Chapter 2 Resource Masters - KTL MATH CLASSES · PDF file©Glencoe/McGraw-Hill v Glencoe...

Chapter 2Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 2 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828005-2 Algebra 2Chapter 2 Resource Masters

2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 2-1Study Guide and Intervention . . . . . . . . . 57–58Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 59Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Reading to Learn Mathematics . . . . . . . . . . . 61Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 62







Chapter 2 AssessmentChapter 2 Test, Form 1 . . . . . . . . . . . . . 99–100Chapter 2 Test, Form 2A . . . . . . . . . . . 101–102Chapter 2 Test, Form 2B . . . . . . . . . . . 103–104Chapter 2 Test, Form 2C . . . . . . . . . . . 105–106Chapter 2 Test, Form 2D . . . . . . . . . . . 107–108Chapter 2 Test, Form 3 . . . . . . . . . . . . 109–110Chapter 2 Open-Ended Assessment . . . . . . 111Chapter 2 Vocabulary Test/Review . . . . . . . 112Chapter 2 Quizzes 1 & 2 . . . . . . . . . . . . . . . 113Chapter 2 Quizzes 3 & 4 . . . . . . . . . . . . . . . 114Chapter 2 Mid-Chapter Test . . . . . . . . . . . . . 115Chapter 2 Cumulative Review . . . . . . . . . . . 116Chapter 2 Standardized Test Practice . . 117–118

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 2 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 2 Resource Masters includes the core materials neededfor Chapter 2. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 2-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 2Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 106–107. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

22

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 2.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

absolute value function

boundary

constant function

family of graphs

function

greatest integer function

identity function

linear equation

line of fit

one-to-one function

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

parent graph

piecewise function

PEES·WYZ

point-slope form

prediction equation

pree·DIHK·shuhn

relation

scatter plot

slope

slope-intercept form

IHN·tuhr·SEHPT

standard form

step function

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

22

Study Guide and InterventionRelations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

© Glencoe/McGraw-Hill 57 Glencoe Algebra 2

Less

on

2-1

Graph Relations A relation can be represented as a set of ordered pairs or as anequation; the relation is then the set of all ordered pairs (x, y) that make the equation true.The domain of a relation is the set of all first coordinates of the ordered pairs, and therange is the set of all second coordinates.A function is a relation in which each element of the domain is paired with exactly oneelement of the range. You can tell if a relation is a function by graphing, then using thevertical line test. If a vertical line intersects the graph at more than one point, therelation is not a function.

Graph the equation y � 2x � 3 and find the domain and range. Doesthe equation represent a function?

Make a table of values to find ordered pairs that satisfy the equation. Then graph the ordered pairs.

The domain and range are both all real numbers. Thegraph passes the vertical line test, so it is function.

Graph each relation or equation and find the domain and range. Then determinewhether the relation or equation is a function.

1. {(1, 3), (�3, 5), 2. {(3, �4), (1, 0), 3. {(0, 4), (�3, �2),(�2, 5), (2, 3)} (2, �2), (3, 2)} (3, 2), (5, 1)}

D � {�3, �2, 1, 2}, D � {1, 2, 3}, D � {�3, 0, 3, 5},R � {3, 5}; yes R � {�4, �2, 0, 2}; no R � {�2, 1, 2, 4}; yes

4. y � x2 � 1 5. y � x � 4 6. y � 3x � 2

D � all reals, D � all reals, D � all reals,R � {yy � �1}; yes R � all reals; yes R � all reals; yes

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ExampleExample

ExercisesExercises


Equations of Functions and Relations Equations that represent functions areoften written in functional notation. For example, y � 10 � 8x can be written as f(x) � 10 � 8x. This notation emphasizes the fact that the values of y, the dependentvariable, depend on the values of x, the independent variable.

To evaluate a function, or find a functional value, means to substitute a given value in thedomain into the equation to find the corresponding element in the range.

Given the function f(x) � x2 � 2x, find each value.

a. f(3)

f(x) � x2 � 2x Original function

f(3) � 32 � 2(3) Substitute.

� 15 Simplify.

b. f(5a)

f(x) � x2 � 2x Original function

f(5a) � (5a)2 � 2(5a) Substitute.

� 25a2 � 10a Simplify.

Find each value if f(x) � �2x � 4.

1. f(12) �20 2. f(6) �8 3. f(2b) �4b � 4

Find each value if g(x) � x3 � x.

4. g(5) 120 5. g(�2) �6 6. g(7c) 343c3 � 7c

Find each value if f(x) � 2x � and g(x) � 0.4x2 � 1.2.

7. f(0.5) 5 8. f(�8) �16 9. g(3) 2.4

10. g(�2.5) 1.3 11. f(4a) 8a � 12. g� � � 1.2

13. f � � 6 14. g(10) 38.8 15. f(200) 400.01

Let f(x) � 2x2 � 1.

16. Find the values of f(2) and f(5). f (2) � 7, f (5) � 49

17. Compare the values of f(2) � f(5) and f(2 � 5). f (2) � f (5) � 343, f (2 � 5) � 199

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Study Guide and Intervention (continued)

Relations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

ExampleExample

ExercisesExercises

Skills PracticeRelations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1


Less

on

2-1

Determine whether each relation is a function. Write yes or no.

1. yes 2. no

3. yes 4. no


5. {(2, �3), (2, 4), (2, �1)} 6. {(2, 6), (6, 2)}

D � {2}, R � {�3, �1, 4}; no D � {2, 6}, R � {2, 6}; yes

7. {(�3, 4), (�2, 4), (�1, �1), (3, �1)} 8. x � �2

D � {�3, �2, �1, 3}, D � {�2}, R � all reals; no R � {�1, 4}; yes

Find each value if f(x) � 2x � 1 and g(x) � 2 � x2.

9. f(0) �1 10. f(12) 23 11. g(4) �14

12. f(�2) �5 13. g(�1) 1 14. f(d) 2d � 1

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Determine whether each relation is a function. Write yes or no.

1. no 2. yes

3. yes 4. no


5. {(�4, �1), (4, 0), (0, 3), (2, 0)} 6. y � 2x � 1

D � {�4, 0, 2, 4}, D � all reals, R � all reals; yesR � {�1, 0, 3}; yes

Find each value if f(x) � and g(x) � �2x � 3.

7. f(3) 1 8. f(�4) � 9. g� � 2

10. f(�2) undefined 11. g(�6) 15 12. f(m � 2)

13. MUSIC The ordered pairs (1, 16), (2, 16), (3, 32), (4, 32), and (5, 48) represent the cost ofbuying various numbers of CDs through a music club. Identify the domain and range ofthe relation. Is the relation a function? D � {1, 2, 3, 4, 5}, R � {16, 32, 48}; yes

14. COMPUTING If a computer can do one calculation in 0.0000000015 second, then thefunction T(n) � 0.0000000015n gives the time required for the computer to do ncalculations. How long would it take the computer to do 5 billion calculations? 7.5 s

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Practice (Average)

Relations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Reading to Learn MathematicsRelations and Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1


Less

on

2-1

Pre-Activity How do relations and functions apply to biology?

Read the introduction to Lesson 2-1 at the top of page 56 in your textbook.

• Refer to the table. What does the ordered pair (8, 20) tell you? For adeer, the average longevity is 8 years and the maximumlongevity is 20 years.

• Suppose that this table is extended to include more animals. Is it possibleto have an ordered pair for the data in which the first number is largerthan the second? Sample answer: No, the maximum longevitymust always be greater than the average longevity.

Reading the Lesson

1. a. Explain the difference between a relation and a function. Sample answer: Arelation is any set of ordered pairs. A function is a special kind ofrelation in which each element of the domain is paired with exactlyone element in the range.

b. Explain the difference between domain and range. Sample answer: The domainof a relation is the set of all first coordinates of the ordered pairs. Therange is the set of all second coordinates.

2. a. Write the domain and range of the relation shown in the graph.

D: {�3, �2, �1, 0, 3}; R: {�5, �4, 0, 1, 2, 4}

b. Is this relation a function? Explain. Sample answer: No, it is not a functionbecause one of the elements of the domain, 3, is paired with twoelements of the range.

Helping You Remember

3. Look up the words dependent and independent in a dictionary. How can the meaning ofthese words help you distinguish between independent and dependent variables in afunction? Sample answer: The variable whose values depend on, or aredetermined by, the values of the other variable is the dependent variable.

(0, 4)

(3, 1)

(3, –4)(–1, –5)

(–2, 0)

(–3, 2)

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MappingsThere are three special ways in which one set can be mapped to another. A setcan be mapped into another set, onto another set, or can have a one-to-onecorrespondence with another set.

State whether each set is mapped into the second set, onto the second set, or has a one-to-one correspondence with the second set.

1. 2. 3. 4.

into, onto into, onto into, onto, into, ontoone-to-one

5. 6. 7. 8.

into into, onto into, onto into, onto,one-to-one

9. Can a set be mapped onto a set with fewer elements than it has? yes

10. Can a set be mapped into a set that has more elements than it has? yes

11. If a mapping from set A into set B is a one-to-one correspondence, what can you conclude about the number of elements in A and B?The sets have the same number of elements.

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Into mappingA mapping from set A to set B where every element of A is mapped to one or more elements of set B, but never to an element not in B.

Onto mappingA mapping from set A to set B where each element of set B has at least one element of set A mapped to it.

One-to-one A mapping from set A onto set B where each element of set A is mapped to exactly one correspondence element of set B and different elements of A are never mapped to the same element of B.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-12-1

Study Guide and InterventionLinear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

on

2-2

Identify Linear Equations and Functions A linear equation has no operationsother than addition, subtraction, and multiplication of a variable by a constant. Thevariables may not be multiplied together or appear in a denominator. A linear equation doesnot contain variables with exponents other than 1. The graph of a linear equation is a line.

A linear function is a function whose ordered pairs satisfy a linear equation. Any linearfunction can be written in the form f(x) � mx � b, where m and b are real numbers.

If an equation is linear, you need only two points that satisfy the equation in order to graphthe equation. One way is to find the x-intercept and the y-intercept and connect these twopoints with a line.

Is f(x) � 0.2 � alinear function? Explain.

Yes; it is a linear function because it canbe written in the formf(x) � � x � 0.2.

Is 2x � xy � 3y � 0 alinear function? Explain.

No; it is not a linear function becausethe variables x and y are multipliedtogether in the middle term.

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x�5

Find the x-intercept and they-intercept of the graph of 4x � 5y � 20.Then graph the equation.

The x-intercept is the value of x when y � 0.

4x � 5y � 20 Original equation

4x � 5(0) � 20 Substitute 0 for y.

x � 5 Simplify.

So the x-intercept is 5.Similarly, the y-intercept is �4. x

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Example 1Example 1 Example 3Example 3

Example 2Example 2

ExercisesExercises

State whether each equation or function is linear. Write yes or no. If no, explain.

1. 6y � x � 7 yes 2. 9x � No; the 3. f(x) � 2 � yes

variable y appears in the denominator.

Find the x-intercept and the y-intercept of the graph of each equation. Then graphthe equation.

4. 2x � 7y � 14 5. 5y � x � 10 6. 2.5x � 5y � 7.5 � 0

x-int: 7; y-int: 2 x-int: �10; y-int: 2 x-int: �3; y-int: 1.5

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Standard Form The standard form of a linear equation is Ax � By � C, where A, B, and C are integers whose greatest common factor is 1.

Write each equation in standard form. Identify A, B, and C.


Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

ExampleExample

a. y � 8x � 5

y � 8x � 5 Original equation

�8x � y � �5 Subtract 8x from each side.

8x � y � 5 Multiply each side by �1.

So A � 8, B � �1, and C � 5.

b. 14x � �7y � 21

14x � �7y � 21 Original equation

14x � 7y � 21 Add 7y to each side.

2x � y � 3 Divide each side by 7.

So A � 2, B � 1, and C � 3.

ExercisesExercises


1. 2x � 4y �1 2. 5y � 2x � 3 3. 3x � �5y � 22x � 4y � �1; A � 2, 2x � 5y � �3; A � 2, 3x � 5y � 2; A � 3,B � �4, C � �1 B � �5, C � �3 B � 5, C � 2

4. 18y � 24x � 9 5. y � x � 5 6. 6y � 8x � 10 � 0

8x � 6y � 3; A � 8, 8x � 9y � �60; A � 8, 4x � 3y � 5; A � 4,B � �6, C � 3 B � �9, C � �60 B � �3, C � 5

7. 0.4x � 3y � 10 8. x � 4y � 7 9. 2y � 3x � 62x � 15y � 50; A � 2, x � 4y � �7; A � 1, 3x � 2y � �6; A � 3,B � 15, C � 50 B � �4, C� �7 B � �2, C � �6

10. x � y �2 � 0 11. 4y � 4x � 12 � 0 12. 3x � �18

6x � 5y � 30; A � 6, x � y � �3; A � 1, x � �6; A � 1,B � 5, C � 30 B � 1, C � �3 B � 0, C � �6

13. x � � 7 14. 3y � 9x � 18 15. 2x � 20 � 8y

9x � y � 63; A � 9, 3x � y � 6; A � 3, x � 4y � 10; A � 1,B � �1, C � 63 B � �1, C � 6 B � 4, C � 10

16. � 3 � 2x 17. � � � y � 8 18. 0.25y � 2x � 0.75

8x � y � �12; A � 8, 10x � 3y � 32; A � 10, 8x � y � 3; A � 8,B � �1, C� �12 B � �3, C � 32 B � �1, C � 3

19. 2y� � 4 � 0 20. 1.6x � 2.4y � 4 21. 0.2x � 100 � 0.4y

x � 12y � �24; A � 1, 2x � 3y � 5; A � 2, x � 2y � 500; A � 1,B � �12, C � �24 B � �3, C � 5 B � 2, C � 500

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Skills PracticeLinear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

on

2-2

State whether each equation or function is linear. Write yes or no. If no, explainyour reasoning.

1. y � 3x 2. y � �2 � 5x

yes yes

3. 2x � y � 10 4. f(x) � 4x2

yes No; the exponent of x is not 1.

5. � � y � 15 6. x � y � 8

No; x is in a denominator. yes

7. g(x) � 8 8. h(x) � �x� � 3

yes No; x is inside a square root.


9. y � x x � y � 0; 1, �1, 0 10. y � 5x � 1 5x � y � �1; 5, �1, �1

11. 2x � 4 � 7y 2x � 7y � 4; 2, 7, 4 12. 3x � �2y � 2 3x � 2y � �2; 3, 2, �2

13. 5y � 9 � 0 5y � 9; 0, 5, 9 14. �6y � 14 � 8x 4x � 3y � 7; 4, 3, 7


15. y � 3x � 6 2, �6 16. y � �2x 0, 0

17. x � y � 5 5, 5 18. 2x � 5y � 10 5, 2

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State whether each equation or function is linear. Write yes or no. If no, explainyour reasoning.

1. h(x) � 23 yes 2. y � x yes

3. y � No; x is a denominator. 4. 9 � 5xy � 2 No; x and y are multiplied.


5. y � 7x � 5 7x � y � 5; 7, �1, 5 6. y � x � 5 3x � 8y � �40; 3, �8, �40

7. 3y � 5 � 0 3y � 5; 0, 3, 5 8. x � � y � 28x � 8y � 21; 28, 8, 21


9. y � 2x � 4 �2, 4 10. 2x � 7y � 14 7, 2

11. y � �2x � 4 �2, �4 12. 6x � 2y � 6 1, 3

13. MEASURE The equation y � 2.54x gives the length in centimeters corresponding to alength x in inches. What is the length in centimeters of a 1-foot ruler? 30.48 cm

LONG DISTANCE For Exercises 14 and 15, use the following information.

For Meg’s long-distance calling plan, the monthly cost C in dollars is given by the linearfunction C(t) � 6 � 0.05t, where t is the number of minutes talked.

14. What is the total cost of talking 8 hours? of talking 20 hours? $30; $66

15. What is the effective cost per minute (the total cost divided by the number of minutestalked) of talking 8 hours? of talking 20 hours? $0.0625; $0.055

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3�4

2�7

3�8

5�x

2�3

Practice (Average)

Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

Reading to Learn MathematicsLinear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2


Less

on

2-2

Pre-Activity How do linear equations relate to time spent studying?


• If Lolita spends 2 hours studying math, how many hours will she have

to study chemistry? 1 hours• Suppose that Lolita decides to stay up one hour later so that she now has

5 hours to study and do homework. Write a linear equation that describesthis situation. x � y � 5

Reading the Lesson

1. Write yes or no to tell whether each linear equation is in standard form. If it is not,explain why it is not.

a. �x � 2y � 5 No; A is negative.

b. 9x � 12y � �5 yes

c. 5x � 7y � 3 yes

d. 2x � y � 1 No; B is not an integer.

e. 0x � 0y � 0 No; A and B are both 0.

f. 2x � 4y � 8 No; The greatest common factor of 2, 4, and 8 is 2, not 1.

2. How can you use the standard form of a linear equation to tell whether the graph is ahorizontal line or a vertical line? If A � 0, then the graph is a horizontal line. IfB � 0, then the graph is a vertical line.


3. One way to remember something is to explain it to another person. Suppose that you are studying this lesson with a friend who thinks that she should let x � 0 to find the x-intercept and let y � 0 to find the y-intercept. How would you explain to her how toremember the correct way to find intercepts of a line? Sample answer: The x-intercept is the x-coordinate of a point on the x-axis. Every point on the x-axis has y-coordinate 0, so let y � 0 to find an x-intercept. The y-intercept is the y-coordinate of a point on the y-axis. Every point on the y-axis has x-coordinate 0, so let x � 0 to find a y-intercept.

4�7

1�2

1�2


Greatest Common FactorSuppose we are given a linear equation ax � by � c where a, b, and c are nonzerointegers, and we want to know if there exist integers x and y that satisfy theequation. We could try guessing a few times, but this process would be timeconsuming for an equation such as 588x � 432y � 72. By using the EuclideanAlgorithm, we can determine not only if such integers x and y exist, but also find them. The following example shows how this algorithm works.

Find integers x and y that satisfy 588x � 432y � 72.

Divide the greater of the two coefficients by the lesser to get a quotient andremainder. Then, repeat the process by dividing the divisor by the remainderuntil you get a remainder of 0. The process can be written as follows.

588 � 432(1) � 156 (1)432 � 156(2) � 120 (2)156 � 120(1) � 36 (3)120 � 36(3) � 12 (4)36 � 12(3)

The last nonzero remainder is the GCF of the two coefficients. If the constantterm 72 is divisible by the GCF, then integers x and y do exist that satisfy theequation. To find x and y, work backward in the following manner.

72 � 6 � 12� 6 � [120 � 36(3)] Substitute for 12 using (4)

� 6(120) � 18(36)� 6(120) � 18[156 � 120(1)] Substitute for 36 using (3)

� �18(156) � 24(120)� �18(156) � 24[432 � 156(2)] Substitute for 120 using (2)

� 24(432) � 66(156)� 24(432) � 66[588 � 432(1)] Substitute for 156 using (1)

� 588(�66) � 432(90)

Thus, x � �66 and y � 90.

Find integers x and y, if they exist, that satisfy each equation.

1. 27x � 65y � 3 2. 45x � 144y � 36

3. 90x � 117y � 10 4. 123x � 36y � 15

5. 1032x � 1001y � 1 6. 3125x � 3087y � 1

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-22-2

ExampleExample

Study Guide and InterventionSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3


Less

on

2-3

Slope

Slope m of a Line For points (x1, y1) and (x2, y2), where x1 � x2, m � �y2 � y1�x2 � x1

change in y��change in x

Determine the slope ofthe line that passes through (2, �1) and(�4, 5).

m � Slope formula

� (x1, y1) � (2, �1), (x2, y2) � (�4, 5)

� � �1 Simplify.

The slope of the line is �1.

6��6

5 � (�1)��4 � 2

y2 � y1�x2 � x1

Graph the line passingthrough (�1, �3) with a slope of .

Graph the ordered pair (�1, �3). Then,according to the slope, go up 4 unitsand right 5 units.Plot the new point(4,1). Connect thepoints and draw the line.

x

y

O

4�5


ExercisesExercises

Find the slope of the line that passes through each pair of points.

1. (4, 7) and (6, 13) 3 2. (6, 4) and (3, 4) 0 3. (5, 1) and (7, �3) �2

4. (5, �3) and (�4, 3) � 5. (5, 10) and (�1,�2) 2 6. (�1, �4) and (�13, 2) �

7. (7, �2) and (3, 3) � 8. (�5, 9) and (5, 5) � 9. (4, �2) and (�4, �8)

Graph the line passing through the given point with the given slope.

10. slope � � 11. slope � 2 12. slope � 0

passes through (0, 2) passes through (1, 4) passes through (�2, �5)

13. slope � 1 14. slope � � 15. slope �

passes through (�4, 6) passes through (�3, 0) passes through (0, 0)

x

y

O

x

y

O

x

y

O

1�5

3�4

x

y

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x

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y

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1�3

3�4

2�5

5�4

1�2

2�3


Parallel and Perpendicular Lines


Slope

NAME ______________________________________________ DATE______________ PERIOD _____

2-32-3

In a plane, nonvertical lines with thesame slope are parallel. All verticallines are parallel.

x

y

O

slope � m

slope � m

In a plane, two oblique lines are perpendicular ifand only if the product of their slopes is �1. Anyvertical line is perpendicular to any horizontal line.

x

y

O

slope � m

slope � � 1m

ExampleExample Are the line passing through (2, 6) and (�2, 2) and the line passingthrough (3, 0) and (0, 4) parallel, perpendicular, or neither?

Find the slopes of the two lines.

The slope of the first line is � 1.

The slope of the second line is � � .

The slopes are not equal and the product of the slopes is not �1, so the lines are neitherparallel nor perpendicular.

Are the lines parallel, perpendicular, or neither?

1. the line passing through (4, 3) and (1, �3) and the line passing through (1, 2) and (�1, 3)perpendicular

2. the line passing through (2, 8) and (�2, 2) and the line passing through (0, 9) and (6, 0)neither

3. the line passing through (3, 9) and (�2, �1) and the graph of y � 2x parallel

4. the line with x-intercept �2 and y-intercept 5 and the line with x-intercept 2 and y-intercept �5 parallel

5. the line with x-intercept 1 and y-intercept 3 and the line with x-intercept 3 and y-intercept 1 neither

6. the line passing through (�2, �3) and (2, 5) and the graph of x � 2y � 10perpendicular

7. the line passing through (�4, �8) and (6, �4) and the graph of 2x � 5y � 5 parallel

4�3

4 � 0�0 � 3

6 � 2��2 � (�2)

ExercisesExercises

Skills PracticeSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3


Less

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1. (1, 5), (�1, �3) 4 2. (0, 2), (3, 0) � 3. (1, 9), (0, 6) 3

4. (8, �5), (4, �2) � 5. (�3, 5), (�3, �1) undefined 6. (�2, �2), (10, �2) 0

7. (4, 5), (2, 7) �1 8. (�2, �4), (3, 2) 9. (5, 2), (�3, 2) 0


10. (0, 4), m � 1 11. (2, �4), m � �1

12. (�3, �5), m � 2 13. (�2, �1), m � �2

Graph the line that satisfies each set of conditions.

14. passes through (0, 1), perpendicular to 15. passes through (0, �5), parallel to the

a line whose slope is graph of y � 1

16. HIKING Naomi left from an elevation of 7400 feet at 7:00 A.M. and hiked to an elevationof 9800 feet by 11:00 A.M. What was her rate of change in altitude? 600 ft /h

x

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x

y

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1�3

x

y

Ox

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O

x

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x

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3�4

2�3



1. (3, �8), (�5, 2) � 2. (�10, �3), (7, 2) 3. (�7, �6), (3, �6) 0

4. (8, 2), (8, �1) undefined 5. (4, 3), (7, �2) � 6. (�6, �3), (�8, 4) �


7. (0, �3), m � 3 8. (2, 1), m � �

9. (0, 2), m � 0 10. (2, �3), m �

Graph the line that satisfies each set of conditions.

11. passes through (3, 0), perpendicular 12. passes through (�3, �1), parallel to a line

to a line whose slope is whose slope is �1

DEPRECIATION For Exercises 13–15, use the following information.A machine that originally cost $15,600 has a value of $7500 at the end of 3 years. The samemachine has a value of $2800 at the end of 8 years.

13. Find the average rate of change in value (depreciation) of the machine between itspurchase and the end of 3 years. �$2700 per year

14. Find the average rate of change in value of the machine between the end of 3 years andthe end of 8 years. �$940 per year

15. Interpret the sign of your answers. It is negative because the value is decreasing.

x

y

O

x

y

O

3�2

xO

y

x

y

O

4�5

x

y

O

x

y

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3�4

7�2

5�3

5�17

5�4

Practice (Average)

Slope

NAME ______________________________________________ DATE______________ PERIOD _____

2-32-3

Reading to Learn MathematicsSlope

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3


Less

on

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Pre-Activity How does slope apply to the steepness of roads?


• What is the grade of a road that rises 40 feet over a horizontal distanceof 1000 feet? 4%

• What is the grade of a road that rises 525 meters over a horizontaldistance of 10 kilometers? (1 kilometer � 1000 meters) 5.25%

Reading the Lesson

1. Describe each type of slope and include a sketch.

Type of Slope Description of Graph Sketch

Positive The line rises to the right.

Zero The line is horizontal.

Negative The line falls to the right.

Undefined The line is vertical.

2. a. How are the slopes of two nonvertical parallel lines related? They are equal.

b. How are the slopes of two oblique perpendicular lines related? Their product is �1.


3. Look up the terms grade, pitch, slant, and slope. How can everyday meanings of thesewords help you remember the definition of slope? Sample answer: All these wordscan be used when you describe how much a thing slants upward ordownward. You can describe this numerically by comparing rise to run.


Aerial Surveyors and AreaMany land regions have irregular shapes. Aerial surveyors supply aerial mappers with lists of coordinates and elevations for the areas that need to be photographed from the air. These maps provide information about the horizontal and vertical features of the land.

Step 1 List the ordered pairs for the vertices in counterclockwise order, repeating the first ordered pair at the bottom of the list.

Step 2 Find D, the sum of the downward diagonal products (from left to right).D � (5 � 5) � (2 � 1) � (2 � 3) � (6 � 7)

� 25 � 2 � 6 � 42 or 75

Step 3 Find U, the sum of the upward diagonal products (from left to right).U � (2 � 7) � (2 � 5) � (6 � 1) � (5 � 3)

� 14 � 10 � 6 � 15 or 45

Step 4 Use the formula A � �12�(D � U) to find the area.

A � �12�(75 � 45)

� �12�(30) or 15

The area is 15 square units. Count the number of square units enclosed by the polygon. Does this result seem reasonable?

Use the coordinate method to find the area of each region in square units.

1. 2. 3.

x

y

O

x

y

Ox

y

O

(5, 7)�

(2, 5)�

(2, 1)�

(6, 3)�

(5, 7)

x

y

O

(2, 1)

(2, 5)

(5, 7)

(6, 3)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-32-3

Study Guide and InterventionWriting Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4


Less

on

2-4

Forms of Equations

Slope-Intercept Form of a Linear Equation

y � mx � b, where m is the slope and b is the y-intercept

Point-Slope Form y � y1 � m(x � x1), where (x1, y1) are the coordinates of a point on the line and of a Linear Equation m is the slope of the line

Write an equation inslope-intercept form for the line thathas slope �2 and passes through thepoint (3, 7).

Substitute for m, x, and y in the slope-intercept form.

y � mx � b Slope-intercept form

7 � (�2)(3) � b (x, y ) � (3, 7), m � �2

7 � �6 � b Simplify.

13 � b Add 6 to both sides.

The y-intercept is 13. The equation in slope-intercept form is y � �2x � 13.

Write an equation inslope-intercept form for the line thathas slope and x-intercept 5.

y � mx � b Slope-intercept form

0 � � �(5) � b (x, y ) � (5, 0), m �

0 � � b Simplify.

� � b Subtract from both sides.

The y-intercept is � . The slope-intercept

form is y � x � .5�3

1�3

5�3

5�3

5�3

5�3

1�3

1�3

1�3


ExercisesExercises

Write an equation in slope-intercept form for the line that satisfies each set ofconditions.

1. slope �2, passes through (�4, 6) 2. slope , y-intercept 4

y � �2x � 2 y � x � 4

3. slope 1, passes through (2, 5) 4. slope � , passes through (5, �7)

y � x � 3 y � � x � 6

Write an equation in slope-intercept form for each graph.

5. 6. 7.

y � �3x � 9 y � x y � x � 1 4�9

1�9

5�4

x

y

O

(–4, 1)(5, 2)

x

y

O

(4, 5)

(0, 0)

x

y

O

(1, 6)

(3, 0)

13�5

13�5

3�2

3�2


Parallel and Perpendicular Lines Use the slope-intercept or point-slope form to findequations of lines that are parallel or perpendicular to a given line. Remember that parallellines have equal slope. The slopes of two perpendicular lines are negative reciprocals, thatis, their product is �1.


Writing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4

Write an equation of theline that passes through (8, 2) and isperpendicular to the line whose equation is y � � x � 3.

The slope of the given line is � . Since the

slopes of perpendicular lines are negativereciprocals, the slope of the perpendicularline is 2.Use the slope and the given point to writethe equation.y � y1 � m(x � x1) Point-slope form

y � 2 � 2(x � 8) (x1, y1) � (8, 2), m � 2

y � 2 � 2x � 16 Distributive Prop.

y � 2x � 14 Add 2 to each side.

An equation of the line is y � 2x � 14.

1�2

1�2

Write an equation of theline that passes through (�1, 5) and isparallel to the graph of y � 3x � 1.

The slope of the given line is 3. Since theslopes of parallel lines are equal, the slopeof the parallel line is also 3.Use the slope and the given point to writethe equation.y �y1 � m(x � x1) Point-slope form

y � 5 � 3(x � (�1)) (x1, y1) � (�1, 5), m � 3

y � 5 � 3x � 3 Distributive Prop.

y � 3x � 8 Add 5 to each side.

An equation of the line is y � 3x � 8.


ExercisesExercises


1. passes through (�4, 2), parallel to the line whose equation is y � x � 5 y � x � 4

2. passes through (3, 1), perpendicular to the graph of y � �3x � 2 y � x

3. passes through (1, �1), parallel to the line that passes through (4, 1) and (2, �3)y � 2x � 3

4. passes through (4, 7), perpendicular to the line that passes through (3, 6) and (3, 15)y � 7

5. passes through (8, �6), perpendicular to the graph of 2x � y � 4 y � � x � 2

6. passes through (2, �2), perpendicular to the graph of x � 5y � 6 y � 5x � 12

7. passes through (6, 1), parallel to the line with x-intercept �3 and y-intercept 5

y � x � 9

8. passes through (�2, 1), perpendicular to the line y � 4x � 11 y � � x � 1�2

1�4

5�3

1�2

1�3

1�2

1�2

Skills PracticeWriting Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4


Less

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State the slope and y-intercept of the graph of each equation.

1. y � 7x � 5 7, �5 2. y � � x � 3 � , 3

3. y � x , 0 4. 3x � 4y � 4 � , 1

5. 7y � 4x � 7 , �1 6. 3x � 2y � 6 � 0 , 3

7. 2x � y � 5 2, �5 8. 2y � 6 � 5x � , 3


9. 10. 11.

y � 3x � 1 y � �1 y � �2x � 3


12. slope 3, passes through (1, �3) 13. slope �1, passes through (0, 0)

y � 3x � 6 y � �x

14. slope �2, passes through (0, �5) 15. slope 3, passes through (2, 0)

y � �2x � 5 y � 3x � 6

16. passes through (�1, �2) and (�3, 1) 17. passes through (�2, �4) and (1, 8)

y � � x � y � 4x � 4

18. x-intercept 2, y-intercept �6 19. x-intercept , y-intercept 5

y � 3x � 6 y � �2x � 5

20. passes through (3, �1), perpendicular to the graph of y � � x � 4. y � 3x � 101�3

5�2

7�2

3�2

x

y

O

(0, 3)

(3, –3)

x

y

O(–3, –1) (4, –1)

x

y

O

(–1, –4)

(1, 2)

5�2

3�2

4�7

3�4

2�3

2�3

3�5

3�5


State the slope and y-intercept of the graph of each equation.

1. y � 8x � 12 8, 12 2. y � 0.25x � 1 0.25, �1 3. y � � x � , 0

4. 3y � 7 0, 5. 3x � �15 � 5y , 3 6. 2x � 3y � 10 , �


7. 8. 9.

y � 2 y � x � 2 y � � x � 1


10. slope �5, passes through (�3, �8) 11. slope , passes through (10, �3)

y � �5x � 23 y � x � 11

12. slope 0, passes through (0, �10) 13. slope � , passes through (6, �8)

y � �10 y � � x � 4

14. passes through (3, 11) and (�6, 5) 15. passes through (7, �2) and (3, �1)

y � x � 9 y � � x �

16. x-intercept 3, y-intercept 2 17. x-intercept �5, y-intercept 7

y � � x � 2 y � x � 7

18. passes through (�8, �7), perpendicular to the graph of y � 4x � 3 y � � x � 9

19. RESERVOIRS The surface of Grand Lake is at an elevation of 648 feet. During thecurrent drought, the water level is dropping at a rate of 3 inches per day. If this trendcontinues, write an equation that gives the elevation in feet of the surface of Grand Lakeafter x days. y � �0.25x � 648

20. BUSINESS Tony Marconi’s company manufactures CD-ROM drives. The company willmake $150,000 profit if it manufactures 100,000 drives, and $1,750,000 profit if itmanufactures 500,000 drives. The relationship between the number of drivesmanufactured and the profit is linear. Write an equation that gives the profit P when n drives are manufactured. P � 4n � 250,000

1�4

7�5

2�3

1�4

1�4

2�3

2�3

2�3

4�5

4�5

2�3

3�2

x

y

O(3, –1)

(–3, 3)

x

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O

(4, 4)

(0, –2)

x

y

O

(0, 2)

10�3

2�3

3�5

7�3

3�5

3�5

Practice (Average)

Writing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4

Reading to Learn MathematicsWriting Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4


Less

on

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Pre-Activity How do linear equations apply to business?


• If the total cost of producing a product is given by the equation y � 5400 � 1.37x, what is the fixed cost? What is the variable cost (for each item produced)? $5400; $1.37

• Write a linear equation that describes the following situation:A company that manufactures computers has a fixed cost of $228,750 anda variable cost of $852 to produce each computer.y � 228,750 � 852x

Reading the Lesson

1. a. Write the slope-intercept form of the equation of a line. Then explain the meaning ofeach of the variables in the equation. y � mx � b; m is the slope and b is they-intercept. The variables x and y are the coordinates of any point onthe line.

b. Write the point-slope form of the equation of a line. Then explain the meaning of eachof the variables in the equation. y � y1 � m(x � x1); m is the slope. x and yare the coordinates of any point on the line. x1 and y1 are the coordinates of one specific point on the line.

2. Suppose that your algebra teacher asks you to write the point-slope form of the equationof the line through the points (�6, 7) and (�3, �2). You write y � 2 � �3(x � 3) andyour classmate writes y � 7 � �3(x � 6). Which of you is correct? Explain. You areboth correct. Either point may be used as (x1, y1) in the point-slope form.You used (�3, �2), and your classmate used (�6, 7).

3. You are asked to write an equation of two lines that pass through (3, �5), one of themparallel to and one of them perpendicular to the line whose equation is y � �3x � 4.The first step in finding these equations is to find their slopes. What is the slope of theparallel line? What is the slope of the perpendicular line? �3;


4. Many students have trouble remembering the point-slope form for a linear equation.How can you use the definition of slope to remember this form? Sample answer:

Write the definition of slope: m � . Multiply both sides of this

equation by x2 � x1. Drop the subscripts in y2 and x2. This gives thepoint-slope form of the equation of a line.

y2 � y1�x2 � x1

1�3


Two-Intercept Form of a Linear EquationYou are already familiar with the slope-intercept form of a linear equation,

y � mx � b. Linear equations can also be written in the form �ax

� � �by

� � 1 with x-intercept a and y-intercept b. This is called two-intercept form.

Draw the graph of ��

x3�

� �6y

� � 1.

The graph crosses the x-axis at �3 and the y-axis at 6. Graph (�3, 0) and (0, 6), then draw a straight line through them.

Write 3x � 4y � 12 in two-intercept form.

�132x� � �1

42y� � �

1122� Divide by 12 to obtain 1 on the right side.

�4x

� � �3y

� � 1 Simplify.

The x-intercept is 4; the y-intercept is 3.

Use the given intercepts a and b, to write an equation in two-intercept form. Then draw the graph. See students’ graphs.

1. a � �2, b � �4 2. a � 1, b � 8

3. a � 3, b � 5 4. a � 6, b � 9

Write each equation in two-intercept form. Then draw the graph.

5. 3x � 2y � �6 6. �12�x � �

14�y � 1 7. 5x � 2y � �10

x

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Ox

y

Ox

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x

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-42-4

Example 1Example 1

Example 2Example 2

Study Guide and InterventionModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

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Scatter Plots When a set of data points is graphed as ordered pairs in a coordinateplane, the graph is called a scatter plot. A scatter plot can be used to determine if there isa relationship among the data.

BASEBALL The table below shows the number of home runs andruns batted in for various baseball players who won the Most Valuable PlayerAward during the 1990s. Make a scatter plot of the data.

Source: New York Times Almanac

Make a scatter plot for the data in each table below.

1. FUEL EFFICIENCY The table below shows the average fuel efficiency in miles per gallon of new cars manufactured during the years listed.

Source: New York Times Almanac

2. CONGRESS The table below shows the number of women serving in the United States Congress during the years 1987�1999.

Source: Wall Street Journal Almanac

Congressional Session Number of Women

100 25

101 31

102 33

103 55

104 58

105 62

Session of Congress

Nu

mb

er o

f W

om

en

100 102 104

70

56

42

28

14

0

Women in Congress

Year Fuel Efficiency (mpg)

1960 15.5

1970 14.1

1980 22.6

1990 26.9 Year

Mile

s p

er G

allo

n

1960 1970 1980 1990

36

30

24

18

12

6

0

Average Fuel Efficiency

Home Runs

MVP HRs and RBIs

Ru

ns

Bat

ted

In

1260 24 3618 30 42 48

150

125

100

75

50

25

Home Runs Runs Batted In

33 114

39 116

40 130

28 61

41 128

47 144

ExampleExample

ExercisesExercises


Prediction Equations A line of fit is a line that closely approximates a set of datagraphed in a scatter plot. The equation of a line of fit is called a prediction equationbecause it can be used to predict values not given in the data set.

To find a prediction equation for a set of data, select two points that seem to represent thedata well. Then to write the prediction equation, use what you know about writing a linearequation when given two points on the line.

STORAGE COSTS According to a certain prediction equation, thecost of 200 square feet of storage space is $60. The cost of 325 square feet ofstorage space is $160.

a. Find the slope of the prediction equation. What does it represent?Since the cost depends upon the square footage, let x represent the amount of storagespace in square feet and y represent the cost in dollars. The slope can be found using the

formula m � . So, m � � � 0.8

The slope of the prediction equation is 0.8. This means that the price of storage increases80¢ for each one-square-foot increase in storage space.

b. Find a prediction equation.Using the slope and one of the points on the line, you can use the point-slope form to finda prediction equation.

y � y1 � m(x � x1) Point-slope form

y � 60 � 0.8(x � 200) (x1, y1) � (200, 60), m � 0.8

y � 60 � 0.8x � 160 Distributive Property

y � 0.8x � 100 Add 60 to both sides.

A prediction equation is y � 0.8x � 100.

SALARIES The table below shows the years of experience for eight technicians atLewis Techomatic and the hourly rate of pay each technician earns. Use the datafor Exercises 1 and 2.

Experience (years) 9 4 3 1 10 6 12 8

Hourly Rate of Pay $17 $10 $10 $7 $19 $12 $20 $15

1. Draw a scatter plot to show how years of experience are related to hourly rate of pay. Draw a line of fit. See graph.

2. Write a prediction equation to show how years of experience(x) are related to hourly rate of pay (y). Sample answerusing (1, 7) and (9, 17): y � 1.25x � 5.75

Experience (years)

Ho

url

y Pa

y ($

)

20 6 104 8 12 14

24

20

16

12

8

4

Technician Salaries

100�125

160 � 60��325 � 200

y2 � y1�x2 � x1


Modeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

ExampleExample

ExercisesExercises

Skills PracticeModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


Less

on

2-5

For Exercises 1–3, complete parts a–c for each set of data.

a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.

1. 1a.

1b. Sample answer using (1, 1) and (8, 15): y � 2x � 11c. Sample answer: 19

2. 2a.

2b. Sample answer using (5, 9) and (40, 44): y � x � 42c. Sample answer: 54

3. 3a.

3b. Sample answer using (2, 16) and (7, 34): y � 3.6x � 8.83c. Sample answer: 19.6

1 3 5 72 4 6 8

36

30

24

18

12

6

0 x

yx y

1 16

2 16

3 ?

4 22

5 30

7 34

8 36

5 15 25 3510 20 30 40

40

32

24

16

8

0 x

yx y

5 9

10 17

20 22

25 30

35 38

40 44

50 ?

1 3 5 72 4 6 8

15

12

9

6

3

0 x

yx y

1 1

3 5

4 7

6 11

7 12

8 15

10 ?


For Exercises 1–3, complete parts a–c for each set of data.a. Draw a scatter plot.b. Use two ordered pairs to write a prediction equation.c. Use your prediction equation to predict the missing value.

1. FUEL ECONOMY The table gives the approximate weights in tons and estimates for overall fuel economy in miles per gallon for several cars.1b. Sample answer using (1.4, 24) and

(2.4, 15): y � �9x � 36.6

1c. Sample answer: 18.6 mi/gal

2. ALTITUDE In most cases, temperature decreases with increasing altitude. As Ancharadrives into the mountains, her car thermometer registers the temperatures (°F) shownin the table at the given altitudes (feet).

2b. Sample answer using (7500, 61) and (9700, 50): y � �0.005x � 98.5

2c. Sample answer: 38.5°F

3. HEALTH Alton has a treadmill that uses the time on the treadmill and the speed of walking or running to estimate the number of Calories he burns during a workout. Thetable gives workout times and Calories burned for several workouts.

3b. Sample answer using (24, 280) and(48, 440): y � 6.67x � 119.92

3c. Sample answer: about 520 calories

Time (min) 18 24 30 40 42 48 52 60

Calories Burned 260 280 320 380 400 440 475 ?

Altitude (ft)

Tem

per

atu

re (�

F)

0 7,000 8,000 9,000 10,000

65

60

55

50

45

TemperatureVersus Altitude

Altitude (ft) 7500 8200 8600 9200 9700 10,400 12,000

Temperature (�F) 61 58 56 53 50 46 ?

Weight (tons)

Fuel

Eco

no

my

(mi/

gal

)

0 0.5 1.0 1.5 2.0 2.5

30

25

20

15

10

5

Fuel Economy Versus Weight

Weight (tons) 1.3 1.4 1.5 1.8 2 2.1 2.4

Miles per Gallon 29 24 23 21 ? 17 15

Practice (Average)

Modeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

Reading to Learn MathematicsModeling Real-World Data: Using Scatter Plots

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5


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on

2-5

Pre-Activity How can a linear equation model the number of Calories you burnexercising?


• If a woman runs 5.5 miles per hour, about how many Calories will sheburn in an hour? Sample answer: 572 Calories

• If a man runs 7.5 miles per hour, about how many Calories will he burnin half an hour? Sample answer: 397 Calories

Reading the Lesson

1. Suppose that a set of data can be modeled by a linear equation. Explain the differencebetween a scatter plot of the data and a graph of the linear equation that models thatdata.Sample answer: The scatter plot is a discrete graph. It is made up just ofthe individual points that represent the data points. The linear equationhas a continuous graph that is the line that best fits the data points.

2. Suppose that tuition at a state college was $3500 per year in 1995 and has beenincreasing at a rate of $225 per year.

a. Write a prediction equation that expresses this information.y � 3500 � 225x

b. Explain the meaning of each variable in your prediction equation.x represents the number of year since 1995 and y represents thetuition in that year.

3. Use this model to predict the tuition at this college in 2007. $6200


4. Look up the word scatter in a dictionary. How can its definition help you to rememberthe meaning of the difference between a scatter plot and the graph of a linear equation?Sample answer: To scatter means to break up and go in many directions.The points on a scatter plot are broken up. In a scatter plot, the pointsare scattered or broken up. In the graph of a linear equation, the pointsare connected to form a continuous line.


Median-Fit Lines A median-fit line is a particular type of line of fit. Follow the steps below to find the equation of the median-fit line for the data.

Approximate Percentage of Violent Crimes Committed by Juveniles That Victims Reported to Law Enforcement

Year 1980 1982 1984 1986 1988 1990 1992 1994 1996

Offenders 36 35 33 32 31 30 29 29 30

Source: U.S. Bureau of Justice Statistics

1. Divide the data into three approximately equal groups. There should always be the same number of points in the first and third groups. In this case, there will be three data points in each group.

Group 1 Group 2 Group 3enders

2. Find x1, x2, and x3, the medians of the x values in groups 1, 2, and 3,respectively. Find y1, y2, and y3, the medians of the y values in groups 1, 2, and 3, respectively. 1982, 1988, 1994; 35, 31, 29

3. Find an equation of the line through (x1, y1) and (x3, y3). y � �0.5x � 1026

4. Find Y, the y-coordinate of the point on the line in Exercise 2 with an x-coordinate of x2. 32

5. The median-fit line is parallel to the line in Exercise 2, but is one-third

closer to (x2, y2). This means it passes through �x2, �23�Y � �

13�y2�. Find this

ordered pair. about (1988, 31.67)

6. Write an equation of the median-fit line. y � �0.5x � 1025.67

7. Use the median-fit line to predict the percentage of juvenile violent crime offenders in 2010 and 2020. 2010: about 21%; 2020: about16%

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-52-5

Study Guide and InterventionSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


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on

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Step Functions, Constant Functions, and the Identity Function The chartbelow lists some special functions you should be familiar with.

Function Written as Graph

Constant f(x) � c horizontal line

Identity f(x) � x line through the origin with slope 1

Greatest Integer Function f(x) � �x�one-unit horizontal segments, with right endpoints missing, arranged like steps

The greatest integer function is an example of a step function, a function with a graph thatconsists of horizontal segments.

Identify each function as a constant function, the identity function,or a step function.

a. b.

a constant function a step function

Identify each function as a constant function, the identity function, a greatestinteger function, or a step function.

1. 2. 3.

a constant function a step function the identity function

x

f(x)

Ox

f(x)

Ox

f(x)

O

x

f(x)

Ox

f(x)

O

ExampleExample

ExercisesExercises


Absolute Value and Piecewise Functions Another special function is theabsolute value function, which is also called a piecewise function.

Absolute Value Function f(x ) � x two rays that are mirror images of each other and meet at a point, the vertex

To graph a special function, use its definition and your knowledge of the parent graph. Findseveral ordered pairs, if necessary.

Graph f(x) � 3x � 4.

Find several ordered pairs. Graph the points andconnect them. You would expect the graph to looksimilar to its parent function, f(x) � x .

Graph f(x) � �2x if x � 2x � 1 if x � 2.

First, graph the linear function f(x) � 2x for x � 2. Since 2 does notsatisfy this inequality, stop with a circle at (2, 4). Next, graph thelinear function f(x) � x � 1 for x � 2. Since 2 does satisfy thisinequality, begin with a dot at (2, 1).

Graph each function. Identify the domain and range.

1. g(x) � � � 2. h(x) � 2x � 1 3. h(x ) �

domain: all real domain: all real domain: all real numbers; range: numbers; range: numbers; range:all integers {yy � 0} {yy 1}

x

y

O

x

y

O

x

y

O

x�3

x

f(x)

O

x

f(x)

O

x 3x � 4

0 �4

1 �1

2 2

�1 �1

�2 2


Special Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

ExercisesExercises

Example 1Example 1

Example 2Example 2

if x 0

2x � 6 if 0 � x � 21 if x � 2

x�3

Skills PracticeSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


Less

on

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Identify each function as S for step, C for constant, A for absolute value, or P forpiecewise.

1. 2. 3.

S C A


4. f(x) � �x � 1� 5. f(x) � �x � 3�

D � all reals, R � all integers D � all reals, R � all integers

6. g(x) � 2 x 7. f(x) � x � 1

D � all reals, D � all reals, R � {yy � 1}R � nonnegative reals

8. f(x) � �x if x � 09. h(x) � �3 if x � �1

2 if x � 0 x � 1 if x > 1

D � all reals, D � {xx � �1 or x 1},R � {yy � 0 or y � 2} R � {yy 2}

x

h(x)

O

x

f(x)

O

x

f(x)

Ox

g(x)

O

x

f(x)

O

x

f(x)

O

x

y

O

x

y

Ox

y

O



1. f(x) � �0.5x� 2. f(x) � �x� � 2

D � all reals, R � all integers D � all reals, R � all integers3. g(x) � �2 x 4. f(x) � x � 1

D � all reals, D � all reals,R � nonpositive reals R � nonnegative reals

5. f(x) � �x � 2 if x � 26. h(x) � �4 � x if x 0

3x if x �2 �2x � 2 if x � 0

D � all reals, R � all reals D � all nonzero reals, R � all reals7. BUSINESS A Stitch in Time charges 8. BUSINESS A wholesaler charges a store $3.00

$40 per hour or any fraction thereof per pound for less than 20 pounds of candy andfor labor. Draw a graph of the step $2.50 per pound for 20 or more pounds. Draw afunction that represents this situation. graph of the function that represents this

situation.

Hours

Tota

l Co

st (

$)

10 3 52 4 6 7

280

240

200

160

120

80

40

Labor Costs

x

f(x)

O

x

g(x)

O

x

f(x)

Ox

f(x)

O

Practice (Average)

Special Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

Reading to Learn MathematicsSpecial Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6


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on

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Pre-Activity How do step functions apply to postage rates?


• What is the cost of mailing a letter that weighs 0.5 ounce?$0.34 or 34 cents

• Give three different weights of letters that would each cost 55 cents tomail. Answers will vary. Sample answer: 1.1 ounces,1.9 ounces, 2.0 ounces

Reading the Lesson

1. Find the value of each expression.

a. �3 � ��3� �

b. 6.2 � �6.2� �

c. �4.01 � ��4.01� �

2. Tell how the name of each kind of function can help you remember what the graph looks like.

a. constant function Sample answer: Something is constant if it does notchange. The y-values of a constant function do not change, so thegraph is a horizontal line.

b. absolute value function Sample answer: The absolute value of a numbertells you how far it is from 0 on the number line. It makes no differencewhether you go to the left or right so long as you go the samedistance each time.

c. step function Sample answer: A step function’s graph looks like stepsthat go up or down.

d. identity function Sample answer: The x- and y-values are alwaysidentically the same for any point on the graph. So the graph is a linethrough the origin that has slope 1.


3. Many students find the greatest integer function confusing. Explain how you can use anumber line to find the value of this function for any real number. Answers will vary.Sample answer: Draw a number line that shows the integers. To find thevalue of the greatest integer function for any real number, place thatnumber on the number line. If it is an integer, the value of the function isthe number itself. If not, move to the integer directly to the left of thenumber you chose. This integer will give the value you need.

�54.01

66.2

�33


Greatest Integer FunctionsUse the greatest integer function � x� to explore some unusual graphs. It will be helpful to make a chart of values for each functions and to use a colored pen or pencil.

Graph each function.

1. y � 2x � � x� 2. y � ��xx

��

�

3. y � ��00..55xx

�

�

11

��

� 4. y � ��xx��

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

x

y

O 1–1–2–3–4 2 3 4

4

3

2

1

–1

–2

–3

–4

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-62-6

Study Guide and InterventionGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7

Graph Linear Inequalities. A linear inequality, like y � 2x � 1, resembles a linearequation, but with an inequality sign instead of an equals sign. The graph of the relatedlinear equation separates the coordinate plane into two half-planes. The line is theboundary of each half-plane.

To graph a linear inequality, follow these steps.

1. Graph the boundary, that is, the related linear equation. If the inequality symbol is or �, the boundary is solid. If the inequality symbol is � or , the boundary is dashed.

2. Choose a point not on the boundary and test it in the inequality. (0, 0) is a good point tochoose if the boundary does not pass through the origin.

3. If a true inequality results, shade the half-plane containing your test point. If a falseinequality results, shade the other half-plane.

Graph x � 2y � 4.

The boundary is the graph of x � 2y � 4.

Use the slope-intercept form, y � � x � 2, to graph the boundary line.

The boundary line should be solid.

Now test the point (0, 0).

0 � 2(0) �? 4 (x, y ) � (0, 0)

0 � 4 false

Shade the region that does not contain (0, 0).

Graph each inequality.

1. y � 3x � 1 2. y � x � 5 3. 4x � y �1

4. y � � 4 5. x � y 6 6. 0.5x � 0.25y � 1.5

x

y

O

x

y

O

x

y

O

x�2

x

y

O

x

y

O

x

y

O

1�2

x

y

O

ExercisesExercises

ExampleExample


Graph Absolute Value Inequalities Graphing absolute value inequalities is similarto graphing linear inequalities. The graph of the related absolute value equation is theboundary. This boundary is graphed as a solid line if the inequality is or �, and dashed ifthe inequality is � or . Choose a test point not on the boundary to determine which regionto shade.

Graph y 3x � 1.

First graph the equation y � 3 x � 1 .Since the inequality is , the graph of the boundary is solid.Test (0, 0).0 ? 3 0 � 1 (x, y) � (0, 0)

0 ? 3 �1 �1 � 1

0 3 true

Shade the region that contains (0, 0).


1. y � x � 1 2. y 2x � 1 3. y � 2 x 3

4. y � � x � 3 5. x � y � 4 6. x � 1 � 2y � 0

7. 2 � x � y �1 8. y � 3 x � 3 9. y 1 � x � 4

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

Ox

y

Ox

y

O

x

y

O


Graphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

ExercisesExercises

ExampleExample

Skills PracticeGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


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on

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1. y 1 2. y x � 2 3. x � y 4

4. x � 3 � y 5. 2 � y � x 6. y � �x

7. x � y �2 8. 9x � 3y � 6 0 9. y � 1 � 2x

10. y � 7 �9 11. x �5 12. y x

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

x

y

O

x

y

Ox

y

O



1. y �3 2. x 2 3. x � y �4

4. y � �3x � 5 5. y � x � 3 6. y � 1 � �x

7. x � 3y 6 8. y x � 1 9. y �3 x � 1 � 2

COMPUTERS For Exercises 10–12, use the following information.

A school system is buying new computers. They will buy desktop computers costing $1000 per unit, andnotebook computers costing $1200 per unit. The total cost of the computers cannot exceed $80,000.

10. Write an inequality that describes this situation.1000d � 1200n 80,000

11. Graph the inequality.

12. If the school wants to buy 50 of the desktop computers and 25 of the notebook computers,will they have enough money? yes

Desktops

No

teb

oo

ks

100 30 5020 40 60 70 80 90 100

80

70

60

50

40

30

20

10

Computers Purchased

x

y

Ox

y

O

x

y

O

1�2

x

y

O

x

y

O

x

y

O

Practice (Average)

Graphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Reading to Learn MathematicsGraphing Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7


Less

on

2-7

Pre-Activity How do inequalities apply to fantasy football?


• Which of the combinations of yards and touchdowns listed would Danaconsider a good game? The first one: 168 yards and 3 touchdowns

• Suppose that in one of the games Dana plays, Moss gets 157 receivingyards. What is the smallest number of touchdowns he must get in orderfor Dana to consider this a good game? 3

Reading the Lesson

1. When graphing a linear inequality in two variables, how do you know whether to makethe boundary a solid line or a dashed line? If the symbol is � or , the line issolid. If the symbol is or �, the line is dashed.

2. How do you know which side of the boundary to shade? Sample answer: If the testpoint gives a true inequality, shade the region containing the test point. Ifthe test point gives a false inequality, shade the region not containingthe test point.

3. Match each inequality with its graph.

a. y 2x � 3 iii b. y � �2x � 3 iv c. y � 2x � 3 ii d. y � �2x � 3 i

i. ii. iii. iv.


4. Describe some ways in which graphing an inequality in one variable on a number line issimilar to graphing an inequality in two variables in a coordinate plane. How can whatyou know about graphing inequalities on a number line help you to graph inequalities ina coordinate plane? Sample answer: A boundary on a coordinate graph issimilar to an endpoint on a number line graph. A dashed line is similar toa circle on a number line: both are open and mean not included; theyrepresent the symbols and �. A solid line is similar to a dot on anumber line: both are closed and mean included; they represent thesymbols � and .

x

y

O

x

y

Ox

y

O

x

y

O


Algebraic ProofThe following paragraph states a result you might be asked to prove in amathematics course. Parts of the paragraph are numbered.

01 Let n be a positive integer.

02 Also, let n1 � s(n1) be the sum of the squares of the digits in n.

03 Then n2 � s(n1) is the sum of the squares of the digits of n1, and n3 � s(n2)is the sum of the squares of the digits of n2.

04 In general, nk � s(nk � 1) is the sum of the squares of the digits of nk � 1.

05 Consider the sequence: n, n1, n2, n3, …, nk, ….

06 In this sequence either all the terms from some k on have the value 1,

07 or some term, say nj, has the value 4, so that the eight terms 4, 16, 37, 58, 89, 145, 42, and 20 keep repeating from that point on.

Use the paragraph to answer these questions.

1. Use the sentence in line 01. List the first five values of n.

2. Use 9246 for n and give an example to show the meaning of line 02.

3. In line 02, which symbol shows a function? Explain the function in a sentence.

4. For n � 9246, find n2 and n3 as described in sentence 03.

5. How do the first four sentences relate to sentence 05?

6. Use n � 31 and find the first four terms of the sequence.

7. Which sentence of the paragraph is illustrated by n � 31?

8. Use n � 61 and find the first ten terms.

9. Which sentence is illustrated by n � 61?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

2-72-7

Write the letter for the correct answer in the blank at the right of each question. 1. Find the domain of the relation {(0, 0), (1, 1), (2, 0)}. Then determine

whether the relation is a function.A. {0, 1, 0}; function B. {0, 1, 0}; not a functionC. {0, 1, 2}; function D. {0, 1, 2}; not a function 1.

2. The table shows the annualized percent return of a mutual fund for several years. Find the range of the relation. Then determine whether the relation is a function.

A. {20.9, 22.8, 20.0, 20.5}; not a function B. {1, 3, 5, 10}; not a functionC. {20.9, 22.8, 20.0, 20.5}; function D. {1, 3, 5, 10}; function 2.

3. Find f(�1) if f(x) � �3x � 5.A. �9 B. �8 C. �2 D. 2 3.

4. Find f(0) if f(t) � t2 � 2t � 2.A. 2 B. �4 C. 0 D. �2 4.

5. Which equation is linear?A. xy � 60 B. 3x � 2y � 5C. y � x2 � 3x � 1 D. y2 � 1 � x 5.

6. Which function is a linear function?A. f(x) � x3 � x B. g(s) � 1 � 4s

C. h(t) � 2t � �1t�

D. f(r) � �r� 6.

7. Write y � 4x � 7 in standard form.A. 4x � y � �7 B. 4x � y � 7 C. y � 4x � 7 D. 4x � y � 7 7.

8. Find the x-intercept of the graph of �5x � 10y � 20.A. �2 B. 2 C. 4 D. �4 8.

9. Find the slope of the line that passes through (0, 2) and (8, 8).

A. 8 B. �43� C. �

34� D. �

54� 9.

10. If a line rises to the right, its slope is ___?____.A. zero B. positive C. negative D. undefined 10.

11. What is the slope of a line that is perpendicular to the graph of y � 2x � 5?

A. ��12� B. �

12� C. 2 D. �2 11.

12. Graph the line through (2, 3) that is parallel to the line with equation y � �1. Which point below also lies on that line?A. (2, 9) B. (�5, 3) C. (0, 1) D. (1, 4) 12.

Chapter 2 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

22

Year 1 3 5 10

Percent Return 20.9 22.8 20.0 20.5


Chapter 2 Test, Form 1 (continued)

13. Write an equation in slope-intercept form for the line that has a slope of

��45� and passes through (0, 7).

A. y � 7x B. y � 7x � �45� C. y � �

45�x � 7 D. y � ��

45�x � 7 13.

14. Write an equation for the line that passes through (0, 1) and is perpendicular to the line whose equation is y � 2x.

A. y � �2x � 1 B. y � 2x � 1 C. y � �12�x � 1 D. y � ��

12�x � 1 14.

15. Use a scatter plot to determine which data point is an outlier.A. (0, 2) B. (1, 3) C. (2, 10) D. (8, 18) 15.

16. The scatter plot shows the area of the floor and the price for certain tents.Which equation could be a prediction equation for this set of data?A. y � x � 50 B. y � 10x � 25C. y � 5x � 50 D. y � 5x � 22 16.

17. A banquet hall has tables that can seat 8 people. The number of tables needed depends on the number of guests.What type of special function models this situation?A. linear function B. step functionC. absolute value function D. constant function 17.

18. Identify the range of y � � x �.A. all real numbers B. {x � x � 0}C. {y � y � 0} D. {y � y � 0} 18.

19. The graph of the linear inequality y � 2x � 1 is the region __?___ the graph of the line y � 2x � 1.A. on or above B. on or below C. above D. below 19.

20. Which inequality is graphed at the right?A. y � � x � � 3 B. y � � x � � 3

C. y � � x � � 3 D. y � � x � � 3 20.

Bonus Find the value of k so that the slope of the line through (4, 2)

and (k, 3) is �16�. B:

NAME DATE PERIOD

22

y

xO 6

50100150200250300350400450500

12 18 24 30 36 42 48 54 60

Pric

e ($

)

Floor Area (ft2)

y

xO

x 0 1 2 5 8

y 2 3 10 12 18

Chapter 2 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question.

1. Find the range of the relation {(�2, 3), (�1, 3), (�1, 5)}. Then determine whether the relation is a function.A. {�2, �1}; function B. {�2, �1}; not a functionC. {3, 5}; function D. {3, 5}; not a function 1.

2. Find f(�1) if f(x) � �xx2

��

24

�.

A. �5 B. �3 C. 1 D. 3 2.

3. Find f(a) if f(t) � t2 � 2t � 2.A. (t � a)2 � 2t � a � 2 B. (t � a)2 � 2(t � a) � 2C. a2 � 2t � 2 D. a2 � 2a � 2 3.

4. Which equation is linear?

A. y � x � 2 B. y � x2 C. y � 3 D. y2 � �12�x � 1 4.

5. Write 3y � �1 � 5x in standard form.A. 5x � 3y � �1 B. �5x � 3y � �1

C. y � ��53�x � 1 D. 3x � 5y � 1 � 0 5.

6. Find the x-intercept and the y-intercept of the graph of 3x � 2y � 12.A. (4, �6) B. 4; �6 C. (2, �3) D. �6; 4 6.

7. Find the slope of the line that passes through (2, 6) and (�7, 8).

A. ��52� B. ��

25� C. ��

29� D. ��

92� 7.

8. What is the slope of the line y � �2?

A. �2 B. 0 C. �12� D. undefined 8.

9. What is the slope of a line that is parallel to the graph of 2x � 3y � 5?

A. �32� B. ��

23� C. �

23� D. ��

32� 9.

10. The graph of the line through (2, 3) that is perpendicular to the line with equation y � �1 also goes through which point?A. (0, 1) B. (1, 4) C. (2, �4) D. (�2, 3) 10.

11. Write an equation in slope-intercept form for the line that has a slope of �4 and passes through (1, 2).A. y � �2x � 4 B. y � �4x � 6 C. y � �4x � 2 D. y � �4x � 9 11.

12. Write an equation in slope-intercept form for the line that passes through (1, �2) and (3, 7).

A. y � �92�x � �

123� B. y � �

92�x � �

527� C. y � �

29�x � �

193� D. y � �

29�x � �

139� 12.

22


Chapter 2 Test, Form 2A (continued)

13. Write an equation for the line that passes through (0, 5) and is parallel to the line whose equation is 4x � y � 3.

A. y � ��14�x � 5 B. y � 4x � 3 C. y � �

14�x � 5 D. y � 4x � 5 13.

14. The table shows the relationship between height and growing times for 8 plants of the same species. Use a scatter plot to determine which data point is an outlier.

A. (15, 6) B. (17, 14) C. (20, 18) D. (25, 24) 14.

15. Which equation could be a prediction equation for the data points shown in the scatter plot at the right?

A. y � �74�x � 400 B. y � �

151�x � 650

C. y � 5x � 600 D. y � �32�x � 800 15.

16. Evaluate f��34�� if f(x) � �1 � 2x�.

A. 0 B. �2 C. �1 D. 1 16.

17. Identify the range of y � � x � � 4.A. {x � x � 4} B. {y � y � �4}C. {y � y � 0} D. all real numbers 17.

18. Which is not part of the definition of the piecewise function shown?A. �3 if x � �2B. x � 2 if � 2 � x � 1C. x � 3 if x � �2D. �x � 1 if x � 1 18.

19. The graph of the linear inequality y � ��23�x � 2 is the region ___?___ the

graph of y � ��23�x � 2.

A. above B. below C. on or above D. on or below 19.

20. Which point satisfies the inequality y � � �x � 3 �?A. (3, 6) B. (�2, 4) C. (5, 7) D. (1, 4) 20.

Bonus Find the value of k so that the slope of the line through B:(2, �k) and (�1, 4) is �3.

NAME DATE PERIOD

22

y

xO

y

xO

200400600800

100012001400160018002000

150 300 450 600 750

Pric

e ($

)

Hard Drive Size (MB)

Height (inches) 15 17 18 19 20 22 23 25

Growing Time (weeks) 6 14 16 17 18 21 23 24

Chapter 2 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Find the range of the relation {(�1, 4), (2, 5), (3, 5)}. Then determine whether the relation is a function.A. {�1, 2, 3}; function B. {�1, 2, 3}; not a functionC. {4, 5}; function D. {4, 5}; not a function 1.

2. Find f(�1) if f(x) � �xx2 �

�62x

�.

A. �5 B. ��53� C. �

73� D. 7 2.

3. Find f(a) if f(t) � 2t2 � t � 2.A. 2(t � a)2 � 2t � a � 2 B. 2(t � a)2 � 2(t � a) � 2C. 2a2 � a � 2 D. 4a2 � 2a � 2 3.

4. Which equation is linear?

A. x � �2 B. y � 3x2 � 1 C. y � 5x � 2 D. y2 � �12�x � 3 4.

5. Write �3y � �1 � 5x in standard form.

A. �5x � 3y � 1 B. 5x � 3y � 1 C. y � ��53�x � 1 D. 3x � 5y � 1 � 0 5.

6. Find the x-intercept and the y-intercept of the graph of 4x � 2y � 8.A. (2, �4) B. �4; 2 C. (4, �2) D. 2; �4 6.

7. Find the slope of a line that passes through (2, 4) and (�7, 8).

A. ��49� B. ��

45� C. �

54� D. ��

94� 7.

8. What is the slope of the line x � �2?

A. �2 B. 0 C. �12� D. undefined 8.

9. What is the slope of a line that is parallel to the graph of 2x � 3y � 6?

A. �32� B. ��

23� C. �

23� D. ��

32� 9.

10. The graph of the line through (2, 3) that is perpendicular to the line with equation x � �1 also goes through which point?A. (0, �1) B. (�2, 3) C. (2, �4) D. (1, 4) 10.

11. Write an equation in slope-intercept form for the line that has a slope of 3 and passes through (�1, 2).A. y � 3x � 1 B. y � 3x � 5 C. y � 5x � 3 D. y � 3x � 5 11.

12. Write an equation in slope-intercept form for the line that passes through (�1, �2) and (3, �7).

A. y � �54�x � �

34� B. y � ��

45�x � �

65� C. y � �

45�x � �

65� D. y � ��

54�x � �

143� 12.

22


Chapter 2 Test, Form 2B (continued)

13. Write an equation for the line that passes through (0, �2) and is parallel to the line whose equation is 3x � 5y � 3.

A. y � ��35�x � 2 B. y � 3x � 2 C. y � �

35�x � 2 D. y � �3x � 2 13.

14. The table shows the relationship between the number of hours practiced and the number of free throws made by 6 players. Use a scatter plot to determine which data point is an outlier.

A. (1, 0) B. (3, 4) C. (7, 16) D. (12, 18) 14.

15. Which equation could be a prediction equation for the data points shown in the scatter plot at the right?A. y � 10x � 6

B. y � ��110�

x � 6

C. y � x � 6

D. y � �110�

x � 6 15.

16. Evaluate f��34�� if f(x) � �2x � 1�.

A. 1 B. �3 C. �1 D. �2 16.

17. Identify the domain of y � 3� x � 2 �.A. all real numbers B. {x � x � 2}C. {y � y � 0} D. {y � y � 2} 17.

18. Which is not part of the definition of the piecewise function shown?A. 2 if x � �1B. x � 1 if �1 � x � 1C. �x � 1 if �1 � x � 1D. 2x if x � 1 18.

19. The graph of the linear inequality y � 3x � 1 is the region ___?___ the graph of y � 3x � 1.A. above B. below C. on or above D. on or below 19.

20. Which point satisfies the inequality y � �� x � 2 �?A. (�1, �1) B. (1, 0) C. (�4, �3) D. (3, 2) 20.

Bonus Find the value of k so that the slope of the line through B:(2, �k) and (�1, 4) is 4.

NAME DATE PERIOD

22

y

xO

1224364860728496

108120

200

Source: The World Almanac

400 600 800 1000

Men

's W

orl

d R

eco

rd (

seco

nd

s)

Distance Run (meters)

y

xO

Hours Practiced 1 3 4 6 7 12

Free Throws Made 0 4 6 9 16 18

Chapter 2 Test, Form 2C


1. Graph the relation {(�3, 3), (�3, 2), (�3, 1), (�3, 0)} and 1.find the domain and range. Then determine whether the relation is a function.

Determine whether each relation is a function.

2. 3.

Find each value if f(x) � 10x � 3x2 and g(x) � 5x2 � 8x.

4. f(�3) 5. g(a)

For Questions 6 and 7, state whether each equation or function is linear. If no, explain your reasoning.

6. f(x) � �x �1

3� 7. y � 3x � 10

8. Write the equation �52�x � 9 � 8y in standard form. Identify

A, B, and C.

9. Find the x-intercept and the y-intercept of the graph of 3y � 2x � 6.

For Questions 10–13, graph each equation or inequality.

10. y � 3x � 2

11. f(x) � �

12. x � 2 � �12�y

1 � x if x � 23 if x � 2

y

xO

y

xO

NAME DATE PERIOD

SCORE 22

Ass

essm

ent

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. y

xO

xO

f (x )

y

xO

y

xO


Chapter 2 Test, Form 2C (continued)

13. y � � �2x � 2 �

14. Find the slope of the line that passes through (�7, 9) and (�6, �5).

15. Graph the line passing through (2, 4) that is perpendicular to the graph of y � �3.

16. Write an equation in slope-intercept form for the line that has a slope of 2 and passes through (1, �5).

17. Write an equation for the line that passes through (�2, 3) and is parallel to the line whose equation is 3x � 2y � 6.

For Questions 18 and 19, use the set of data in the table.

The table below shows the relationship between the number of field goals attempted and the number of points scored by one basketball player over a 6-game period.

18. Draw a scatter plot for the data.

19. Use two ordered pairs to write a prediction equation. Then use your prediction equation to predict the number of points scored when 20 field goals are attempted.

20. Determine whether the graph represents a step function,a constant function, the identity function, an absolute value function, or a piecewise function.Then identify the domain and range.

Bonus Find the value of k so that the slope of the line through B:(2, �k) and (�1, 4) is 1.

NAME DATE PERIOD

22

13.

14.

15.

16.

17.

18.

19.

20.

p

a80

91011121314151617

5 6 7 8 9 10 11 12 13 14

Poin

ts S

core

d

Field Goals Attempted

y

xO

y

xO

xO

y

Field Goals Attempted (a) 8 6 10 9 7 10

Points Scored (p) 12 9 14 14 11 15

Chapter 2 Test, Form 2D


1. Graph the relation {(0, 0), (2, 4), (�4, 0), (4, 0)} and find the domain and range. Then determine whether the relation is a function.

Determine whether each relation is a function.

2. 3.

Find each value if f(x) � �3x � 2x2 and g(x) � �4x2 � 2x � 3.

4. f(�2) 5. g(a)

For Questions 6 and 7, state whether each equation or function is linear. If no, explain your reasoning.

6. f(x) � 100x � 37 7. xy � 60 � 0

8. Write �2x7� 1� � 8y in standard form. Identify A, B, and C.

9. Find the x-intercept and the y-intercept of the graph of 4y � 12 � 3x.

For Questions 10–13, graph each equation or inequality.

10. 3y � 2x � 9

11. f(x) � �

12. x � 2y � 4

�2 if x � �2x � 3 if x � �2

y

xO

y

xO

NAME DATE PERIOD

SCORE 22

Ass

essm

ent

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. y

xO

xO

f (x )

y

xO

y

xO


Chapter 2 Test, Form 2D (continued)

13. y � � x � 1 �

14. Find the slope of the line that passes through (2, 18) and (4, �2).

15. Graph the line passing through (3, �2) that is perpendicular to the graph of x � �3.

16. Write an equation in slope-intercept form for the line that has a slope of �1 that passes through (�4, 3).

17. Write an equation for the line that passes through (2, �5) and is parallel to the line whose equation is 5x � 2y � 6.

For Questions 18 and 19, use the set of data in the table.

The table below shows the relationship between the number of phone calls made and the number of tickets sold during a fundraising campaign by 6 callers.

18. Draw a scatter plot for the data.

19. Use two ordered pairs to write a prediction equation.Then use your prediction equation to predict the number of tickets sold when 16 calls are made.

20. Determine whether the graph represents a step function,a constant function, the identity function, an absolute value function, or a piecewise function.Then identify the domain and range.

Bonus Find the value of k so that the slope of the line through B:(2, �k) and (4, �1) is �2.

NAME DATE PERIOD

22

xO

f(x )

13.

14.

15.

16.

17.

18.

19.

20.

t

n0

12141618202224262830

6 7 8 9 10 11 12 13 14 15

Tick

ets

Sold

Calls Made

y

xO

y

xO

Calls Made (n) 8 9 7 8 6 12

Tickets Sold (t) 16 17 15 15 12 25

Chapter 2 Test, Form 3


1. Graph the relation x � 1 � y2 and find the domain and range. Then determine whether the relation is a function.

2. Determine whether the 2.relation shown at the right is a function.

3. Find f(�2) if f(x) � �2� 1x3

�

�

x42x�

�. 3.

4. If f(2b � 1) � 6b � 2, find f(x). 4.

5. State whether each equation or function is linear. 5.

A. f(x) � �3x �

514

� B. 3x � xy � y

6. Write �2.5x3� 0.3� � �

16� y in standard form. Identify A, B, and C. 6.

7. Find the x-intercept and the y-intercept of the graph of 7.2( y � 0.5) � 3.5x � 2y.

8. Determine whether the graph at the 8.right represents a step function, a constant function, an absolute value function, or a piecewise function.

For Questions 9 and 10, graph each equation.

9. 2y � 1 � 0.8x

10. y � ��12�x � 1�

11. Determine the value of t so that the line through (1.6, t)

and (2, 5) has slope ��32�.

12. The median weekly earnings for American workers in 1990 was $412 and in 1999 it was $549. Calculate the average rate of change between 1990 and 1999.

NAME DATE PERIOD

SCORE 22

Ass

essm

ent

y

xO

1.

9.

10.

11.

12.

y

xO

y

xO

y

xO

xO

f(x )


Chapter 2 Test, Form 3 (continued)

13. Write an equation for the line that passes through (5, �4) and is perpendicular to the graph of 5x � 2y � �6.

14. Write an equation in slope-intercept form of the line

through ��14�, �

53�� and ��

12�, ��

43��.

15. Sweets Bakery charges $12 for each pie and $15 for each cake. Yesterday, the bakery took in no more than $360 for sales of pies and cakes. Write an inequality to represent the situation, where p is the number of pies sold and c is the number of cakes sold. Then graph the inequality.

16. Write an equation in standard form for the line that is

perpendicular to the graph of �15�x � �

25�y � 0.05 and has the

same y-intercept as the graph of �0.8x � 1.2y � �0.6.

For Questions 17 and 18, use the set of data in the table.The table below shows the relationship between distance traveled and elapsed time.

17. Draw a scatter plot for the data. Then identify any outliers.

18. Use two ordered pairs to write a prediction equation.Then use your prediction equation to predict the time for a distance of 160 kilometers. Compare your prediction to the one given in the table.

19. Write an absolute value inequality for the graph at the right.

20. Write the function shown in the graph at the right.

Bonus Find the value of k so that the graph of kx � 3y � 4 is parallel to the line through (2, �k) and (4, �1).

NAME DATE PERIOD

22

y

xO

xO

f (x )

13.

14.

15.

16.

17.

18.

19.

t

d0

20406080

100120140160

60 100 140 180 220Ti

me

(min

)Distance (km)

c

p

Distance d (km) 40 75 110 150 160 200

Time t (min) 30 60 80 110 150 150

20.

B:

Chapter 2 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. Explain two ways to determine whether a relation is a function.Use specific examples. Then write a relation that is not afunction.

2. Give an example of a real-world situation for which there wouldbe a negative rate of change.

3. The point-slope form of the equation of a line is y � 2 � �12�(x � 6).

Write the equation in slope-intercept form, then write theequation in standard form. Which of the three forms of theequation is most useful? Explain your choice.

4. Suppose you are looking at a scatter plot and the graph of a lineof fit for the data points. The horizontal axis is labeled 1990,1991, …, 2000. The vertical axis is labeled 0, 10, …, 100. You usea prediction equation to predict values for the years 1994 and2005. Which prediction do you think would be more accurate?Why ?

5. Compare the graph of the parent function f(x) � � x � with thegraphs of the functions g(x) � � x � 2 � and h(x) � � x � 3 �. How arethe graphs similar? How are they different? How would thegraph of y � � x � 500 � compare with the graph of the parentfunction?

6. When graphing the linear inequality y � �2 � 5, Alessia firstgraphed the line y � �2x � 5. She then selected the test point(�1, 7) in order to complete her graph. Why did Alessia need atest point? What information did the point (�1, 7) give Alessiaabout her graph?

7. Is the graph of the relation y � � x � 3 � a function? Explain.

NAME DATE PERIOD

SCORE 22

Ass

essm

ent


Chapter 2 Vocabulary Test/Review

Write the letter of the term that best describes each example.

1. f(x) � 6

2. 3x � 5y � 2

3. f(x) � 4x � 3

4. y � �5x � 10

5. (�12, 8)

6. f(x) � �x� � 1

7. y � 5 � �2(x � 3)

8. f(x) � � x � 3 if x � 02 � x if x � 0

9. �38�

�(�

51)� � �

34�

10. {3, 4, 5} for the function {(0, 4), (2, 5), (3, 3)}

In your own words—Define each term.

11. vertical line test

12. linear function

absolute value functionboundaryCartesian coordinate planeconstant functiondependent variabledomainfamily of graphsfunctionfunctional notationgreatest integer functionidentity functionindependent variable

linear equationlinear functionline of fitmappingone-to-one functionordered pairparent graphpiecewise functionpoint-slope formprediction equationquadrantrange

rate of changerelationscatter plotslopeslope-intercept formstandard formstep functionvertical line testx-intercepty-intercept

NAME DATE PERIOD

SCORE 22

a. ordered pair

b. point-slope form

c. step function

d. range

e. constant function

f. piecewise function

g. slope-intercept form

h. absolute value function

i. standard form

j. slope

Chapter 2 Quiz (Lessons 2–1 and 2–2)

22


1. Find the domain and range of x � y � 1. Then determine 1.whether it is a function.

Determine whether the relation shown in the graph is a function.

2. 3. Find f(4) if f(x) � �xx2

��

19

�.

4. Write x � 2 � �15�y in standard form. Identify A, B, and C.

5. Find the x-intercept and the y-intercept of the graph of 3x � 4y � 12. Then graph the equation.

y

xO

NAME DATE PERIOD

SCORE


For Questions 1 and 2, find the slope of the line that passes through each pair of points. 1.

1. (2, 4), (4, 7) 2. ��12�, �4�, ��

12�, 5� 2.

3. Graph the line passing through (2, 4) that is parallel to the 3.graph of x � 3y � 6.

4. Standardized Test Practice Which is an equation of the

line that has a slope of ��23� and passes through (�1, 3)?

A. y � �32�x � �

73� B. y � ��

23�x � �

73�

C. y � ��23�x � 1 D. y � ��

23�x � �

131� 4.

5. Write an equation for the line that passes through (3, 5) 5.and is perpendicular to the line whose equation is

y � �12�(x � 2).

y

xO

NAME DATE PERIOD

SCORE 22

Ass

essm

ent

2.

3.

4.

5.

y

xO


For Questions 1 and 2, use the set of data in the table. 1.

1. Draw a scatter plot for the data. Then state which of the data points is an outlier.

2. Use two ordered pairs to write a prediction equation.Then use your prediction equation to predict the missing value.

3. f(x) � � x � 2 �. Identify the domain and range.

Chapter 2 Quiz (Lesson 2–7)

1. Write an inequality for the graph shown.


2. y � �2

3. 6 � 2y � 3x

4. y � � � 2x �

y

xO

(0, 1)

(1, 4)

NAME DATE PERIOD

SCORE


22

NAME DATE PERIOD

SCORE

22

1.

2.

3. y

xO

y

x

20

1520253035404550

4 6 8 10 12 14 16 18 20

1.2.

3.

4. y

xO

y

xO

y

xO

x 2 5 10 15 20 30

y 1 25 21 32 41 ?

Chapter 2 Mid-Chapter Test (Lessons 2–1 through 2–4)



1. Which of the following relations is not a function?A. B. C. D. 1.

2. Which equation or function is linear?

A. y � �x �3

1� B. f(x) � �23�(1 � x)2 C. 2y � �

2x4� 1� D. 3xy � 4 2.

3. Write an equation in standard form for the line that is parallel to the graph of �8x � 5 � 4y and has y-intercept �0.5.

A. x � 0.5y � 0.25 B. 10x � 5y � 2.5 C. 4x � 2y � 1 D. 2x � y � 1 3.

4. Find the slope of the line that passes through ��4.5, �72�� and (3, 3.5).

A. ��16� B. �6 C. undefined D. 0 4.

5. The graphs of which pair of lines are perpendicular?

A. 2x � 3y � 12, y � ��23�x � 5 B. 3x � 2y � 6, 2x � 3y � 7

C. y � 4x � 13, y � �14�x � 13 D. x � y � 1, 2y � �2x � 2 5.

Graph each relation and find the domain and range.Then determine whether the relation is a function.

6. {(2, 4), (4, �2), (1, 3), (0, 3)}

7. y � 2x � 3

For Questions 8 and 9, find each value if f(x) � �3x3 � 2x2.

8. f(�1) 9. f��12��

10. Write an equation in slope-intercept form for the line that

has a slope of ��13� and passes through (�6, 1).

Part II

y

xO

y

xO

y

xO

y

xO

Part I

NAME DATE PERIOD

SCORE 22

Ass

essm

ent

6.

7.

8.

9.

10.

y

xO

y

xO


Chapter 2 Cumulative Review (Chapters 1 and 2)

1. Evaluate �7aa2

��

2bc

� if a � 3, b � 2, and c � 5. (Lesson 1-1) 1.

2. Name the sets of numbers to which �42.1 belongs. (Lesson 1-2) 2.

3. Solve 3�7 � a� � 12. Check each solution. (Lesson 1-4) 3.

For Questions 4 and 5, solve each inequality. Graph the solution set.

4. 2(3x � 1) � 5x � 3 (Lesson 1-5) 4.

5. �6 � 2(y � 1) � 10 (Lesson 1-6) 5.

6. Find the domain and range of the relation. Then determine 6.whether the relation is a function.{(4, �7), (3, �7), (2, 0), (4, 0)} (Lesson 2-1)

7. Find f(�7) if f(x) � 2x2 � 3x. (Lesson 2-1) 7.

8. Find the x-intercept and the y-intercept of the graph of 8.3x � 4y � 8. (Lesson 2-2)

9. Find the slope of the line whose graph is perpendicular to 9.the graph of 2x � 5y � 7. (Lesson 2-3)

10. Write an equation in slope-intercept form for the line that 10.has a slope of �4 and passes through (3, �5). (Lesson 2-4)

11. The prediction equation y � 5.92x � 119.21 models the 11.median selling price, in thousands of dollars, of new homes in a certain area since 1995. Predict the median selling price in 2015. (Lesson 2-5)

12. Identify the domain and range of the piecewise function 12.

h(x) � � . (Lesson 2-6)

13. Graph y � ��45�x � 1. (Lesson 2-7) 13.

y

xO

x � 5 if x � �2�4x if x � �2

�1�2 0 1 2 4 5 63

�1�2�3�4 0 1 2 43

NAME DATE PERIOD

22

Standardized Test Practice (Chapters 1 and 2)


1. If the perimeter of a rectangle is 96 inches and the length is 4 inches longer than the width, what is the area?A. 22 in2 B. 26 in2 C. 230 in2 D. 572 in2 1.

2. For what values of a will 3a � 1 be equal to 3a � 10?E. all negative values F. 0G. all positive values H. no values 2.

3. In the figure at the right, if RSTV is a square with perimeter 24, what is the area of the circle with center R?A. 6 B. 36

C. 12 D. 144 3.

4. In a group of 20 students, 12 belonged to the band,7 belonged to the choir, and 5 belonged to both the band and the choir. How many students did not belong to either the band or the choir?E. 1 F. 2 G. 6 H. 14 4.

5. A point on the graph of 2x � 2y � 12 is __?___.A. (�3, �3) B. (�3, 3) C. (3, 3) D. (3, �3) 5.

6. If x � y � 6 and 3x � 10 � 2y, what is the value of y?E. �8 F. �4 G. 4 H. 8 6.

7. Which is equal to x3 � 8?A. (x � 2)(x2 � 4x � 4) B. (x � 2)(x2 � 2x � 4)C. (x � 2)(x2 � 4x � 4) D. (x � 2)(x2 � 2x � 4) 7.

8. In the sequence 1, 3, 12, 60, 360, ___, ___, ___, the eighth term is __?___.E. 2520 F. 2880 G. 20,160 H. 181,440 8.

9. If m�ABD � 65, m�EBC � 70, and m�ABC � 115, find m�EBD.A. 5° B. 20°C. 45° D. 50° 9.

10. 8 less than a is 6 more than c. Thus, cexpressed in terms of a is __?___.E. �

a �6

8� F. a � 2 G. a � 14 H. 2 � a 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

NAME DATE PERIOD

22

Ass

essm

ent

V T

R S

B

C

AE D


Standardized Test Practice (continued)

11. A shoe salesperson sold 20 pairs of shoes for 11. 12.$640. A brown pair of shoes sells for $30 and a black pair for $35. How many brown pairs were sold?

12. How many integers between 299 and 501 are divisible by 2 or 5?

13. The histogram 13. 14.shows the distribution of mid-term exam scores for Ms. Hawkins’ three algebra classes.What percent of her students scored at least 70?

14. If 3x�2 � 81, what is the value of 22x�7?

Column A Column B

15. 15.

16. 8 � 7 � y

16.

17. 17.

18.

, a � b � c � 4a

18. DCBAbc

a˚ c˚b˚

DCBA(c)(c)(c)(c)�4(c � c)

DCBA4y � 102y � 2

DCBA8(4 � 3) � 63(6)(0) � 14

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

22

NAME DATE PERIOD

10

15

5

0

25

20

Nu

mb

er o

f St

ud

ents

Exam Scores50 60 70 80 90 100

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

A

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 106–107 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

22

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 9

Solve the problem and write your answer in the blank.

For Questions 11–17, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

10 12 14 16

11 13 15 17

Select the best answer from the choices given and fill in the corresponding oval.

18 20 22

19 21 DCBADCBA

DCBADCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 2-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Rel

atio

ns

and

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1

©G

lenc

oe/M

cGra

w-H

ill57

Gle

ncoe

Alg

ebra

2

Lesson 2-1

Gra

ph

Rel

atio

ns

A r

elat

ion

can

be

repr

esen

ted

as a

set

of

orde

red

pair

s or

as

aneq

uat

ion

;th

e re

lati

on i

s th

en t

he

set

of a

ll o

rder

ed p

airs

(x,

y) t

hat

mak

e th

e eq

uat

ion

tru

e.T

he

dom

ain

of a

rel

atio

n i

s th

e se

t of

all

fir

st c

oord

inat

es o

f th

e or

dere

d pa

irs,

and

the

ran

geis

th

e se

t of

all

sec

ond

coor

din

ates

.A

fu

nct

ion

is a

rel

atio

n i

n w

hic

h e

ach

ele

men

t of

th

e do

mai

n i

s pa

ired

wit

h e

xact

ly o

ne

elem

ent

of t

he

ran

ge.Y

ou c

an t

ell

if a

rel

atio

n i

s a

fun

ctio

n b

y gr

aph

ing,

then

usi

ng

the

vert

ical

lin

e te

st.I

f a

vert

ical

lin

e in

ters

ects

th

e gr

aph

at

mor

e th

an o

ne

poin

t,th

ere

lati

on i

s n

ot a

fu

nct

ion

.

Gra

ph

th

e eq

uat

ion

y�

2x�

3 an

d f

ind

th

e d

omai

n a

nd

ran

ge.D

oes

the

equ

atio

n r

epre

sen

t a

fun

ctio

n?

Mak

e a

tabl

e of

val

ues

to

fin

d or

dere

d pa

irs

that

sa

tisf

y th

e eq

uat

ion

.Th

en g

raph

th

e or

dere

d pa

irs.

Th

e do

mai

n a

nd

ran

ge a

re b

oth

all

rea

l n

um

bers

.Th

egr

aph

pas

ses

the

vert

ical

lin

e te

st,s

o it

is

fun

ctio

n.

Gra

ph

eac

h r

elat

ion

or

equ

atio

n a

nd

fin

d t

he

dom

ain

an

d r

ange

.Th

en d

eter

min

ew

het

her

th

e re

lati

on o

r eq

uat

ion

is

a fu

nct

ion

.

1.{(

1,3)

,(�

3,5)

,2.

{(3,

�4)

,(1,

0),

3.{(

0,4)

,(�

3,�

2),

(�2,

5),(

2,3)

}(2

,�2)

,(3,

2)}

(3,2

),(5

,1)}

D �

{�3,

�2,

1,2}

,D

�{1

,2,3

},D

�{�

3,0,

3,5}

,R

�{3

,5};

yes

R �

{�4,

�2,

0,2}

;n

oR

�{�

2,1,

2,4}

;ye

s

4.y

�x2

�1

5.y

�x

�4

6.y

�3x

�2

D �

all r

eals

,D

�al

l rea

ls,

D �

all r

eals

,R

�{y

y�

�1}

;ye

sR

�al

l rea

ls;

yes

R �

all r

eals

;ye

sx

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O

xy

�1

�5

0�

3

1�

1

21

33

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill58

Gle

ncoe

Alg

ebra

2

Equ

atio

ns

of

Fun

ctio

ns

and

Rel

atio

ns

Equ

atio

ns

that

rep

rese

nt

fun

ctio

ns

are

ofte

n w

ritt

en i

n f

un

ctio

nal

not

atio

n.F

or e

xam

ple,

y�

10 �

8xca

n b

e w

ritt

en a

s f(

x) �

10 �

8x.T

his

not

atio

n e

mph

asiz

es t

he

fact

th

at t

he

valu

es o

f y,

the

dep

end

ent

vari

able

,dep

end

on t

he

valu

es o

f x,

the

ind

epen

den

t va

riab

le.

To

eval

uat

e a

fun

ctio

n,o

r fi

nd

a fu

nct

ion

al v

alu

e,m

ean

s to

su

bsti

tute

a g

iven

val

ue

in t

he

dom

ain

in

to t

he

equ

atio

n t

o fi

nd

the

corr

espo

ndi

ng

elem

ent

in t

he

ran

ge.

Giv

en t

he

fun

ctio

n f

(x)

�x2

�2x

,fin

d e

ach

val

ue.

a.f(

3)

f(x)

�x2

�2x

Orig

inal

fun

ctio

n

f(3)

�32

�2(

3)S

ubst

itute

.

�15

Sim

plify

.

b.

f(5a

)

f(x)

�x2

�2x

Orig

inal

fun

ctio

n

f(5a

) �

(5a)

2�

2(5a

)S

ubst

itute

.

�25

a2�

10a

Sim

plify

.

Fin

d e

ach

val

ue

if f

(x)

��

2x�

4.

1.f(

12)

�20

2.f(

6)�

83.

f(2b

)�

4b�

4

Fin

d e

ach

val

ue

if g

(x)

�x3

�x.

4.g(

5)12

05.

g(�

2)�

66.

g(7c

)34

3c3

�7c

Fin

d e

ach

val

ue

if f

(x)

�2x

�an

d g

(x)

�0.

4x2

�1.

2.

7.f(

0.5)

58.

f(�

8)�

169.

g(3)

2.4

10.g

(�2.

5)1.

311

.f(4

a)8a

�12

.g�

��

1.2

13.f

��6

14.g

(10)

38.8

15.f

(200

)40

0.01

Let

f(x

) �

2x2

�1.

16.F

ind

the

valu

es o

f f(

2) a

nd

f(5)

.f(

2) �

7,f(

5) �

49

17.C

ompa

re t

he

valu

es o

f f(

2) �

f(5)

an

d f(

2 �

5).

f(2)

�f(

5) �

343,

f(2

�5)

�19

9

2 � 31 � 3

b2

� 10b � 2

1 � 2a1 � 4

2 � x

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Rel

atio

ns

and

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Rel

atio

ns

and

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1

©G

lenc

oe/M

cGra

w-H

ill59

Gle

ncoe

Alg

ebra

2

Lesson 2-1

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.W

rite

yes

or n

o.

1.ye

s2.

no

3.ye

s4.

no

Gra

ph

eac

h r

elat

ion

or

equ

atio

n a

nd

fin

d t

he

dom

ain

an

d r

ange

.Th

en d

eter

min

ew

het

her

th

e re

lati

on o

r eq

uat

ion

is

a fu

nct

ion

.

5.{(

2,�

3),(

2,4)

,(2,

�1)

}6.

{(2,

6),(

6,2)

}

D �

{2},

R �

{�3,

�1,

4};

no

D �

{2,6

},R

�{2

,6};

yes

7.{(

�3,

4),(

�2,

4),(

�1,

�1)

,(3,

�1)

}8.

x�

�2

D �

{�3,

�2,

�1,

3},

D �

{�2}

,R �

all r

eals

;n

o

R �

{�1,

4};

yes

Fin

d e

ach

val

ue

if f

(x)

�2x

�1

and

g(x

) �

2 �

x2.

9.f(

0)�

110

.f(1

2)23

11.g

(4)

�14

12.f

(�2)

�5

13.g

(�1)

114

.f(d

)2d

�1

x

y

O

( –2,

4)

( –3,

4)

( –1,

–1)

( 3, –

1)x

y

O

( 2, 6

) ( 6, 2

)

x

y

O

( 2, 4

)

( 2, –

1)

( 2, –

3)

x

y

O

x

y

O

xy

12

24

36

D 3

R 1 5

D 100

200

300

R 50

100

150

©G

lenc

oe/M

cGra

w-H

ill60

Gle

ncoe

Alg

ebra

2

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.W

rite

yes

or n

o.

1.n

o2.

yes

3.ye

s4.

no

Gra

ph

eac

h r

elat

ion

or

equ

atio

n a

nd

fin

d t

he

dom

ain

an

d r

ange

.Th

en d

eter

min

ew

het

her

th

e re

lati

on o

r eq

uat

ion

is

a fu

nct

ion

.

5.{(

�4,

�1)

,(4,

0),(

0,3)

,(2,

0)}

6.y

�2x

�1

D �

{�4,

0,2,

4},

D �

all r

eals

,R �

all r

eals

;ye

sR

�{�

1,0,

3};

yes

Fin

d e

ach

val

ue

if f

(x)

�an

d g

(x)

��

2x�

3.

7.f(

3)1

8.f(

�4)

�9.

g ��2

10.f

(�2)

un

def

ined

11.g

(�6)

1512

.f(m

�2)

13.M

USI

CT

he

orde

red

pair

s (1

,16)

,(2,

16),

(3,3

2),(

4,32

),an

d (5

,48)

rep

rese

nt

the

cost

of

buyi

ng

vari

ous

nu

mbe

rs o

f C

Ds

thro

ugh

a m

usi

c cl

ub.

Iden

tify

th

e do

mai

n a

nd

ran

ge o

fth

e re

lati

on.I

s th

e re

lati

on a

fu

nct

ion

?D

�{1

,2,3

,4,5

},R

�{1

6,32

,48}

;ye

s

14.C

OM

PUTI

NG

If a

com

pute

r ca

n d

o on

e ca

lcu

lati

on i

n 0

.000

0000

015

seco

nd,

then

th

efu

nct

ion

T(n

) �

0.00

0000

0015

ngi

ves

the

tim

e re

quir

ed f

or t

he

com

pute

r to

do

nca

lcu

lati

ons.

How

lon

g w

ould

it

take

th

e co

mpu

ter

to d

o 5

bill

ion

cal

cula

tion

s?7.

5 s

5 � m

1 � 25 � 2

5� x

�2

x

y

O

( –4,

–1)

( 2, 0

)

( 0, 3

)

( 4, 0

) x

y

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x

y

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xy

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0

�1

�1

00

2�

2

34

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R 105

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R 21 25 30

Pra

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e (

Ave

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e)

Rel

atio

ns

and

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

__P

ER

IOD

____

_

2-1

2-1



Readin

g t

o L

earn

Math

em

ati

csR

elat

ion

s an

d F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1

©G

lenc

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w-H

ill61

Gle

ncoe

Alg

ebra

2

Lesson 2-1

Pre-

Act

ivit

yH

ow d

o re

lati

ons

and

fu

nct

ion

s ap

ply

to

bio

logy

?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-1

at

the

top

of p

age

56 i

n y

our

text

book

.

•R

efer

to

the

tabl

e.W

hat

doe

s th

e or

dere

d pa

ir (

8,20

) te

ll y

ou?

Fo

r a

dee

r,th

e av

erag

e lo

ng

evit

y is

8 y

ears

an

d t

he

max

imu

mlo

ng

evit

y is

20

year

s.•

Su

ppos

e th

at t

his

tab

le i

s ex

ten

ded

to i

ncl

ude

mor

e an

imal

s.Is

it

poss

ible

to h

ave

an o

rder

ed p

air

for

the

data

in

wh

ich

th

e fi

rst

nu

mbe

r is

lar

ger

than

th

e se

con

d?S

amp

le a

nsw

er:

No

,th

e m

axim

um

lon

gev

ity

mu

st a

lway

s b

e g

reat

er t

han

th

e av

erag

e lo

ng

evit

y.

Rea

din

g t

he

Less

on

1.a.

Exp

lain

th

e di

ffer

ence

bet

wee

n a

rel

atio

n a

nd

a fu

nct

ion

.S

amp

le a

nsw

er:

Are

lati

on

is a

ny s

et o

f o

rder

ed p

airs

.A f

un

ctio

n is

a s

pec

ial k

ind

of

rela

tio

n in

wh

ich

eac

h e

lem

ent

of

the

do

mai

n is

pai

red

wit

h e

xact

lyo

ne

elem

ent

in t

he

ran

ge.

b.

Exp

lain

th

e di

ffer

ence

bet

wee

n d

omai

n a

nd

ran

ge.

Sam

ple

an

swer

:Th

e d

om

ain

of

a re

lati

on

is t

he

set

of

all f

irst

co

ord

inat

es o

f th

e o

rder

ed p

airs

.Th

era

ng

e is

th

e se

t o

f al

l sec

on

d c

oo

rdin

ates

.

2.a.

Wri

te t

he

dom

ain

an

d ra

nge

of

the

rela

tion

sh

own

in

th

e gr

aph

.

D:

{�3,

�2,

�1,

0,3}

;R

:{�

5,�

4,0,

1,2,

4}

b.

Is t

his

rel

atio

n a

fu

nct

ion

? E

xpla

in.

Sam

ple

an

swer

:N

o,i

t is

no

t a

fun

ctio

nb

ecau

se o

ne

of

the

elem

ents

of

the

do

mai

n,3

,is

pai

red

wit

h t

wo

elem

ents

of

the

ran

ge.

Hel

pin

g Y

ou

Rem

emb

er

3.L

ook

up

the

wor

ds d

epen

den

tan

d in

dep

end

ent

in a

dic

tion

ary.

How

can

th

e m

ean

ing

ofth

ese

wor

ds h

elp

you

dis

tin

guis

h b

etw

een

in

depe

nde

nt

and

depe

nde

nt

vari

able

s in

afu

nct

ion

?S

amp

le a

nsw

er:T

he

vari

able

wh

ose

val

ues

dep

end

on

,or

are

det

erm

ined

by,

the

valu

es o

f th

e o

ther

var

iab

le is

th

e d

epen

den

t va

riab

le.

( 0, 4

)

( 3, 1

)

( 3, –

4)( –

1, –

5)

( –2,

0)

( –3,

2)

x

y

O

©G

lenc

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ebra

2

Map

pin

gs

Th

ere

are

thre

e sp

ecia

l w

ays

in w

hic

h o

ne

set

can

be

map

ped

to a

not

her

.A s

etca

n b

e m

appe

d in

toan

oth

er s

et,o

nto

anot

her

set

,or

can

hav

e a

one-

to-o

ne

corr

espo

nd

ence

wit

h a

not

her

set

.

Sta

te w

het

her

eac

h s

et i

s m

app

ed i

nto

th

e se

con

d s

et,o

nto

th

e se

con

d

set,

or h

as a

on

e-to

-on

e co

rres

pon

den

ce w

ith

th

e se

con

d s

et.

1.2.

3.4.

into

,on

toin

to,o

nto

into

,on

to,

into

,on

too

ne-

to-o

ne

5.6.

7.8.

into

into

,on

toin

to,o

nto

into

,on

to,

on

e-to

-on

e

9.C

an a

set

be

map

ped

onto

a se

t w

ith

few

er e

lem

ents

th

an i

t h

as?

yes

10.C

an a

set

be

map

ped

into

a se

t th

at h

as m

ore

elem

ents

th

an i

t h

as?

yes

11.I

f a

map

pin

g fr

om s

et A

into

set

Bis

a o

ne-

to-o

ne

corr

espo

nde

nce

,wh

at

can

you

con

clu

de a

bou

t th

e n

um

ber

of e

lem

ents

in

Aan

d B

?T

he

sets

hav

e th

e sa

me

nu

mb

er o

f el

emen

ts.

–2 9 12 5

1 4 –7 0

–2 9 12 5

1 4 –7 0

–3

15 10 2

–2 9 12 5

1 4 –7 0

10 –6 24 2

3

1 3 7 9–

5

a g k l q

0–

3 9 7

4 12 6

2 4 –1 –4

7 0 2

Into

map

pin

gA

map

ping

fro

m s

et A

to s

et B

whe

re e

very

ele

men

t of

Ais

map

ped

to o

ne o

r m

ore

elem

ents

of

set

B,

but

neve

r to

an

elem

ent

not

in B

.

On

to m

app

ing

Am

appi

ng f

rom

set

Ato

set

Bw

here

eac

h el

emen

t of

set

Bha

s at

leas

t on

e el

emen

t of

se

t A

map

ped

to it

.

On

e-to

-on

e A

map

ping

fro

m s

et A

onto

set

Bw

here

eac

h el

emen

t of

set

Ais

map

ped

to e

xact

ly o

ne

corr

esp

on

den

ceel

emen

t of

set

Ban

d di

ffere

nt e

lem

ents

of

Aar

e ne

ver

map

ped

to t

he s

ame

elem

ent

of B

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-1

2-1


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

©G

lenc

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ill63

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ncoe

Alg

ebra

2

Lesson 2-2

Iden

tify

Lin

ear

Equ

atio

ns

and

Fu

nct

ion

sA

lin

ear

equ

atio

nh

as n

o op

erat

ion

sot

her

th

an a

ddit

ion

,su

btra

ctio

n,a

nd

mu

ltip

lica

tion

of

a va

riab

le b

y a

con

stan

t.T

he

vari

able

s m

ay n

ot b

e m

ult

ipli

ed t

oget

her

or

appe

ar i

n a

den

omin

ator

.A l

inea

r eq

uat

ion

doe

sn

ot c

onta

in v

aria

bles

wit

h e

xpon

ents

oth

er t

han

1.T

he

grap

h o

f a

lin

ear

equ

atio

n i

s a

lin

e.

A l

inea

r fu

nct

ion

is a

fu

nct

ion

wh

ose

orde

red

pair

s sa

tisf

y a

lin

ear

equ

atio

n.A

ny

lin

ear

fun

ctio

n c

an b

e w

ritt

en i

n t

he

form

f(x

) �

mx

�b,

wh

ere

man

d b

are

real

nu

mbe

rs.

If a

n e

quat

ion

is

lin

ear,

you

nee

d on

ly t

wo

poin

ts t

hat

sat

isfy

th

e eq

uat

ion

in

ord

er t

o gr

aph

the

equ

atio

n.O

ne

way

is

to f

ind

the

x-in

terc

ept

and

the

y-in

terc

ept

and

con

nec

t th

ese

two

poin

ts w

ith

a l

ine. Is

f(x

) �

0.2

�a

lin

ear

fun

ctio

n?

Exp

lain

.

Yes;

it i

s a

lin

ear

fun

ctio

n b

ecau

se i

t ca

nbe

wri

tten

in

th

e fo

rmf(

x) �

�x

�0.

2. Is 2

x�

xy�

3y�

0 a

lin

ear

fun

ctio

n?

Exp

lain

.

No;

it i

s n

ot a

lin

ear

fun

ctio

n b

ecau

seth

e va

riab

les

xan

d y

are

mu

ltip

lied

toge

ther

in

th

e m

iddl

e te

rm.

1 � 5

x � 5F

ind

th

e x-

inte

rcep

t an

d t

he

y-in

terc

ept

of t

he

grap

h o

f 4x

�5y

�20

.T

hen

gra

ph

th

e eq

uat

ion

.

Th

e x-

inte

rcep

t is

th

e va

lue

of x

wh

en y

�0.

4x�

5y�

20O

rigin

al e

quat

ion

4x�

5(0)

�20

Sub

stitu

te 0

for

y.

x�

5S

impl

ify.

So

the

x-in

terc

ept

is 5

.S

imil

arly

,th

e y-

inte

rcep

t is

�4.

x

y

O

Exam

ple1

Exam

ple1

Exam

ple3

Exam

ple3

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sta

te w

het

her

eac

h e

qu

atio

n o

r fu

nct

ion

is

lin

ear.

Wri

te y

esor

no.

If n

o,ex

pla

in.

1.6y

�x

�7

yes

2.9x

�N

o;

the

3.f(

x) �

2 �

yes

vari

able

yap

pea

rs

in t

he

den

om

inat

or.

Fin

d t

he

x-in

terc

ept

and

th

e y-

inte

rcep

t of

th

e gr

aph

of

each

eq

uat

ion

.Th

en g

rap

hth

e eq

uat

ion

.

4.2x

�7y

�14

5.5y

�x

�10

6.2.

5x�

5y�

7.5

�0

x-in

t:7;

y-in

t:2

x-in

t:�

10;

y-in

t:2

x-in

t:�

3;y-

int:

1.5

x

y

Ox

y

Ox

y

O

x � 1118 � y

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2

Stan

dar

d F

orm

Th

e st

and

ard

for

mof

a l

inea

r eq

uat

ion

is

Ax

�B

y�

C,w

her

e A

,B,a

nd

Car

e in

tege

rs w

hos

e gr

eate

st c

omm

on f

acto

r is

1.

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.I

den

tify

A,B

,an

d C

.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

Exam

ple

Exam

ple

a.y

�8x

�5

y�

8x�

5O

rigin

al e

quat

ion

�8x

�y

��

5S

ubtr

act

8xfr

om e

ach

side

.

8x�

y�

5M

ultip

ly e

ach

side

by

�1.

So

A�

8,B

��

1,an

d C

�5.

b.

14x

��

7y�

21

14x

��

7y�

21O

rigin

al e

quat

ion

14x

�7y

�21

Add

7y

to e

ach

side

.

2x�

y�

3D

ivid

e ea

ch s

ide

by 7

.

So

A�

2,B

�1,

and

C�

3.

Exer

cises

Exer

cises

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.I

den

tify

A,B

,an

d C

.

1.2x

�4y

�1

2.5y

�2x

�3

3.3x

��

5y�

22x

�4y

��

1;A

�2,

2x�

5y�

�3;

A�

2,3x

�5y

�2;

A�

3,B

��

4,C

��

1B

��

5,C

��

3B

�5,

C�

2

4.18

y�

24x

�9

5.y

�x

�5

6.6y

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�10

�0

8x�

6y�

3;A

�8,

8x�

9y�

�60

;A

�8,

4x�

3y�

5;A

�4,

B�

�6,

C�

3B

��

9,C

��

60

B�

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C�

5

7.0.

4x�

3y�

108.

x�

4y�

79.

2y�

3x�

62x

�15

y�

50;

A�

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A�

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50B

��

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��

7 B

��

2,C

��

6

10.

x�

y�

2 �

011

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12.3

x�

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30;

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B �

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16.

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C �

500

x � 6

3 � 45x � 2

y � 4

y � 9

1 � 32 � 5

2 � 33 � 4



Skil

ls P

ract

ice

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

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het

her

eac

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qu

atio

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r fu

nct

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is

lin

ear.

Wri

te y

esor

no.

If n

o,ex

pla

inyo

ur

reas

onin

g.

1.y

�3x

2.y

��

2 �

5x

yes

yes

3.2x

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4.f(

x) �

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yes

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;th

e ex

po

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t o

f x

is n

ot

1.

5.�

�y

�15

6.x

�y

�8

No

;x

is in

a d

eno

min

ato

r.ye

s

7.g(

x) �

88.

h(x

) �

�x��

3

yes

No

;x

is in

sid

e a

squ

are

roo

t.

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.I

den

tify

A,B

,an

d C

.

9.y

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x�

y�

0;1,

�1,

010

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1;5,

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11.2

x�

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412

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2

13.5

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9 �

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4x�

3y�

7;4,

3,7

Fin

d t

he

x-in

terc

ept

and

th

e y-

inte

rcep

t of

th

e gr

aph

of

each

eq

uat

ion

.Th

en g

rap

hth

e eq

uat

ion

.

15.y

�3x

�6

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616

.y�

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17.x

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18.2

x�

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2

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)

( 0, 2

)

x

y

O( 5

, 0)

( 0, 5

)

x

y

O

( 0, 0

)x

y

O

( 2, 0

)

( 0, –

6)

x

y

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1 � 33 � x

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Sta

te w

het

her

eac

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qu

atio

n o

r fu

nct

ion

is

lin

ear.

Wri

te y

esor

no.

If n

o,ex

pla

inyo

ur

reas

onin

g.

1.h

(x)

�23

yes

2.y

�x

yes

3.y

�N

o;

xis

a d

eno

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ato

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9 �

5xy

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No

;x

and

yar

e m

ult

iplie

d.

Wri

te e

ach

eq

uat

ion

in

sta

nd

ard

for

m.I

den

tify

A,B

,an

d C

.

5.y

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y�

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y�

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7.3y

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��

y�

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�8y

�21

;28

,8,2

1

Fin

d t

he

x-in

terc

ept

and

th

e y-

inte

rcep

t of

th

e gr

aph

of

each

eq

uat

ion

.Th

en g

rap

hth

e eq

uat

ion

.

9.y

�2x

�4

�2,

410

.2x

�7y

�14

7,2

11.y

��

2x�

4�

2,�

412

.6x

�2y

�6

1,3

13.M

EASU

RE

Th

e eq

uat

ion

y�

2.54

xgi

ves

the

len

gth

in

cen

tim

eter

s co

rres

pon

din

g to

ale

ngt

h x

in i

nch

es.W

hat

is

the

len

gth

in

cen

tim

eter

s of

a 1

-foo

t ru

ler?

30.4

8 cm

LON

G D

ISTA

NC

EF

or E

xerc

ises

14

and

15,

use

th

e fo

llow

ing

info

rmat

ion

.

For

Meg

’s l

ong-

dist

ance

cal

lin

g pl

an,t

he

mon

thly

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t C

in d

olla

rs i

s gi

ven

by

the

lin

ear

fun

ctio

n C

(t)

�6

�0.

05t,

wh

ere

tis

th

e n

um

ber

of m

inu

tes

talk

ed.

14.W

hat

is

the

tota

l co

st o

f ta

lkin

g 8

hou

rs?

of t

alki

ng

20 h

ours

?$3

0;$6

6

15.W

hat

is

the

effe

ctiv

e co

st p

er m

inu

te (

the

tota

l co

st d

ivid

ed b

y th

e n

um

ber

of m

inu

tes

talk

ed)

of t

alki

ng

8 h

ours

? of

tal

kin

g 20

hou

rs?

$0.0

625;

$0.0

55

( 1, 0

)

( 0, 3

)

x

y

O

x

y

(–2,

0)

( 0, –

4)

O

( 7, 0

)

( 0, 2

)

x

y

O( –

2, 0

)

( 0, 4

)

x

y

O

3 � 42 � 7

3 � 8

5 � x

2 � 3

Pra

ctic

e (

Ave

rag

e)

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csL

inea

r E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

©G

lenc

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cGra

w-H

ill67

Gle

ncoe

Alg

ebra

2

Lesson 2-2

Pre-

Act

ivit

yH

ow d

o li

nea

r eq

uat

ion

s re

late

to

tim

e sp

ent

stu

dyi

ng?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-2

at

the

top

of p

age

63 i

n y

our

text

book

.

•If

Lol

ita

spen

ds 2

hou

rs s

tudy

ing

mat

h,h

ow m

any

hou

rs w

ill

she

hav

e

to s

tudy

ch

emis

try?

1h

ou

rs•

Su

ppos

e th

at L

olit

a de

cide

s to

sta

y u

p on

e h

our

late

r so

th

at s

he

now

has

5 h

ours

to

stu

dy a

nd

do h

omew

ork.

Wri

te a

lin

ear

equ

atio

n t

hat

des

crib

esth

is s

itu

atio

n.

x�

y�

5

Rea

din

g t

he

Less

on

1.W

rite

yes

or n

oto

tel

l w

het

her

eac

h l

inea

r eq

uat

ion

is

in s

tan

dard

for

m.I

f it

is

not

,ex

plai

n w

hy

it i

s n

ot.

a.�

x�

2y�

5N

o;

Ais

neg

ativ

e.

b.

9x�

12y

��

5ye

s

c.5x

�7y

�3

yes

d.

2x�

y�

1N

o;

Bis

no

t an

inte

ger

.

e.0x

�0y

�0

No

;A

and

Bar

e b

oth

0.

f.2x

�4y

�8

No

;Th

e g

reat

est

com

mo

n f

acto

r o

f 2,

4,an

d 8

is 2

,no

t 1.

2.H

ow c

an y

ou u

se t

he

stan

dard

for

m o

f a

lin

ear

equ

atio

n t

o te

ll w

het

her

th

e gr

aph

is

ah

oriz

onta

l li

ne

or a

ver

tica

l li

ne?

If A

�0,

then

th

e g

rap

h is

a h

ori

zon

tal l

ine.

IfB

�0,

then

th

e g

rap

h is

a v

erti

cal l

ine.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to a

not

her

per

son

.Su

ppos

e th

at y

ou

are

stu

dyin

g th

is l

esso

n w

ith

a f

rien

d w

ho

thin

ks t

hat

sh

e sh

ould

let

x�

0 to

fin

d th

e x-

inte

rcep

t an

d le

t y

�0

to f

ind

the

y-in

terc

ept.

How

wou

ld y

ou e

xpla

in t

o h

er h

ow t

ore

mem

ber

the

corr

ect

way

to

fin

d in

terc

epts

of

a li

ne?

Sam

ple

an

swer

:Th

e x-

inte

rcep

t is

th

e x-

coo

rdin

ate

of

a p

oin

t o

n t

he

x-ax

is.E

very

po

int

on

th

e x-

axis

has

y-c

oo

rdin

ate

0,so

let

y�

0 to

fin

d a

n x

-in

terc

ept.

Th

e y-

inte

rcep

t is

th

e y-

coo

rdin

ate

of

a p

oin

t o

n t

he

y-ax

is.E

very

po

int

on

th

e y-

axis

has

x-c

oo

rdin

ate

0,so

let

x�

0 to

fin

d a

y-i

nte

rcep

t.

4 � 7

1 � 2

1 � 2

©G

lenc

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Alg

ebra

2

Gre

ates

t C

om

mo

n F

acto

rS

uppo

se w

e ar

e gi

ven

a li

near

equ

atio

n ax

�by

�c

whe

re a

,b,a

nd c

are

nonz

ero

inte

gers

,and

we

wan

t to

kno

w if

the

re e

xist

int

eger

s x

and

yth

at s

atis

fy t

heeq

uati

on.W

e co

uld

try

gues

sing

a f

ew t

imes

,but

thi

s pr

oces

s w

ould

be

tim

eco

nsum

ing

for

an e

quat

ion

such

as

588x

�43

2y�

72.B

y us

ing

the

Euc

lide

anA

lgor

ithm

,we

can

dete

rmin

e no

t on

ly if

suc

h in

tege

rs x

and

yex

ist,

but

also

fin

d th

em.T

he f

ollo

win

g ex

ampl

e sh

ows

how

thi

s al

gori

thm

wor

ks.

Fin

d i

nte

gers

xan

d y

that

sat

isfy

588

x�

432y

�72

.

Div

ide

the

grea

ter

of t

he

two

coef

fici

ents

by

the

less

er t

o ge

t a

quot

ien

t an

dre

mai

nde

r.T

hen

,rep

eat

the

proc

ess

by d

ivid

ing

the

divi

sor

by t

he

rem

ain

der

un

til

you

get

a r

emai

nde

r of

0.T

he

proc

ess

can

be

wri

tten

as

foll

ows.

588

�43

2(1)

�15

6(1

)43

2 �

156(

2) �

120

(2)

156

�12

0(1)

�36

(3)

120

�36

(3)

�12

(4)

36 �

12(3

)

Th

e la

st n

onze

ro r

emai

nde

r is

th

e G

CF

of

the

two

coef

fici

ents

.If

the

con

stan

tte

rm 7

2 is

div

isib

le b

y th

e G

CF,

then

in

tege

rs x

and

ydo

exi

st t

hat

sat

isfy

th

eeq

uat

ion

.To

fin

d x

and

y,w

ork

back

war

d in

th

e fo

llow

ing

man

ner

.

72�

6 �

12�

6 �

[120

�36

(3)]

Sub

stitu

te f

or 1

2 us

ing

(4)

�6(

120)

�18

(36)

�6(

120)

�18

[156

�12

0(1)

]S

ubst

itute

for

36

usin

g (3

)

��

18(1

56)

�24

(120

)�

�18

(156

) �

24[4

32 �

156(

2)]

Sub

stitu

te f

or 1

20 u

sing

(2)

�24

(432

) �

66(1

56)

�24

(432

) �

66[5

88 �

432(

1)]

Sub

stitu

te f

or 1

56 u

sing

(1)

�58

8(�

66)

�43

2(90

)

Th

us,

x�

�66

an

d y

�90

.

Fin

d i

nte

gers

xan

d y

,if

they

exi

st,t

hat

sat

isfy

eac

h e

qu

atio

n.

1.27

x�

65y

�3

2.45

x�

144y

�36

x�

�36

an

d y

�15

x�

�12

an

d y

�4

3.90

x�

117y

�10

4.12

3x�

36y

�15

no

inte

gra

l so

luti

on

s ex

ist

x�

25 a

nd

y�

�85

5.10

32x

�10

01y

�1

6.31

25x

�30

87y

�1

x�

�22

6 an

d y

�23

3x

��

1381

an

d y

�13

98

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-2

2-2

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

Slo

pe

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill69

Gle

ncoe

Alg

ebra

2

Lesson 2-3

Slo

pe

Slo

pe

mo

f a

Lin

eF

or p

oint

s (x

1, y

1) a

nd (

x 2,

y 2),

whe

re x

1�

x 2,

m�

�y 2

�y 1

� x 2�

x 1

chan

ge in

y�

�ch

ange

in x

Det

erm

ine

the

slop

e of

the

lin

e th

at p

asse

s th

rou

gh (

2,�

1) a

nd

(�4,

5).

m�

Slo

pe f

orm

ula

�(x

1, y

1) �

(2,

�1)

, (x

2, y

2) �

(�4,

5)

��

�1

Sim

plify

.

Th

e sl

ope

of t

he

lin

e is

�1.

6� �

6

5 �

(�1)

��

�4

�2

y 2�

y 1� x 2

�x 1

Gra

ph

th

e li

ne

pas

sin

gth

rou

gh (

�1,

�3)

wit

h a

slo

pe

of

.

Gra

ph t

he

orde

red

pair

(�

1,�

3).T

hen

,ac

cord

ing

to t

he

slop

e,go

up

4 u

nit

san

d ri

ght

5 u

nit

s.P

lot

the

new

poi

nt

(4,1

).C

onn

ect

the

poin

ts a

nd

draw

th

e li

ne.

x

y

O

4 � 5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.(4

,7)

and

(6,1

3)3

2.(6

,4)

and

(3,4

)0

3.(5

,1)

and

(7,�

3)�

2

4.(5

,�3)

an

d (�

4,3)

�5.

(5,1

0) a

nd

(�1,

�2)

26.

(�1,

�4)

and

(�

13,2

)�

7.(7

,�2)

an

d (3

,3)

�8.

(�5,

9) a

nd

(5,5

)�

9.(4

,�2)

an

d (�

4,�

8)

Gra

ph

th

e li

ne

pas

sin

g th

rou

gh t

he

give

n p

oin

t w

ith

th

e gi

ven

slo

pe.

10.s

lope

��

11.s

lope

�2

12.s

lope

�0

pass

es t

hro

ugh

(0,

2)pa

sses

th

rou

gh (

1,4)

pass

es t

hro

ugh

(�

2,�

5)

13.s

lope

�1

14.s

lope

��

15.s

lope

�

pass

es t

hro

ugh

(�

4,6)

pass

es t

hro

ugh

(�

3,0)

pass

es t

hro

ugh

(0,

0) x

y

O

x

y

O

x

y

O

1 � 53 � 4

x

y

O

x

y

Ox

y

O

1 � 3

3 � 42 � 5

5 � 4

1 � 22 � 3

©G

lenc

oe/M

cGra

w-H

ill70

Gle

ncoe

Alg

ebra

2

Para

llel a

nd

Per

pen

dic

ula

r Li

nes

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Slo

pe

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

In a

pla

ne,

non

vert

ical

lin

es w

ith

th

esa

me

slop

e ar

e p

aral

lel.

All

ver

tica

lli

nes

are

par

alle

l.

x

y

O

slop

e �

m

slop

e �

m

In a

pla

ne,

two

obli

que

lin

es a

re p

erp

end

icu

lar

ifan

d on

ly i

f th

e pr

odu

ct o

f th

eir

slop

es i

s �

1.A

ny

vert

ical

lin

e is

per

pen

dicu

lar

to a

ny

hor

izon

tal

lin

e.

x

y

O

slop

e �

m

slop

e �

�1 m

Exam

ple

Exam

ple

Are

th

e li

ne

pas

sin

g th

rou

gh (

2,6)

an

d (

�2,

2) a

nd

th

e li

ne

pas

sin

gth

rou

gh (

3,0)

an

d (

0,4)

par

alle

l,p

erp

end

icu

lar,

or n

eith

er?

Fin

d th

e sl

opes

of

the

two

lin

es.

Th

e sl

ope

of t

he

firs

t li

ne

is

�1.

Th

e sl

ope

of t

he

seco

nd

lin

e is

�

�.

Th

e sl

opes

are

not

equ

al a

nd

the

prod

uct

of

the

slop

es i

s n

ot �

1,so

th

e li

nes

are

nei

ther

para

llel

nor

per

pen

dicu

lar.

Are

th

e li

nes

par

alle

l,p

erp

end

icu

lar,

or n

eith

er?

1.th

e li

ne

pass

ing

thro

ugh

(4,

3) a

nd

(1,�

3) a

nd

the

lin

e pa

ssin

g th

rou

gh (

1,2)

an

d (�

1,3)

per

pen

dic

ula

r

2.th

e li

ne

pass

ing

thro

ugh

(2,

8) a

nd

(�2,

2) a

nd

the

lin

e pa

ssin

g th

rou

gh (

0,9)

an

d (6

,0)

nei

ther

3.th

e li

ne

pass

ing

thro

ugh

(3,

9) a

nd

(�2,

�1)

an

d th

e gr

aph

of

y�

2xp

aral

lel

4.th

e li

ne

wit

h x

-in

terc

ept

�2

and

y-in

terc

ept

5 an

d th

e li

ne

wit

h x

-in

terc

ept

2 an

d y-

inte

rcep

t �

5p

aral

lel

5.th

e li

ne

wit

h x

-in

terc

ept

1 an

d y-

inte

rcep

t 3

and

the

lin

e w

ith

x-i

nte

rcep

t 3

and

y-in

terc

ept

1n

eith

er

6.th

e li

ne

pass

ing

thro

ugh

(�

2,�

3) a

nd

(2,5

) an

d th

e gr

aph

of

x�

2y�

10p

erp

end

icu

lar

7.th

e li

ne

pass

ing

thro

ugh

(�

4,�

8) a

nd

(6,�

4) a

nd

the

grap

h o

f 2x

�5y

�5

par

alle

l

4 � 34

�0

� 0 �

3

6 �

2�

�2

�(�

2)

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Slo

pe

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill71

Gle

ncoe

Alg

ebra

2

Lesson 2-3

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.(1

,5),

(�1,

�3)

42.

(0,2

),(3

,0)

�3.

(1,9

),(0

,6)

3

4.(8

,�5)

,(4,

�2)

�5.

(�3,

5),(

�3,

�1)

un

def

ined

6.(�

2,�

2),(

10,�

2)0

7.(4

,5),

(2,7

)�

18.

(�2,

�4)

,(3,

2)9.

(5,2

),(�

3,2)

0

Gra

ph

th

e li

ne

pas

sin

g th

rou

gh t

he

give

n p

oin

t w

ith

th

e gi

ven

slo

pe.

10.(

0,4)

,m�

111

.(2,

�4)

,m�

�1

12.(

�3,

�5)

,m�

213

.(�

2,�

1),m

��

2

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

14.p

asse

s th

rou

gh (

0,1)

,per

pen

dicu

lar

to15

.pas

ses

thro

ugh

(0,

�5)

,par

alle

l to

th

e

a li

ne

wh

ose

slop

e is

gr

aph

of

y�

1

16.H

IKIN

GN

aom

i le

ft f

rom

an

ele

vati

on o

f 74

00 f

eet

at 7

:00

A.M

.an

d h

iked

to

an e

leva

tion

of 9

800

feet

by

11:0

0 A.M

.Wh

at w

as h

er r

ate

of c

han

ge i

n a

ltit

ude

?60

0 ft

/h

(0 ,–

5)

x

y

O(0

,1)

x

y

O

1 � 3

(–2,

–1)

x

y

O

(–3,

–5)

x

y

O

( 2, –

4)x

y

O( 0

, 4)

x

y

O

6 � 5

3 � 4

2 � 3

©G

lenc

oe/M

cGra

w-H

ill72

Gle

ncoe

Alg

ebra

2

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.(3

,�8)

,(�

5,2)

�2.

(�10

,�3)

,(7,

2)3.

(�7,

�6)

,(3,

�6)

0

4.(8

,2),

(8,�

1)u

nd

efin

ed5.

(4,3

),(7

,�2)

�6.

(�6,

�3)

,(�

8,4)

�

Gra

ph

th

e li

ne

pas

sin

g th

rou

gh t

he

give

n p

oin

t w

ith

th

e gi

ven

slo

pe.

7.(0

,�3)

,m�

38.

(2,1

),m

��

9.(0

,2),

m�

010

.(2,

�3)

,m�

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

11.p

asse

s th

rou

gh (

3,0)

,per

pen

dicu

lar

12.p

asse

s th

rou

gh (

�3,

�1)

,par

alle

l to

a l

ine

to a

lin

e w

hos

e sl

ope

is

wh

ose

slop

e is

�1

DEP

REC

IATI

ON

For

Exe

rcis

es 1

3–15

,use

th

e fo

llow

ing

info

rmat

ion

.A

mac

hin

e th

at o

rigi

nal

ly c

ost

$15,

600

has

a v

alu

e of

$75

00 a

t th

e en

d of

3 y

ears

.Th

e sa

me

mac

hin

e h

as a

val

ue

of $

2800

at

the

end

of 8

yea

rs.

13.F

ind

the

aver

age

rate

of

chan

ge i

n v

alu

e (d

epre

ciat

ion

) of

th

e m

ach

ine

betw

een

its

purc

has

e an

d th

e en

d of

3 y

ears

.�

$270

0 p

er y

ear

14.F

ind

the

aver

age

rate

of

chan

ge i

n v

alu

e of

th

e m

ach

ine

betw

een

th

e en

d of

3 y

ears

an

dth

e en

d of

8 y

ears

.�

$940

per

yea

r

15.I

nter

pret

the

sig

n of

you

r an

swer

s.It

is n

egat

ive

bec

ause

th

e va

lue

is d

ecre

asin

g.

( –3,

–1)

x

y

O

(3, 0

)x

y

O

3 � 2

( 2, –

3)

xO

y

( 0, 2

)

x

y

O

4 � 5

(2, 1

)

x

y

O

(0, –

3)

x

y

O

3 � 4

7 � 25 � 3

5 � 175 � 4

Pra

ctic

e (

Ave

rag

e)

Slo

pe

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3



Readin

g t

o L

earn

Math

em

ati

csS

lop

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3

©G

lenc

oe/M

cGra

w-H

ill73

Gle

ncoe

Alg

ebra

2

Lesson 2-3

Pre-

Act

ivit

yH

ow d

oes

slop

e ap

ply

to

the

stee

pn

ess

of r

oad

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-3

at

the

top

of p

age

68 i

n y

our

text

book

.

•W

hat

is

the

grad

e of

a r

oad

that

ris

es 4

0 fe

et o

ver

a h

oriz

onta

l di

stan

ceof

100

0 fe

et?

4%•

Wh

at i

s th

e gr

ade

of a

roa

d th

at r

ises

525

met

ers

over

a h

oriz

onta

ldi

stan

ce o

f 10

kil

omet

ers?

(1

kilo

met

er �

1000

met

ers)

5.25

%

Rea

din

g t

he

Less

on

1.D

escr

ibe

each

typ

e of

slo

pe a

nd

incl

ude

a s

ketc

h.

Typ

e o

f S

lop

eD

escr

ipti

on

of

Gra

ph

Ske

tch

Pos

itive

Th

e lin

e ri

ses

to t

he

rig

ht.

Zer

oT

he

line

is h

ori

zon

tal.

Neg

ativ

eT

he

line

falls

to

th

e ri

gh

t.

Und

efin

edT

he

line

is v

erti

cal.

2.a.

How

are

th

e sl

opes

of

two

non

vert

ical

par

alle

l li

nes

rel

ated

?T

hey

are

eq

ual

.

b.

How

are

the

slo

pes

of t

wo

obli

que

perp

endi

cula

r li

nes

rela

ted?

Th

eir

pro

du

ct is

�1.

Hel

pin

g Y

ou

Rem

emb

er

3.L

ook

up

the

term

s gr

ade,

pitc

h,s

lan

t,an

d sl

ope.

How

can

eve

ryda

y m

ean

ings

of

thes

ew

ords

hel

p yo

u r

emem

ber

the

defi

nit

ion

of

slop

e?S

amp

le a

nsw

er:

All

thes

e w

ord

sca

n b

e u

sed

wh

en y

ou

des

crib

e h

ow

mu

ch a

th

ing

sla

nts

up

war

d o

rd

ow

nw

ard

.Yo

u c

an d

escr

ibe

this

nu

mer

ical

ly b

y co

mp

arin

g r

ise

to r

un

.

x

y

O

x

y

O

x

y

O

x

y

O

©G

lenc

oe/M

cGra

w-H

ill74

Gle

ncoe

Alg

ebra

2

Aer

ial S

urv

eyo

rs a

nd

Are

aM

any

lan

d re

gion

s h

ave

irre

gula

r sh

apes

.Aer

ial

surv

eyor

s

supp

ly a

eria

l m

appe

rs w

ith

lis

ts o

f co

ordi

nat

es a

nd

elev

atio

ns

for

the

area

s th

at n

eed

to b

e ph

otog

raph

ed f

rom

th

e ai

r.T

hes

e m

aps

prov

ide

info

rmat

ion

abo

ut

the

hor

izon

tal

and

vert

ical

fe

atu

res

of t

he

lan

d.

Ste

p 1

Lis

t th

e or

dere

d pa

irs

for

the

vert

ices

in

co

un

terc

lock

wis

e or

der,

repe

atin

g th

e fi

rst

orde

red

pair

at

the

bott

om o

f th

e li

st.

Ste

p 2

Fin

d D

,th

e su

m o

f th

e do

wn

war

d di

agon

al p

rodu

cts

(fro

m l

eft

to r

igh

t).

D�

(5 �

5) �

(2 �

1) �

(2 �

3) �

(6 �

7)�

25 �

2 �

6 �

42 o

r 75

Ste

p 3

Fin

d U

,th

e su

m o

f th

e u

pwar

d di

agon

al p

rodu

cts

(fro

m l

eft

to r

igh

t).

U�

(2 �

7) �

(2 �

5) �

(6 �

1) �

(5 �

3)�

14 �

10 �

6 �

15 o

r 45

Ste

p 4

Use

th

e fo

rmu

la A

��1 2� (

D�

U)

to f

ind

the

area

.

A�

�1 2� (75

�45

)

��1 2� (

30)

or 1

5

Th

e ar

ea i

s 15

squ

are

un

its.

Cou

nt

the

nu

mbe

r of

squ

are

un

its

encl

osed

by

the

poly

gon

.Doe

s th

is r

esu

lt s

eem

rea

son

able

?

Use

th

e co

ord

inat

e m

eth

od t

o fi

nd

th

e ar

ea o

f ea

ch r

egio

n i

n s

qu

are

un

its.

1.2.

3.

20 u

nit

s214

un

its2

34 u

nit

s2

x

y

O

x

y

Ox

y

O

(5, 7

)

(2, 5

)

(2, 1

)

(6, 3

)

(5, 7

)

x

y

O

(2, 1

)

(2, 5

)

(5, 7

)

(6, 3

)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-3

2-3


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Wri

tin

g L

inea

r E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill75

Gle

ncoe

Alg

ebra

2

Lesson 2-4

Form

s o

f Eq

uat

ion

s

Slo

pe-

Inte

rcep

t F

orm

o

f a

Lin

ear

Eq

uat

ion

y�

mx

�b,

whe

re m

is t

he s

lope

and

bis

the

y-in

terc

ept

Po

int-

Slo

pe

Fo

rm

y�

y 1�

m(x

�x 1

), w

here

(x 1

, y 1

) ar

e th

e co

ordi

nate

s of

a p

oint

on

the

line

and

of

a L

inea

r E

qu

atio

nm

is t

he s

lope

of

the

line

Wri

te a

n e

qu

atio

n i

nsl

ope-

inte

rcep

t fo

rm f

or t

he

lin

e th

ath

as s

lop

e �

2 an

d p

asse

s th

rou

gh t

he

poi

nt

(3,7

).

Su

bsti

tute

for

m,x

,an

d y

in t

he

slop

e-in

terc

ept

form

.y

�m

x�

bS

lope

-inte

rcep

t fo

rm

7 �

(�2)

(3)

�b

(x,

y)

�(3

, 7)

, m

��

2

7 �

�6

�b

Sim

plify

.

13 �

bA

dd 6

to

both

sid

es.

Th

e y-

inte

rcep

t is

13.

Th

e eq

uat

ion

in

sl

ope-

inte

rcep

t fo

rm i

s y

��

2x�

13.

Wri

te a

n e

qu

atio

n i

nsl

ope-

inte

rcep

t fo

rm f

or t

he

lin

e th

ath

as s

lop

e an

d x

-in

terc

ept

5.

y�

mx

�b

Slo

pe-in

terc

ept

form

0 �

��(5

) �

b(x

, y

) �

(5,

0),

m�

0 �

�b

Sim

plify

.

��

bS

ubtr

act

from

bot

h si

des.

Th

e y-

inte

rcep

t is

�.T

he

slop

e-in

terc

ept

form

is

y�

x�

.5 � 3

1 � 3

5 � 3

5 � 35 � 3

5 � 3

1 � 31 � 3

1 � 3

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

1.sl

ope

�2,

pass

es t

hro

ugh

(�

4,6)

2.sl

ope

,y-i

nte

rcep

t 4

y�

�2x

�2

y�

x�

4

3.sl

ope

1,pa

sses

th

rou

gh (

2,5)

4.sl

ope

�,p

asse

s th

rou

gh (

5,�

7)

y�

x�

3y

��

x�

6

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

eac

h g

rap

h.

5.6.

7.

y�

�3x

�9

y�

xy

�x

�1

4 � 91 � 9

5 � 4

x

y

O

( –4,

1)

( 5, 2

)

x

y O

( 4, 5

)

( 0, 0

)

x

y

O

( 1, 6

)

( 3, 0

)

13 � 513 � 5

3 � 23 � 2

©G

lenc

oe/M

cGra

w-H

ill76

Gle

ncoe

Alg

ebra

2

Para

llel a

nd

Per

pen

dic

ula

r Li

nes

Use

th

e sl

ope-

inte

rcep

t or

poi

nt-

slop

e fo

rm t

o fi

nd

equ

atio

ns

of l

ines

th

at a

re p

aral

lel

or p

erpe

ndi

cula

r to

a g

iven

lin

e.R

emem

ber

that

par

alle

lli

nes

hav

e eq

ual

slo

pe.T

he

slop

es o

f tw

o pe

rpen

dicu

lar

lin

es a

re n

egat

ive

reci

proc

als,

that

is,t

hei

r pr

odu

ct i

s �

1.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Wri

tin

g L

inea

r E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

Wri

te a

n e

qu

atio

n o

f th

eli

ne

that

pas

ses

thro

ugh

(8,

2) a

nd

is

per

pen

dic

ula

r to

th

e li

ne

wh

ose

equ

atio

n i

s y

��

x�

3.

Th

e sl

ope

of t

he

give

n l

ine

is �

.Sin

ce t

he

slop

es o

f pe

rpen

dicu

lar

lin

es a

re n

egat

ive

reci

proc

als,

the

slop

e of

th

e pe

rpen

dicu

lar

lin

e is

2.

Use

th

e sl

ope

and

the

give

n p

oin

t to

wri

teth

e eq

uat

ion

.y

� y

1�

m(x

�x 1

)P

oint

-slo

pe f

orm

y�

2 �

2(x

�8)

(x1,

y1)

�(8

, 2)

, m

�2

y�

2 �

2x�

16D

istr

ibut

ive

Pro

p.

y�

2x�

14A

dd 2

to

each

sid

e.

An

equ

atio

n o

f th

e li

ne

is y

�2x

�14

.

1 � 2

1 � 2

Wri

te a

n e

qu

atio

n o

f th

eli

ne

that

pas

ses

thro

ugh

(�

1,5)

an

d i

sp

aral

lel

to t

he

grap

h o

f y

�3x

�1.

Th

e sl

ope

of t

he

give

n l

ine

is 3

.Sin

ce t

he

slop

es o

f pa

rall

el l

ines

are

equ

al,t

he

slop

eof

th

e pa

rall

el l

ine

is a

lso

3.U

se t

he

slop

e an

d th

e gi

ven

poi

nt

to w

rite

the

equ

atio

n.

y�

y 1�

m(x

�x 1

)P

oint

-slo

pe f

orm

y�

5 �

3(x

�(�

1))

(x1,

y1)

�(�

1, 5

), m

�3

y�

5 �

3x�

3D

istr

ibut

ive

Pro

p.

y�

3x�

8A

dd 5

to

each

sid

e.

An

equ

atio

n o

f th

e li

ne

is y

�3x

�8.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

1.pa

sses

th

rou

gh (

�4,

2),p

aral

lel

to t

he

lin

e w

hos

e eq

uat

ion

is

y�

x�

5y

�x

�4

2.pa

sses

th

rou

gh (

3,1)

,per

pen

dicu

lar

to t

he

grap

h o

f y

��

3x�

2y

�x

3.pa

sses

th

rou

gh (

1,�

1),p

aral

lel

to t

he

lin

e th

at p

asse

s th

rou

gh (

4,1)

an

d (2

,�3)

y�

2x�

3

4.pa

sses

th

rou

gh (

4,7)

,per

pen

dicu

lar

to t

he

lin

e th

at p

asse

s th

rou

gh (

3,6)

an

d (3

,15)

y�

7

5.pa

sses

th

rou

gh (

8,�

6),p

erpe

ndi

cula

r to

th

e gr

aph

of

2x�

y�

4y

��

x�

2

6.pa

sses

th

rou

gh (

2,�

2),p

erpe

ndi

cula

r to

th

e gr

aph

of

x�

5y�

6y

�5x

�12

7.pa

sses

th

rou

gh (

6,1)

,par

alle

l to

th

e li

ne

wit

h x

-in

terc

ept

�3

and

y-in

terc

ept

5

y�

x�

9

8.pa

sses

th

rou

gh (

�2,

1),p

erpe

ndi

cula

r to

th

e li

ne

y�

4x�

11y

��

x�

1 � 21 � 4

5 � 3

1 � 2

1 � 3

1 � 21 � 2



Skil

ls P

ract

ice

Wri

tin

g L

inea

r E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill77

Gle

ncoe

Alg

ebra

2

Lesson 2-4

Sta

te t

he

slop

e an

d y

-in

terc

ept

of t

he

grap

h o

f ea

ch e

qu

atio

n.

1.y

�7x

�5

7,�

52.

y�

�x

�3

�,3

3.y

�x

,04.

3x�

4y�

4�

,1

5.7y

�4x

�7

,�1

6.3x

�2y

�6

�0

,3

7.2x

�y

�5

2,�

58.

2y�

6 �

5x�

,3

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

eac

h g

rap

h.

9.10

.11

.

y�

3x�

1y

��

1y

��

2x�

3

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

12.s

lope

3,p

asse

s th

rou

gh (

1,�

3)13

.slo

pe �

1,pa

sses

th

rou

gh (

0,0)

y�

3x�

6y

��

x

14.s

lope

�2,

pass

es t

hro

ugh

(0,

�5)

15.s

lope

3,p

asse

s th

rou

gh (

2,0)

y�

�2x

�5

y�

3x�

6

16.p

asse

s th

rou

gh (

�1,

�2)

an

d (�

3,1)

17.p

asse

s th

rou

gh (

�2,

�4)

an

d (1

,8)

y�

�x

�y

�4x

�4

18.x

-in

terc

ept

2,y-

inte

rcep

t �

619

.x-i

nte

rcep

t ,y

-in

terc

ept

5

y�

3x�

6y

��

2x�

5

20.p

asse

s th

rou

gh (

3,�

1),p

erpe

ndi

cula

r to

th

e gr

aph

of

y�

�x

�4.

y�

3x�

101 � 3

5 � 2

7 � 23 � 2

x

y

O

( 0, 3

)

( 3, –

3)

x

y

O( –

3, –

1)( 4

, –1)

x

y

O

( –1,

–4)

( 1, 2

)

5 � 2

3 � 24 � 7

3 � 42 � 3

2 � 3

3 � 53 � 5

©G

lenc

oe/M

cGra

w-H

ill78

Gle

ncoe

Alg

ebra

2

Sta

te t

he

slop

e an

d y

-in

terc

ept

of t

he

grap

h o

f ea

ch e

qu

atio

n.

1.y

�8x

�12

8,12

2.y

�0.

25x

�1

0.25

,�1

3.y

��

x�

,0

4.3y

�7

0,5.

3x�

�15

�5y

,36.

2x�

3y�

10,�

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

eac

h g

rap

h.

7.8.

9.

y�

2y

�x

�2

y�

�x

�1

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

th

e li

ne

that

sat

isfi

es e

ach

set

of

con

dit

ion

s.

10.s

lope

�5,

pass

es t

hro

ugh

(�

3,�

8)11

.slo

pe

,pas

ses

thro

ugh

(10

,�3)

y�

�5x

�23

y�

x�

11

12.s

lope

0,p

asse

s th

rou

gh (

0,�

10)

13.s

lope

�,p

asse

s th

rou

gh (

6,�

8)

y�

�10

y�

�x

�4

14.p

asse

s th

rou

gh (

3,11

) an

d (�

6,5)

15.p

asse

s th

rou

gh (

7,�

2) a

nd

(3,�

1)

y�

x�

9y

��

x�

16.x

-in

terc

ept

3,y-

inte

rcep

t 2

17.x

-in

terc

ept

�5,

y-in

terc

ept

7

y�

�x

�2

y�

x�

7

18.p

asse

s th

rou

gh (

�8,

�7)

,per

pen

dicu

lar

to t

he

grap

h o

f y

�4x

�3

y�

�x

�9

19.R

ESER

VO

IRS

Th

e su

rfac

e of

Gra

nd

Lak

e is

at

an e

leva

tion

of

648

feet

.Du

rin

g th

ecu

rren

t dr

ough

t,th

e w

ater

lev

el i

s dr

oppi

ng

at a

rat

e of

3 i

nch

es p

er d

ay.I

f th

is t

ren

dco

nti

nu

es,w

rite

an

equ

atio

n t

hat

giv

es t

he

elev

atio

n i

n f

eet

of t

he

surf

ace

of G

ran

d L

ake

afte

r x

days

.y

��

0.25

x�

648

20.B

USI

NES

ST

ony

Mar

con

i’s c

ompa

ny

man

ufa

ctu

res

CD

-RO

M d

rive

s.T

he

com

pan

y w

ill

mak

e $1

50,0

00 p

rofi

t if

it

man

ufa

ctu

res

100,

000

driv

es,a

nd

$1,7

50,0

00 p

rofi

t if

it

man

ufa

ctu

res

500,

000

driv

es.T

he

rela

tion

ship

bet

wee

n t

he

nu

mbe

r of

dri

ves

man

ufa

ctu

red

and

the

prof

it i

s li

nea

r.W

rite

an

equ

atio

n t

hat

giv

es t

he

prof

it P

wh

en

ndr

ives

are

man

ufa

ctu

red.

P�

4n�

250,

000

1 � 4

7 � 52 � 3

1 � 41 � 4

2 � 3

2 � 32 � 3

4 � 54 � 5

2 � 33 � 2

x

y

O( 3

, –1)

( –3,

3)

x

y

O

( 4, 4

)

( 0, –

2)

x

y

O

( 0, 2

)

10 � 32 � 3

3 � 57 � 3

3 � 53 � 5

Pra

ctic

e (

Ave

rag

e)

Wri

tin

g L

inea

r E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csW

riti

ng

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-4

2-4

©G

lenc

oe/M

cGra

w-H

ill79

Gle

ncoe

Alg

ebra

2

Lesson 2-4

Pre-

Act

ivit

yH

ow d

o li

nea

r eq

uat

ion

s ap

ply

to

bu

sin

ess?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-4

at

the

top

of p

age

75 i

n y

our

text

book

.

•If

th

e to

tal

cost

of

prod

uci

ng

a pr

odu

ct i

s gi

ven

by

the

equ

atio

n

y�

5400

�1.

37x,

wh

at i

s th

e fi

xed

cost

? W

hat

is

the

vari

able

cos

t (f

or e

ach

ite

m p

rodu

ced)

?$5

400;

$1.3

7•

Wri

te a

lin

ear

equ

atio

n t

hat

des

crib

es t

he

foll

owin

g si

tuat

ion

:A

com

pan

y th

at m

anu

fact

ure

s co

mpu

ters

has

a f

ixed

cos

t of

$22

8,75

0 an

da

vari

able

cos

t of

$85

2 to

pro

duce

eac

h c

ompu

ter.

y�

228,

750

�85

2x

Rea

din

g t

he

Less

on

1.a.

Wri

te t

he

slop

e-in

terc

ept

form

of

the

equ

atio

n o

f a

lin

e.T

hen

exp

lain

th

e m

ean

ing

ofea

ch o

f th

e va

riab

les

in t

he

equ

atio

n.

y�

mx

�b

;m

is t

he

slo

pe

and

bis

th

ey-

inte

rcep

t.T

he

vari

able

s x

and

yar

e th

e co

ord

inat

es o

f an

y p

oin

t o

nth

e lin

e.

b.

Wri

te t

he

poin

t-sl

ope

form

of

the

equ

atio

n o

f a

lin

e.T

hen

exp

lain

th

e m

ean

ing

of e

ach

of t

he

vari

able

s in

th

e eq

uat

ion

.y

�y 1

�m

(x�

x 1);

mis

th

e sl

op

e.x

and

yar

e th

e co

ord

inat

es o

f an

y p

oin

t o

n t

he

line.

x 1an

d y

1ar

e th

e co

ord

inat

es o

f o

ne

spec

ific

po

int

on

th

e lin

e.

2.S

upp

ose

that

you

r al

gebr

a te

ach

er a

sks

you

to

wri

te t

he

poin

t-sl

ope

form

of

the

equ

atio

nof

th

e li

ne

thro

ugh

th

e po

ints

(�

6,7)

an

d (�

3,�

2).Y

ou w

rite

y�

2 �

�3(

x�

3) a

nd

you

r cl

assm

ate

wri

tes

y�

7 �

�3(

x�

6).W

hic

h o

f yo

u i

s co

rrec

t?E

xpla

in. Y

ou

are

bo

th c

orr

ect.

Eit

her

po

int

may

be

use

d a

s (x

1,y 1

) in

th

e p

oin

t-sl

op

e fo

rm.

You

use

d (

�3,

�2)

,an

d y

ou

r cl

assm

ate

use

d (

�6,

7).

3.Yo

u a

re a

sked

to

wri

te a

n e

quat

ion

of

two

lin

es t

hat

pas

s th

rou

gh (

3,�

5),o

ne

of t

hem

para

llel

to

and

one

of t

hem

per

pen

dicu

lar

to t

he

lin

e w

hos

e eq

uat

ion

is

y�

�3x

�4.

Th

e fi

rst

step

in

fin

din

g th

ese

equ

atio

ns

is t

o fi

nd

thei

r sl

opes

.Wh

at i

s th

e sl

ope

of t

he

para

llel

lin

e? W

hat

is

the

slop

e of

th

e pe

rpen

dicu

lar

lin

e?�

3;

Hel

pin

g Y

ou

Rem

emb

er

4.M

any

stu

den

ts h

ave

trou

ble

rem

embe

rin

g th

e po

int-

slop

e fo

rm f

or a

lin

ear

equ

atio

n.

How

can

you

use

th

e de

fin

itio

n o

f sl

ope

to r

emem

ber

this

for

m?

Sam

ple

an

swer

:

Wri

te t

he

def

init

ion

of

slo

pe:

m�

.Mu

ltip

ly b

oth

sid

es o

f th

is

equ

atio

n b

y x 2

�x 1

.Dro

p t

he

sub

scri

pts

in y

2an

d x

2.T

his

giv

es t

he

po

int-

slo

pe

form

of

the

equ

atio

n o

f a

line.

y 2�

y 1� x 2

�x 1

1 � 3

©G

lenc

oe/M

cGra

w-H

ill80

Gle

ncoe

Alg

ebra

2

Two

-In

terc

ept

Fo

rm o

f a

Lin

ear

Eq

uat

ion

You

are

alre

ady

fam

ilia

r w

ith

the

slop

e-in

terc

ept

form

of

a li

near

equ

atio

n,

y�

mx

�b.

Lin

ear

equa

tion

s ca

n al

so b

e w

ritt

en i

n th

e fo

rm � ax �

�� by �

�1

wit

h

x-in

terc

ept

aan

d y-

inte

rcep

t b.

Thi

s is

cal

led

two-

inte

rcep

t fo

rm.

Dra

w t

he

grap

h o

f � �x 3�

�� 6y �

�1.

The

gra

ph c

ross

es t

he x

-axi

s at

�3

and

the

y-ax

is a

t 6.

Gra

ph

(�3,

0) a

nd (

0,6)

,the

n dr

aw a

str

aigh

t li

ne t

hrou

gh t

hem

.

Wri

te 3

x�

4y�

12 i

n t

wo-

inte

rcep

t fo

rm.

� 13 2x ��

� 14 2y ��

�1 12 2�D

ivid

e by

12

to o

btai

n 1

on t

he r

ight

sid

e.

� 4x ��

� 3y ��

1S

impl

ify.

Th

e x-

inte

rcep

t is

4;t

he

y-in

terc

ept

is 3

.

Use

th

e gi

ven

in

terc

epts

a a

nd

b,t

o w

rite

an

eq

uat

ion

in

tw

o-in

terc

ept

form

.Th

en d

raw

th

e gr

aph

.S

ee s

tud

ents

’gra

ph

s.

1.a

��

2,b

��

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war

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r th

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ata

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ach

tab

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elow

.

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EL E

FFIC

IEN

CY

Th

e ta

ble

belo

w s

how

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e av

erag

e fu

el e

ffic

ien

cy i

n m

iles

per

gal

lon

of

new

car

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anu

fact

ure

d du

rin

g th

e ye

ars

list

ed.

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ce:N

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ork

Times

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anac

2.C

ON

GR

ESS

Th

e ta

ble

belo

w s

how

s th

e n

um

ber

of

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en s

ervi

ng

in t

he

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ited

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tes

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gres

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rin

g th

e ye

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1987

�19

99.

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all S

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ac

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25

101

31

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100

102

104

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men

in

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ng

ress

Year

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el E

ffic

ien

cy (

mp

g)

1960

15.5

1970

14.1

1980

22.6

1990

26.9

Year

Miles per Gallon

1960

1970

1980

1990

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icie

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Rs

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d R

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ay e

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icia

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arn

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dat

afo

r E

xerc

ises

1 a

nd

2.

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nce

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94

31

106

128

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rite

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peri

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ted

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ourl

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20

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2-5

2-5

Exam

ple

Exam

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An

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____

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Lesson 2-5

For

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ts a

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ach

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a.

a.D

raw

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lot.

b.

Use

tw

o or

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airs

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pre

dic

tion

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.c.

Use

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r p

red

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on e

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atio

n t

o p

red

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the

mis

sin

g va

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1.1a

.

1b.

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ple

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ng

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1) a

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.

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.

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57

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68

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yx

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2

For

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Dra

w a

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t.b

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se t

wo

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rs t

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rite

a p

red

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on e

qu

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n.

c.U

se y

our

pre

dic

tion

eq

uat

ion

to

pre

dic

t th

e m

issi

ng

valu

e.

1.FU

EL E

CO

NO

MY

Th

e ta

ble

give

s th

e ap

prox

imat

e w

eigh

ts i

n t

ons

and

esti

mat

es

for

over

all

fuel

eco

nom

y in

mil

es p

er g

allo

n

for

seve

ral

cars

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ple

an

swer

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ng

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4,24

) an

d

(2.4

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:y

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36.6

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amp

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mi/g

al

2.A

LTIT

UD

EIn

mos

t ca

ses,

tem

pera

ture

dec

reas

es w

ith

in

crea

sin

g al

titu

de.A

s A

nch

ara

driv

es i

nto

th

e m

oun

tain

s,h

er c

ar t

her

mom

eter

reg

iste

rs t

he

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ture

s (°

F)

show

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th

e ta

ble

at t

he

give

n a

ltit

ude

s (f

eet)

.

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amp

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7500

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(970

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EALT

HA

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a t

read

mil

l th

at u

ses

the

tim

e on

th

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eadm

ill

and

the

spee

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alki

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or r

un

nin

g to

est

imat

e th

e n

um

ber

of C

alor

ies

he

burn

s du

rin

g a

wor

kou

t.T

he

tabl

e gi

ves

wor

kou

t ti

mes

an

d C

alor

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burn

ed f

or s

ever

al w

orko

uts

.

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amp

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sin

g (

24,2

80)

and

(48,

440)

:y

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67x

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3c.S

amp

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ut

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in)

Calories Burned

010

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555

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500

400

300

200

100

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rnin

g C

alo

ries

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ori

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000

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00

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pera

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us

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itu

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per

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re (

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gh

t (t

on

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Fuel Economy (mi/gal)

00.

51.

01.

52.

02.

5

30 25 20 15 10 5Fuel

Eco

no

my V

ers

us

Weig

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t (t

on

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31.

41.

51.

82

2.1

2.4

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2-5

2-5



Readin

g t

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Math

em

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csM

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____

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Lesson 2-5

Pre-

Act

ivit

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ow c

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lin

ear

equ

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n m

odel

th

e n

um

ber

of

Cal

orie

s yo

u b

urn

exer

cisi

ng?

Rea

d th

e in

trod

uct

ion

to

Les

son

2-5

at

the

top

of p

age

81 i

n y

our

text

book

.

•If

a w

oman

ru

ns

5.5

mil

es p

er h

our,

abou

t h

ow m

any

Cal

orie

s w

ill

she

burn

in

an

hou

r?S

amp

le a

nsw

er:

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Cal

ori

es

•If

a m

an r

un

s 7.

5 m

iles

per

hou

r,ab

out

how

man

y C

alor

ies

wil

l h

e bu

rnin

hal

f an

hou

r?S

amp

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nsw

er:

397

Cal

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Rea

din

g t

he

Less

on

1.S

upp

ose

that

a s

et o

f da

ta c

an b

e m

odel

ed b

y a

lin

ear

equ

atio

n.E

xpla

in t

he

diff

eren

cebe

twee

n a

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plo

t of

th

e da

ta a

nd

a gr

aph

of

the

lin

ear

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els

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data

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amp

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t is

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.It

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up

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the

ind

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th

at r

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po

ints

.Th

e lin

ear

equ

atio

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as a

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nu

ou

s g

rap

h t

hat

is t

he

line

that

bes

t fi

ts t

he

dat

a p

oin

ts.

2.S

upp

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that

tu

itio

n a

t a

stat

e co

lleg

e w

as $

3500

per

yea

r in

199

5 an

d h

as b

een

incr

easi

ng

at a

rat

e of

$22

5 pe

r ye

ar.

a.W

rite

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at e

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ach

var

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you

r pr

edic

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equ

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xre

pre

sen

ts t

he

nu

mb

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f ye

ar s

ince

199

5 an

d y

rep

rese

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th

etu

itio

n in

th

at y

ear.

3.U

se t

his

mod

el t

o pr

edic

t th

e tu

itio

n a

t th

is c

olle

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1980

1982

1984

1986

1988

1990

1992

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1996

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3332

3130

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2-5

2-5


An

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2-6

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Lesson 2-6

Step

Fu

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s, C

on

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s, a

nd

th

e Id

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ty F

un

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he

char

tbe

low

lis

ts s

ome

spec

ial

fun

ctio

ns

you

sh

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be

fam

ilia

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ith

.

Fu

nct

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Wri

tten

as

Gra

ph

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tf(

x) �

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orig

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1

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e gr

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exa

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e of

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th

atco

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l se

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h f

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s a

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t fu

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,th

e id

enti

ty f

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p f

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a.b

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nct

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ates

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1.2.

3.

a co

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nct

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iden

tity

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f (x)

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aph

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ph

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ph t

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poin

ts a

nd

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nec

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em.Y

ou w

ould

exp

ect

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grap

h t

o lo

oksi

mil

ar t

o it

s pa

ren

t fu

nct

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x.

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f(x

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ce 2

doe

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ity,

stop

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aph

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2-6

2-6

Exer

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ple1

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0

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x) �

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�al

l rea

ls,

R �

no

np

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tive

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ls

R �

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nn

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ive

real

s

5.f(

x) �

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x

�2

6.h

(x)

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Total Cost ($)

10

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240

200

160

120 80 40

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2-6

2-6


An

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Readin

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o L

earn

Math

em

ati

csS

pec

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un

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2-6

2-6

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2

Lesson 2-6

Pre-

Act

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ow d

o st

ep f

un

ctio

ns

app

ly t

o p

osta

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ates

?

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d th

e in

trod

uct

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to

Les

son

2-6

at

the

top

of p

age

89 i

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our

text

book

.

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hat

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the

cost

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ould

eac

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55 c

ents

to

mai

l.A

nsw

ers

will

var

y.S

amp

le a

nsw

er:

1.1

ou

nce

s,1.

9 o

un

ces,

2.0

ou

nce

s

Rea

din

g t

he

Less

on

1.F

ind

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valu

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nct

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u r

emem

ber

wh

at t

he

grap

h

look

s li

ke.

a.co

nst

ant

fun

ctio

nS

amp

le a

nsw

er:

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met

hin

g is

co

nst

ant

if it

do

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ot

chan

ge.

Th

e y-

valu

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f a

con

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t fu

nct

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no

t ch

ang

e,so

th

eg

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he

abso

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yo

u h

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far

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fro

m 0

on

th

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um

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lin

e.It

mak

es n

o d

iffe

ren

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het

her

yo

u g

o t

o t

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left

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ht

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ng

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go

th

e sa

me

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tan

ce e

ach

tim

e.

c.st

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amp

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er:

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tep

fu

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rap

h lo

oks

like

ste

ps

that

go

up

or

do

wn

.

d.

iden

tity

fu

nct

ion

Sam

ple

an

swer

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e x-

an

d y

-val

ues

are

alw

ays

iden

tica

lly t

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sam

e fo

r an

y p

oin

t o

n t

he

gra

ph

.So

th

e g

rap

h is

a li

ne

thro

ug

h t

he

ori

gin

th

at h

as s

lop

e 1.

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts f

ind

the

grea

test

in

tege

r fu

nct

ion

con

fusi

ng.

Exp

lain

how

you

can

use

an

um

ber

lin

e to

fin

d th

e va

lue

of t

his

fu

nct

ion

for

an

y re

al n

um

ber.

An

swer

s w

ill v

ary.

Sam

ple

an

swer

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raw

a n

um

ber

lin

e th

at s

ho

ws

the

inte

ger

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est

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fu

nct

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fo

r an

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al n

um

ber

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ce t

hat

nu

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er o

n t

he

nu

mb

er li

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is a

n in

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e o

f th

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nct

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isth

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____

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____

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2-6

2-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

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ph

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itie

s

NA

ME

____

____

____

____

____

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lin

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ity,

like

y�

2x�

1,re

sem

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a l

inea

req

uat

ion

,bu

t w

ith

an

in

equ

alit

y si

gn i

nst

ead

of a

n e

qual

s si

gn.T

he

grap

h o

f th

e re

late

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nea

r eq

uat

ion

sep

arat

es t

he

coor

din

ate

plan

e in

to t

wo

hal

f-pl

anes

.Th

e li

ne

is t

he

bou

nda

ry o

f ea

ch h

alf-

plan

e.

To

grap

h a

lin

ear

ineq

ual

ity,

foll

ow t

hes

e st

eps.

1.G

raph

th

e bo

un

dary

,th

at i

s,th

e re

late

d li

nea

r eq

uat

ion

.If

the

ineq

ual

ity

sym

bol

is

or

�,t

he

bou

nda

ry i

s so

lid.

If t

he

ineq

ual

ity

sym

bol

is �

or

,th

e bo

un

dary

is

dash

ed.

2.C

hoo

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poi

nt

not

on

th

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un

dary

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d te

st i

t in

th

e in

equ

alit

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,0)

is a

goo

d po

int

toch

oose

if

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nda

ry d

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pas

s th

rou

gh t

he

orig

in.

3.If

a t

rue

ineq

ual

ity

resu

lts,

shad

e th

e h

alf-

plan

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nta

inin

g yo

ur

test

poi

nt.

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fal

sein

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sult

s,sh

ade

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oth

er h

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ph

x�

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rm,y

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x�

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ph t

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imil

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g li

nea

r in

equ

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Th

e gr

aph

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rela

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abso

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s th

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grap

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as

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lin

e if

th

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equ

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or

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fth

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equ

alit

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.C

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th

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dary

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rmin

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hic

h r

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nto

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ade.

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st g

raph

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.S

ince

th

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equ

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,t

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grap

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un

dary

is

soli

d.T

est

(0,0

).0

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0 �

1(x

, y)

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, 0)

0 ?

3�

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0

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ue

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____

____

____

____

____

____

____

____

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____

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____

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____

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2-7

2-7

Exer

cises

Exer

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ple

Exam

ple


An

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s


Skil

ls P

ract

ice

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ph

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____

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2

CO

MPU

TER

SF

or E

xerc

ises

10–

12,u

se t

he

foll

owin

g in

form

atio

n.

A s

choo

l sy

stem

is

buyi

ng

new

com

pute

rs.T

hey

wil

l bu

y de

skto

p co

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g $1

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per

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it,a

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it.T

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not

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000.

10.W

rite

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at d

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th

is s

itu

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1000

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f th

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to

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50 o

f th

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p co

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d 25

of

the

not

eboo

k co

mpu

ters

,w

ill

they

hav

e en

ough

mon

ey?

yes

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kto

ps

Notebooks

100

3050

2040

6070

8090

100

80 70 60 50 40 30 20 10

Co

mp

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d

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____

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2-7

2-7



Readin

g t

o L

earn

Math

em

ati

csG

rap

hin

g In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

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____

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2-7

2-7

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Lesson 2-7

Pre-

Act

ivit

yH

ow d

o in

equ

alit

ies

app

ly t

o fa

nta

sy f

ootb

all?

Rea

d th

e in

trod

uct

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to

Les

son

2-7

at

the

top

of p

age

96 i

n y

our

text

book

.

•W

hic

h o

f th

e co

mbi

nat

ion

s of

yar

ds a

nd

tou

chdo

wn

s li

sted

wou

ld D

ana

con

side

r a

good

gam

e?T

he

firs

t o

ne:

168

yard

s an

d

3 to

uch

do

wn

s•

Su

ppos

e th

at i

n o

ne

of t

he

gam

es D

ana

play

s,M

oss

gets

157

rec

eivi

ng

yard

s.W

hat

is

the

smal

lest

nu

mbe

r of

tou

chdo

wn

s h

e m

ust

get

in

ord

erfo

r D

ana

to c

onsi

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this

a g

ood

gam

e?3

Rea

din

g t

he

Less

on

1.W

hen

gra

phin

g a

lin

ear

ineq

ual

ity

in t

wo

vari

able

s,h

ow d

o yo

u k

now

wh

eth

er t

o m

ake

the

bou

nda

ry a

sol

id l

ine

or a

das

hed

lin

e? If

th

e sy

mb

ol i

s �

or

,t

he

line

isso

lid.

If t

he

sym

bo

l is

o

r �

,th

e lin

e is

das

hed

.

2.H

ow d

o yo

u k

now

wh

ich

sid

e of

th

e bo

un

dary

to

shad

e?S

amp

le a

nsw

er:

If t

he

test

po

int

giv

es a

tru

e in

equ

alit

y,sh

ade

the

reg

ion

co

nta

inin

g t

he

test

po

int.

Ifth

e te

st p

oin

t g

ives

a f

alse

ineq

ual

ity,

shad

e th

e re

gio

n n

ot

con

tain

ing

the

test

po

int.

3.M

atch

eac

h i

neq

ual

ity

wit

h i

ts g

raph

.

a.y

2x

�3

iiib

.y

��

2x�

3iv

c.y

�2x

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iid

.y

��

2x�

3i

i.ii

.ii

i.iv

.

Hel

pin

g Y

ou R

emem

ber

4.D

escr

ibe

som

e w

ays

in w

hic

h g

raph

ing

an i

neq

ual

ity

in o

ne

vari

able

on

a n

um

ber

lin

e is

sim

ilar

to

grap

hin

g an

in

equ

alit

y in

tw

o va

riab

les

in a

coo

rdin

ate

plan

e.H

ow c

an w

hat

you

kn

ow a

bou

t gr

aph

ing

ineq

ual

itie

s on

a n

um

ber

lin

e h

elp

you

to

grap

h i

neq

ual

itie

s in

a co

ordi

nat

e pl

ane?

Sam

ple

an

swer

:A

bo

un

dar

y o

n a

co

ord

inat

e g

rap

h is

sim

ilar

to a

n e

nd

po

int

on

a n

um

ber

lin

e g

rap

h.A

das

hed

lin

e is

sim

ilar

toa

circ

le o

n a

nu

mb

er li

ne:

bo

th a

re o

pen

an

d m

ean

no

t in

clu

ded

;th

eyre

pre

sen

t th

e sy

mb

ols

an

d �

.A s

olid

lin

e is

sim

ilar

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do

t o

n a

nu

mb

er li

ne:

bo

th a

re c

lose

d a

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mea

n in

clu

ded

;th

ey r

epre

sen

t th

esy

mb

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d

.

x

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Alg

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he

foll

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ragr

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sta

tes

a re

sult

you

mig

ht

be a

sked

to

prov

e in

am

ath

emat

ics

cou

rse.

Par

ts o

f th

e pa

ragr

aph

are

nu

mbe

red.

01L

et n

be a

pos

itiv

e in

tege

r.

02A

lso,

let

n1

�s(

n1)

be

the

sum

of

the

squ

ares

of

the

digi

ts i

n n

.

03T

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n1)

is

the

sum

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1,an

d n

3�

s(n

2)is

th

e su

m o

f th

e sq

uar

es o

f th

e di

gits

of

n2.

04In

gen

eral

,nk

�s(

nk

�1)

is

the

sum

of

the

squ

ares

of

the

digi

ts o

f n

k�

1.

05C

onsi

der

the

sequ

ence

:n,n

1,n

2,n

3,…

,nk,

….

06In

th

is s

equ

ence

eit

her

all

th

e te

rms

from

som

e k

on h

ave

the

valu

e 1,

07or

som

e te

rm,s

ay n

j,h

as t

he

valu

e 4,

so t

hat

th

e ei

ght

term

s 4,

16,3

7,58

,89,

145,

42,a

nd

20 k

eep

repe

atin

g fr

om t

hat

poi

nt

on.

Use

th

e p

arag

rap

h t

o an

swer

th

ese

qu

esti

ons.

1.U

se t

he

sen

ten

ce i

n l

ine

01.L

ist

the

firs

t fi

ve v

alu

es o

f n

.1,

2,3,

4,5

2.U

se 9

246

for

nan

d gi

ve a

n e

xam

ple

to s

how

th

e m

ean

ing

of l

ine

02.

n1

�s

(924

6) �

137,

bec

ause

137

�81

�4

�16

�36

3.In

lin

e 02

,whi

ch s

ymbo

l sh

ows

a fu

ncti

on?

Exp

lain

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fun

ctio

n in

a s

ente

nce.

s(n

);th

e su

m o

f th

e sq

uar

es o

f th

e d

igit

s o

f a

nu

mb

er is

a f

un

ctio

n

of

the

nu

mb

er

4.F

or n

�92

46,f

ind

n2

and

n3

as d

escr

ibed

in

sen

ten

ce 0

3.n

2�

59,n

3�

106

5.H

ow d

o th

e fi

rst

fou

r se

nte

nce

s re

late

to

sen

ten

ce 0

5?T

hey

exp

lain

ho

w t

o c

om

pu

te t

he

term

s o

f th

e se

qu

ence

.

6.U

se n

�31

an

d fi

nd

the

firs

t fo

ur

term

s of

th

e se

quen

ce.

31,1

0,1,

1

7.W

hic

h s

ente

nce

of

the

para

grap

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s il

lust

rate

d by

n�

31?

sen

ten

ce 0

6

8.U

se n

�61

an

d fi

nd

the

firs

t te

n t

erm

s.61

,37,

58,8

9,14

5,42

,20,

4,16

,37

9.W

hic

h s

ente

nce

is

illu

stra

ted

by n

�61

?se

nte

nce

07

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

2-7

2-7


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. A

B

C

B

B

C

B

A

C

D

B

D

k � 10

D

A

C

B

D

C

D

D

B

A

B

C

D

A

B

B

D

C

C

C

Chapter 2 Assessment Answer KeyForm 1 Form 2APage 99 Page 100 Page 101

An

swer

s

(continued on the next page)


13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: k � �16

C

C

B

A

B

D

C

A

D

D

B

C

D

A

D

B

A

C

D

C

k � 5

B

D

C

B

C

B

A

D

Chapter 2 Assessment Answer KeyForm 2A (continued) Form 2BPage 102 Page 103 Page 104


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: k � �7

p

a80

91011121314151617

5 6 7 8 9 10 11 12 13 14

Poin

ts S

core

d

Field Goals Attempted

y � ��32

�x

y � 2x � 7

y

xO

�14

y

xO

y

xO

xO

f (x )

y

xO

yes

5a2 � 8a

�3

no

yes

D � {�3}; R � {0, 1, 2, 3}; no

y

xO

(�3, 1)

(�3, 0)

(�3, 2)(�3, 3)

Chapter 2 Assessment Answer KeyForm 2CPage 105 Page 106

An

swer

s

No, because a variable appears in the denominator.

5x � 16y � 18; A � 5,B � �16, C � 18

x-intercept is 3; y-intercept is �2

Sample answer using(6, 9) and (10, 15):

p � �32

�a; 30

step function; D � all reals, R � all integers


1.

D � {�4, 0, 2, 4}; R � {0, 4}; yes

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: k � �3

t

n0

12141618202224262830

6 7 8 9 10 11 12 13 14 15

Tick

ets

Sold

Calls Made

y � ��52

�x

y � �x � 1

y

xO

�10

y

xO

y

xO

xO

f (x )

y

xO

yes

�4a2 � 2a � 3

14

yes

no

y

xO

(�4, 0)

(0, 0)

(4, 0)

(2, 4)

Chapter 2 Assessment Answer KeyForm 2DPage 107 Page 108

No, because the variablesare multiplied together.

2x � 56y � 1; A � 2, B � �56, C � 1x-intercept is �4; y-intercept is 3

Sample answer using (6, 12) and (8, 16):t � 2n; 32

step function; D � all reals, R � all integers


1.

D � {x � x � 1}; R � all reals; no

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

(160,150)

18.

19.

20.

B:k � �

35

�

y � �� x � 2 �

t

d0

20406080

100120140160

60 100 140 180 220

Tim

e (m

in)

Distance (km)

4x � 2y � 1

c

p

0048

1216202428

5 10 15 20 25 30 35

12p � 15c � 360

y � �4x � �23

�

y � ��25

�x � 2

$15.22 per year

5.6

y

xO

y

xO

absolute value function

x-intercept is �27

�;

no y-intercept

25x � 5y � 3;A � 25, B � �5, C � 3

A. yesB. no

3x � 5

�8

no

y

xO

Chapter 2 Assessment Answer KeyForm 3Page 109 Page 110

An

swer

s

Sample answer using

(40, 30) and (200, 150):

t � �34

�d; 120 min; much

lower

f (x) � ��2x if x � �1

0 if �1 � x � 2x if x � 2


Chapter 2 Assessment Answer KeyPage 111, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofrelations and functions, linear equations and inequalities,scatter plots, and prediction equations.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of relations andfunctions, linear equations and inequalities, scatter plots,and prediction equations.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofrelations and functions, linear equations and inequalities,scatter plots, and prediction equations.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the conceptsof relations and functions, linear equations andinequalities, scatter plots, and prediction equations.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

Chapter 2 Assessment Answer KeyPage 111, Open-Ended Assessment

Sample Answers


1. Students should describe two situations:If given as a mapping, a set of orderedpairs, or a table, determine whethereach member of the domain is pairedwith exactly one member of the range.If given as a graph, determine whetherthe graph passes the vertical line test.Functions must satisfy both of theseconditions.

2. Sample answer: The speed of a cardecreases as you apply the brakes. Thus,the rate of change of the speed withrespect to time is negative.

3. slope-intercept form: y � �12�x � 5

standard form: x � 2y � �10Sample answer: The slope-interceptform is most useful when graphing sincethe slope and the y-intercept can beeasily determined.

4. Students should indicate that the valuefor 1994 is likely to be more accuratethan the value for 2005 because valuesin the future may vary considerablyfrom the known data.

5. Students should state that all of thegraphs have the same shape, that thegraph of g(x) is the graph of the parentfunction f(x) translated, or shifted, left 2 units, and that the graph of h(x) is thegraph of f(x) translated right 3 units.The graph of y � � x � 500 � is the graphof f(x) translated left 500 units.

6. Alessia needed a test point to determinewhich side of the line to shade. Studentsshould indicate that Alessia made apoor choice since the point (�1, 7) lieson the graph of the boundary line and,therefore, does not provide theinformation she needs to complete thegraph.

7. The graph of the relation is an infiniteset of points represented graphically asa shaded region. Any vertical line willtherefore pass through an infinitenumber of points in the region. Thus,the relation is not a function.

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.

An

swer

s


Chapter 2 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 2–1 and 2–2) Quiz (Lessons 2–5 and 2–6)

Page 112 Page 113 Page 114

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. Sample answer: Thevertical line test letsyou use the graphof a relation to tellwhether the relationis a function. Eachvertical line mustintersect the graphin at most onepoint.

12. Sample answer: Alinear function is afunction that can bewritten in the formf(x) � mx � b,where m and b arereal numbers.

1.

2.

3.

4.

5.

Quiz (Lessons 2–3 and 2–4)

Page 113

1.

2.

3.

4.

5.

1.

(5, 25)

2.

3.

D � all reals; R � {y � y � 0}

Quiz (Lesson 2–7)

Page 114

1.2.

3.

4. y

xO

y

xO

y

xO

y � 3x � 1

y

xO

y

x

20

1520253035404550

4 6 8 10 12 14 16 18 20

y � �2x � 11

B

y

xO

undefined

�32

�

5

no

d

j

f

b

c

a

g

h

i

e D � all reals; R � all reals; yes

x-intercept is 4; y-intercept is 3;

y

xO

5x � y � �10; A � 5,B � �1, C � �10

Sample answer using (10, 21) and (20, 41): y � 2x � 1; 61


1.

2.

3.

4.

5.

6.

D � {0, 1, 2, 4}; R � {�2, 3, 4}; yes

7.

D � all reals; R � all reals; yes

8.

9.

10.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

y

xO

D � all real numbers;R � {y � y � 8}

about $237,610

��52

�

x-intercept is �83

�;

y-intercept is �2

119

D � {2, 3, 4}; R � {�7, 0}; no

�1�2 0 1 2 4 5 63

{y � �2 � y � 6} or [�2, 6)

�1�2�3�4 0 1 2 43

{x � x � �1} or (��, �1]

{3, 11}

Q, R

1

y � ��13

�x � 1

�18

�

5

y

xO

y

xO

(4, �2)

(1, 3)

(2, 4)

(0, 3)

B

D

C

C

D

Chapter 2 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 115 Page 116

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17.

18. DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

3 2

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

7 5

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2 1

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 2

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 2 Assessment Answer KeyStandardized Test Practice

Page 117 Page 118

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