-max = 50,

y-scl = 10

relative max. ��2.64, 41.5� relative min. ��6.7,�25.2� increasing interval:

�6.7 � � � �2.64decreasing intervals:

� � �6.7, � � �2.64

x y

60 700

80 1700

100 3100

120 4900

140 7100

160 9700

!��

!+(��

� � �


154 APPENDICES

UNIT 2 REVIEW

1) Table A is not a function. The

values (1, 2) and (1, 3) have a

repeated input with different

outputs. Table B is a function. Each

input has one unique output.

2) Set B. Each input value has

exactly one output value.

Domain: {–2, 0, 1, 2, 3, 4}

Range: {–1, 1, 2}

3) Graph b and Graph c

4) Table A is a function because

no country is repeated.

Table B is not a function

because there are repeated

populations (inputs) for

different countries (outputs).

5) a. Answers can vary.Yes, if

each name has exactly one

social security number.

No, if there are two people

with the same name but

different social security

numbers. i.e. Two people

named John Lewis

Anderson with different SS

#s

b. No, an address can have

multiple residents.

c. Function; each x-value has

no more than one y-value

d. Not a function; x = 4 has

two y-values

e. Function; each x-value has

no more than one y-value

6) a. Yes. Inputs are all different

names of vehicles

b. No. The input “2” is paired

with both bikes and

motorcycles.

7) Answers will vary. He walks

towards the bus stop, turned

around and started walking

faster towards home. Then

he turns around and walks

even faster toward the bus

stop.

8) a. Tom starts at home and runs

away from home increasing

speed quickly in the

beginning and then slowing

speeds.

b. Tom starts at home and

walks at a slow constant rate

to the bus stop. He waits at

the bus stop for a short

amount of time and then

begins walking at a faster

constant rate further away

from home.

c. Tom starts at home and

walks at a constant rate to

the bus stop. He waits at the

bus stop for a while and

then walks back home at a

faster rate than toward the

bus stop.

9) a. 8�8� � 62; I have $62

after 8 hours of babysitting.

b. x = 3; I have $37 after 3

hours of babysitting.

10) a. x-intercepts: none

y-intercept = (0, 120)

b. Domain: 0 ≤ � ≤ 12

Range: 112 ≤ � ≤ 120

c. 0 � � � 4

d. 4 � � � 12

e. �4, 112� f. 116

g. x = 2 and x = 10

h. 2 pounds per month

i. 0.625 or 19 pounds per month

11) a. All real numbers

b. y ≥ –16.25

c. (2, 0)

d. ( –0.2, –16.25) and (3.5, –

5.5)

e. –0.2 < x < 2, and x > 3.5

f. x < –0.2 and 2 < x < 3.5

g. �0, −16)

h. (−1, 0)(2, 0)and�4, 0)

i. x < –1 and x > 4

j. –1 < x < 4 such that x ≠ 2;

or

–1 < x < 2 and 2 < x < 4

12) a. y-intercept (0, 285). In

1985 there were 285 cell

phone subscribers.

b. No x-intercept. The graph

does not show any time

before cell phones existed.

c. Domain: 0 ≤ � ≤ 9

or1985 ≤ � ≤ 1994

Range: 285 ≤ � ≤ 44,000

d. 44,000 cell phone

subscribers in 1994 (9 years

after 1985)

e. x = 6; 6 years after 1985

(1991) there were 8000 cell

phone subscribers

f. Between years 4 and 5

(1989 to 1990)

13) a. ≈ $0.99 ≤ g(t) ≤ $4.20; The

lowest price for a gallon of

gas was $0.99 and the

highest was $4.20.

b. The year 2008

c. ≈ $2.80; At the start of the

year 2010, the price of a

gallon of gas was $2.80.

d. ≈ 1994 and 1998; In 1994

and 1999, the price of a

gallon of gas was $1.00.

e. ≈ $0.166/year; Gas went up

on average $0.166 per

year… or 16.6¢ per year.



SECTION 3.1

1) Linear; the pattern is:

4, 10, 16, 22… because there is

a common difference of 6.

2) Linear; the change in both x and

y is constant; x increases by 5

and y increases by 10 each time

3) Linear; the distance traveled

remains constant as long as the

speed remains constant.

4) Linear; graph forms a straight

line and constant slope.

5) Exponential; the number of

players remaining is cut in half

after each round. Relationship

has a common ratio of (

).

6) Exponential; the number of

rabbits added every month is not

the same. Relationship has a

common ratio of 2.

7) Linear; The amount of money

withdrawn is constant.

8) Exponential; the slope or rate of

change is not constant, and the

line is curved.

9) Exponential; to get the next

number in the sequence, you

have to multiply the previous

term by 5. The relationship has a

common ratio of 5.

10) Linear: constant rate of 3, and

the graph of this would form a

straight line.

11) Linear: this involves adding a

constant amount to each

previous term. This relationship

has a common difference of 5.

12) Exponential: this involves

multiplying which is not a

constant change. The common

ratio is 1/3.

13) Linear: the change in x and y

both remain constant; as x

increases by 2, y decreases by 5.

14) Exponential: 7% is not a

constant dollar amount because

it is taken from the value of the

previous year. Common ratio is

0.93 or 93%.

15) Exponential: you have to

multiply to get the next term.

The common ratio is 1.5.

16) Linear: constant rate of change;

adding 3 to each previous term;

would create a straight line

graph; the common difference is

3.

17) Exponential: all answers will be

powers of 5, not adding 5 each

time. The common ratio is 5.

18) Answers will vary. Ex: A linear

model with have a constant rate

of change or a common

difference. Add the same

amount to each previous term.

The graph will be a straight line.

An exponential model grows by

multiplying each previous term

and has a common ratio. The

graph of an exponential function

will create a curved line with an

asymptote.

19) a. average rate of change for

r(x) is 1/1 or 1,

average rate of change for

s(x) is 0.5/1 or 0.5

b. Graph r(x) is changing faster

from 0 to 1 because the line

is steeper, and 1 is greater

than 0.5 or ½

c. The average rate of change of

r(x) is 1, and the average rate

of change of s(x) is

approximately 1, so r(x) and

s(x) are essentially growing

at the same rate fro.

d. The average rate of change of

r(x) is 1, and the average rate

of change of s(x) is

approximately 1.5, so s(x) is

steeper and therefore is

growing faster between

x = 3 and x = 4.

20) Both Malcolm and Marietta can

be correct depending on the

interval referred to. For

example, in #19 part a, between

0 and 1, the linear function was

growing more quickly and had a

greater rate of change. However,

beginning about when x = 2, the

exponential function began

increasing more quickly than the

linear, and it will continue to do

so forever.

SECTION 3.2A

1) a. 100 grams; the y-intercept

b. 40 grams

c. after 50 years

d. after 65 years

2) a. approximately 7300.

b. 6000; It represents the population in Coleman in the year 2000.

c. When x = –5, that means going back in time 5 years so the population in 1995.

d. 16,000 people in 2050

3) a. plot these points; should look

like exponential decay

3) b. 4 athletes will be left after

5 rounds

c. 6.5 rounds would be in

between rounds 6 and 7, and

it is not possible to know

how many players are left as

a game would be in play.

d. 1, there is one winner of the

tournament, and it is not

possible to split the athletes

in half after the last round.

Rounds # of athletes

0 128

1 64

2 32

3 16

4 8

5 4

6 2

7 1


156 APPENDICES

SECTION 3.2A (continued)

4)

Hours Passed # of Bacteria

0 8

1 16

2 32

3 64

4 128

5 256

6 512

a. 8 bacteria

b. 512 bacteria

c. student graph

d. approximately 6 hours;

should create a horizontal

line through 500 on the y-axis

5)

Years Owed Value

0 32,000 32,000

1 28,400 27,200

2 24,800 23,120

3 21,200 19,652

4 17,600 16,704.20

5 14,000 14,198.57

6 10,400 12,068.78

a. linear, the straight line

b. exponential, the curved line

c. No, what is owed is more

than what the car is worth.

d. between year 4 and year 5 is

when the amount owed and

the car’s value are equal.

This would be the first

opportunity for the car to be

worth more than what is

owed. On the graph, the

break-even point is where the

2 lines cross.

6) a.

� 0 1 2 3 4 5 6

!�� 1 2 4 8 16 32 64

b.

� 0 1 2 3 4 5 6

!�� 1 2 4 8 16 32 64

Difference

between

function values

1 2 4 8 16 32

� 0 1 2 3 4 5 6

!�� 1 2 4 8 16 32 64

Ratio between

function values2 2 2 2 2 2

The common ratio between

each function value is 2.

The function is increasing

The function has a common

ratio

The function models

exponential growth

SECTION 3.2B

1) Solving Methods will vary

One possible method: Create a

system of equations (each

‘side’ creates one equation of

the system) Ex: 3� + 5 = 17;

�( = 3� + 5, �) = 17 To

solve a system of equations on

a graph, you look for where

the lines intersect.

Another reasonable method:

To solve using one equation,

all terms must be moved to

one side and set equal to 0. In

the example of 3x + 5 = 17,

subtract 17 from both sides to

get 3� − 12 = 0 and find the

x-intercept of this single line.

(which is the intersection of

� = 3� − 12 and y = 0)

2) a. � = −4 b. � � 2

c. � = 3

d. � = 4and� � 5

3) 16 years

4) 4 years

5) a. just a little past 15 years

b. after 15 years, the

investment with the higher

rate will earn more money,

so I would choose to start

with $5000 and earn 1.12%

6) a. � = 1

b. � = 3

c. � = 1.209

d. � = 0.701

SECTION 3.3

1) a. the boat is worth $15,000

because

!(0) = 15,000(0.65)>

= $15,000

b. !(5) = 15,000(0.65)1

= $1740.44

c. This is exponential decay

because the rate at which the

value of the boat decreases

causes the dollar amount

lost each year to be less and

less. The common ratio of

the relationship is 0.65.

1) d. !(7) is the value of the boat

after 7 years. !(7) =15,000(0.65)2 = $735.33

e. !(−3) would be the value

of the boat 3 years prior.

That value is $54,619.94.

This value does not make

sense given that the boat did

not cost this much.

2) a. $2000

b. $2805.10

c. @(10) = $3934.30, and

this is the value of the

painting after 10 years.

d. This is exponential growth

because the rate at which the

value of the painting

increases causes the dollar

amount gained each year to

be more and more. The

common ratio of the

relationship is 1.07.



SECTION 3.3 (continued)

3) a. 100%

b. 0.000000000085 or

8.5383 ∙ 10+((so �8.5383 ∙ 10+0)% of the

light remains at 65 meters.

Effectively, none of the

light remains.

c. B(10) � 0.02825 so

2.82% of the light remains

at 10 meters

4) a. 20,000 when t = 0

b. exponential growth because

a percentage is being added

each year. The common

ratio is 1.06 which is greater

than 1.

c. 20,000(1 + 0.06))>; find

when t = 20 from a table;

$64,142.71

d. $65,837.41; this is the

difference between what the

value is after 25 years and

20,000.

e. graph a horizontal line on

the graph at y = 1,500,000

and at y = 20,000 ·1.06x find

where the lines intersect.

The lines intersect at 74.1

years and 1,500,000.

4) f. The domain must be when x

(or t) ≥ 0. This is true since

the domain represents time,

and time cannot be negative.

The range of the situation

must be when y (or A(t)) ≥

20,000 because the range

represents the amount of

money in the account, and

since it is growing, it will

never drop below 20,000.

5) a.

Years (x) Value (y)

0 42,000

2 29,635

4 20,911

6 14,755

8 10,411

10 7,345.90

b. $20,000 is between years 4

and 5. This is when I would

be able to afford the car.

6) a. the size of the bacteria at

d = 0 (the start) is 2,000,000

b. �(10) � 1.9073

c. Decay, the size of the

bacteria population is

decreasing by 75% (or ¾)

each day. The common ratio

is ¼ or .25.

6) d. The number of bacteria will

be 1.8 x 10 or 0.0000018.

Since you cannot have a

fraction of a bacteria, they

are all gone.

e. A little over 6 days,

according to the table.

7) a. g(3) = 192

b. 1/25 or 0.04

c. 3/16 or 0.1875

d. 625

e. 1.5

8) Neither student is correct.

Jenna made her first mistake by

multiplying the exponent by

the base to make –4.

Computing 22 gives a fraction,

not a negative number. Joua

did not follow the order of

operations and multiplied the

bases, the 4 and 2, together

before doing the power.

The correct answer and work:

( )22 4 2

14

4

1

H −

− = ⋅

=

=

9) a. C(0) = 20(0.75)>� 20(1) = 20feet

b. H(3)= 20(.75)3 = 8.4375 feet

SECTION 3.4

1) a. Exponential growth; 3 > 1,

common ratio is 3.

b. Exponential decay; ½ < 1,

common ratio is ½.

c. Exponential decay; .97 < 1,

common ratio is 0.97.

d. Exponential growth - the

graph is increasing, and the


e. Exponential growth; 1.06>1,


f. Exponential decay; the


1) g. This function is linear

because it has a constant

rate of change of 3,

therefore, it is neither

exponential growth or decay

h. This function is a quadratic

function, therefore, it is

neither exponential growth

or decay.

i. This function has 2

asymptotes, not a

characteristic of exponential

functions. It increases and

decreases over its entire

domain, therefore it is

neither exponential growth

or decay

2) a. exponential growth

b. neither; linear

c. neither; quadratic

d. exponential decay

e. exponential growth

f. exponential decay

3) a.

Interval Rate of

Change

x = 0 to x = 1 1

x = 1 to x = 2 2

x = 2 to x = 3 4

x = 3 to x = 4 8


158 APPENDICES


3) b. The absolute value of the

rate of change is increasing

c. The rate of change is

doubling for every x value

d. The function is increasing

from x = 0 to x = 4.As the

x-values increase so do the

y-values.

4) a.

b. The absolute value of the

rate of change is increasing.

c. The function values are

decreasing and doubling for

every x-value

d. decreasing. As x-values

increase y-values decrease.

5) a.

x y

–2 0.22

–1 0.67

0 2

1 6

2 18

Domain: all real numbers;

Range: y > 0;

Asymptote: y = 0;

y-int: (0, 2)

5) b.

x y

–2 1

–1 –1

0 –2

1 –2.5

2 –2.75


Range: y > –3;

Asymptote: y = –3;

y-int: (0, –2)

6) a.

x y

–1 –13

0 –1

1 2

2 2.75

3 2.94


Range: y < 3;

Asymptote: y = 3

y-intercept: (0, –1)

6) b.

x y

0 –0.0625

1 –0.25

2 –1

3 –4

4 –16


Range: y < 0;

Asymptote: y = 0

y-intercept: (0, –0.0625)

7) a. The original value is

$20,000

b. The asymptote is $80,000.

The maximum value of the

policy is approaching

$80,000.

c. @��3) would be 3 years

before the start. It does not

make sense to answer this

question because we cannot

go back in time. Also, the

value of the policy would be

negative which is

impossible.

8) a. $1000

b. The asymptote of the graph

shows where the maximum

value of the painting will be;

this is at $25,000

c. @(−3) would be 3 years

before the start. It does not

make sense to answer this

question because we cannot

go back in time. Also, the

value of the painting would

be negative which is

impossible.

Interval Rate of

Change

x = 0 to x = 1 –1

x = 1 to x = 2 –2

x = 2 to x = 3 –4

x = 3 to x = 4 –8




9) a. !(0) is the number of

people who are sick at the

beginning which is 2000.

This number is the

y-intercept (0, 2000) on the

graph for the function.

b. !(7) represents the number

of people well on day 7.

This value can be found by

!(7) = 2000(1 − 0.38)2. !(7) = 70.4 or 70 people.

c. 11.08 days. Solve by

graphing

2000(1 − 0.38)3 = 10. Graph the left side of the

equation is a y1 and the right

side of the equation in a y2.

Find where these lines

intersect.

9) d. Domain: � ≥ 0

Range: 0 ≤ � ≤ 2000.

The domain represents time,

and time can never be

negative. The range

represents people and

people can never be

negative but the number of

people in this situation will

always decrease from 2000,

meaning there will never be

more than 2000 people.

10) a. The annual percent increase

of students is 17%.

b. The attendance in 2010 is

15,000, the starting

population. This point

(0, 15,000) is the y-intercept

on the graph.

10) c. !(−2) represents the

student population 2 years

prior to the start (in 2008). It

is feasible to believe that

there were students

attending prior to 2010, but

we do not have enough

information to determine.

Therefore, having a negative

value for time does not

make sense. The value of

!(−2) = 15,000(1.17)+)

!(−2) = 10,957

d. !(10) = 15,000(1.17)(>

!(10) = 72,102 students

Unit 3 Review

1) Linear means that as x (the input)

changes, y (the output) changes

at a constant rate. Its graph will

be a straight line.

Real-life situations

will vary. Example:

Joe starts a game on

the playground, and

3 more people come

every minute to

play. At the

beginning, it is just

Joe, but then after 1 minute,

there are 4 people total, after 2

minutes, there are 7 people... and

so on.

2) An exponential function means

that there will be a constant ratio

between each number. Its graph

will be curved and also have an

asymptote.

Real-life situation

will vary. Example:

Anna has a secret to

tell, and she tells 4

friends. Those 4

friends each tell 4

more friends, and

this pattern

continues. In the

beginning, it was only Anna that

knew the secret. Then after she

tells, 4 new people know the

secret. After the friends tell, 16

new people know the secret…

and so on.

3) a. !(0) = 5

b. !(2) � 2 112G ≈

2.08333…

c. !(−1) = 3(⅙)+( + 2

= 20

4) a. ( )1 1h − =

b. ( )4 0.03125h =

c. ( )14

1 0.25h = =

5) a.

x y

0 60

1 120

2 240

3 480

4 960

5 1920

6 3840

7 7680

8 15360

between 6 and 7 time periods

(half hours), so between 3 and

3.5 hours

b. ( )0,60 There are 60 at time

equals 0 (the start of the

culture).

c. 960 bacteria

d. 15 bacteria; 60 cut in half

twice

x y

–2 –5

–1 –2

0 1

1 4

2 7

3 10

4 13

x y

–2 1/16

–1 ¼

0 1

1 4

2 16

3 64

4 256


160 APPENDICES

Unit 3 Review (continued)

6) a.

# –2 –1 0 1 2 3 4

*�#� 4 2 1 12 1

4 18 1

16

b.

# –2 –1 0 1 2 3 4

*�#� 4 2 1 12 1

4 18 1

16

Difference

between

function

values

–2 –1 12−

14−

18−

116−

# –2 –1 0 1 2 3 4

*�#� 4 2 1 12 1

4 18 1

16

Ratio

between

function

values

12 1

2 12 1

2 12 1

2

c. Common Ratio

Exponential Decay

decreasing

7) a.

# –2 –1 0 1 2 3 4

*�#� 49 23 1 1.5 2.253.3755.063

7) b.

# –2 –1 0 1 2 3 4

*�#� 49 2

3 1 1.5 2.25 3.375 5.063

Difference

between

function

values

0.2

2

0.3

3

0.5

0.7

5

1.2

5

1.6

88

7) b.

# –2 –1 0 1 2 3 4

*�#� 49 2

3 1 1.5 2.25 3.375 5.063

Ratio

between

function

values

1.5

1.5

1.5

1.5

1.5

1.5

c. Common Ratio

Exponential Growth

increasing

8) a.

# –2 –1 0 1 2 3 4

*�#� 0 1 2.8 5.2 8

b.

# –2 –1 0 1 2 3 4

*�#� 0 1 2.8 5.2 8 Difference

between

function

values

1

1.8

2.4

1.5

4

# –2 –1 0 1 2 3 4

*�#� 0 1 2.8 5.2 8 Ratio

between

function

values

2.8

1.8

6

1.5

4

c. neither

neither

increasing

9) a. !�0) = 40,000(0.94)>

= 40,000

b. !(0) represents the starting

value of the truck at the time

Mr. Smith purchased it.

c. !(6) = 40,000(0.94)I

= $27,594.79 d. In 6 years the truck will be

worth $27,594.79.

9) e. Domain: ( ) 0f x ≥

Range: ( )0 40,000f x≤ ≤

The domain represents time,

and time can never be

negative. In this case, it

would represent the value of

the truck before it was new

which is impossible. The

range represents the value of

the truck. The value of the

truck will never be negative

but it will continue to lose

value from $40,000.

10) Graph includes the following

points:

( ) ( ) ( ) ( )1

1, 2 , 0,2 , 1, 1 , 2,1 , 3,52

− − − and has a horizontal asymptote at

� = −3

Growth

Domain: all real numbers

Range: � > −3

Asymptote: � = −3

y-intercept: (0, −2)

11) a. !(1) = −10

b. !(0) = −7

c. !(−1) = −5.5

d. !(−10) = −4.0029

e. !(−100) = −4.000. …

f. No, the curve will get

infinitely close to the line

� = −4 but it will never

touch or cross. This is where

the asymptote occurs, and the

function values will only

approach −4.

g. Domain: all real numbers

h. Range: � < −4

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

x

y

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

x

y

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

x

y

-4 -3 -2 -1 1 2 3 4 5 6

-4

-3

-2

-1

1

2

3

4

5

6

x

y



Unit 3 Review (continued)

12) a. !��1) = �5

b. !(0) = 1

c. !(1) = 2.5

d. !(10) = 2.99

e. !(100) = 2.99

f. The values of the function

will never reach 3 because

there is an asymptote on the

graph at � = 3.

g. Domain: all real numbers

h. Range: � < 3

13) a. 3x =

b. 2x =

c. 0.98x =

d. 8.02x = −

e. 0.095x =

f. 1.09x =

14) a.

n 0 1 2 3 4

M 0 4 8 12 16

14) b. Mother: Linear model

Daughter: Exponential model

c. Domain for Mom: K ≥ 0

Domain for Daughter: K ≥ 0

Range for Mom: L ≥ 0

Range for Daughter: L ≥ 1

d. Mom receives more messages

by day 3, she will have 12

while her daughter only has 8.

e. Mom and daughter have the

same number of messages on

day 4 with 16 messages.

1 2 3 4 5 6

2

4

6

8

10

12

14

16

0 x

y

(4th ed) Preface Table of Contents (Units 0-3) - no CW no WS · 3 (Topic 2) Intercepts and Graphing...

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Transcript of (4th ed) Preface Table of Contents (Units 0-3) - no CW no WS · 3 (Topic 2) Intercepts and Graphing...