Chapter 4 Resource Masters - Morgan Park High School

Chapter 4Resource Masters

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

Study Guide and Intervention Workbook 0-07-827753-1Study Guide and Intervention Workbook (Spanish) 0-07-827754-XSkills Practice Workbook 0-07-827747-7Skills Practice Workbook (Spanish) 0-07-827749-3Practice Workbook 0-07-827748-5Practice Workbook (Spanish) 0-07-827750-7

ANSWERS FOR WORKBOOKS The answers for Chapter 4 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, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 1. 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: 0-07-827728-0 Algebra 1Chapter 4 Resource Masters

2 3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 1

Contents

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

Lesson 4-1Study Guide and Intervention . . . . . . . . 213–214Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 215Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Reading to Learn Mathematics . . . . . . . . . . 217Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 218








Chapter 4 AssessmentChapter 4 Test, Form 1 . . . . . . . . . . . . 261–262Chapter 4 Test, Form 2A . . . . . . . . . . . 263–264Chapter 4 Test, Form 2B . . . . . . . . . . . 265–266Chapter 4 Test, Form 2C . . . . . . . . . . . 267–268Chapter 4 Test, Form 2D . . . . . . . . . . . 269–270Chapter 4 Test, Form 3 . . . . . . . . . . . . 271–272Chapter 4 Open-Ended Assessment . . . . . . 273Chapter 4 Vocabulary Test/Review . . . . . . . 274Chapter 4 Quizzes 1 & 2 . . . . . . . . . . . . . . . 275Chapter 4 Quizzes 3 & 4 . . . . . . . . . . . . . . . 276Chapter 4 Mid-Chapter Test . . . . . . . . . . . . 277Chapter 4 Cumulative Review . . . . . . . . . . . 278Chapter 4 Standardized Test Practice . . 279–280

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

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A35

© Glencoe/McGraw-Hill iv Glencoe Algebra 1

Teacher’s Guide to Using theChapter 4 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 4 Resource Masters includes the core materials neededfor Chapter 4. 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 1 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 4-1.Encourage them to add these pages to theirAlgebra Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 1 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 1

Assessment OptionsThe assessment masters in the Chapter 4Resources 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 1. 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 252–253. 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 _____

44

© Glencoe/McGraw-Hill vii Glencoe Algebra 1

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.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

arithmetic sequence

axes

common difference

coordinate plane

koh·AWRD·nuht

dilation

dy·LA·shuhn

function

image

inductive reasoning

ihn·DUHK·tihv

inverse

linear equation

mapping

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 1

Vocabulary Term Found on Page Definition/Description/Example

origin

quadrant

KWAH·druhnt

reflection

rotation

sequence

standard form

terms

transformation

translation

vertical line test

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Study Guide and InterventionThe Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

© Glencoe/McGraw-Hill 213 Glencoe Algebra 1

Less

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4-1

Identify Points In the diagram at the right, points are located in reference to two perpendicular number lines called axes. The horizontal number line is the x-axis, and the verticalnumber line is the y-axis. The plane containing the x- and y-axes is called the coordinate plane. Points in the coordinate plane arenamed by ordered pairs of the form (x, y). The first number, or x-coordinate corresponds to a number on the x-axis. The secondnumber, or y-coordinate, corresponds to a number on the y-axis.

The axes divide the coordinate plane into Quadrants I, II, III,and IV, as shown. The point where the axes intersect is called the origin. The origin has coordinates (0, 0).

x

y

O

PQ

R

Quadrant III

Quadrant II Quadrant I

Quadrant IV

Write an orderedpair for point R above.

The x-coordinate is 0 and the y-coordinate is 4. Thus the orderedpair for R is (0, 4).

Write ordered pairs for pointsP and Q above. Then name the quadrant inwhich each point is located.

The x-coordinate of P is �3 and the y-coordinate is�2. Thus the ordered pair for P is (�3, �2). P is inQuadrant III.

The x-coordinate of Q is 4 and the y-coordinate is�1. Thus the ordered pair for Q is (4, �1). Q is inQuadrant IV.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located.

1. N 2. P

3. Q 4. R

5. S 6. T

7. U 8. V

9. W 10. Z

11. A 12. B

13. Write the ordered pair that describes a point 4 units down from and 3 units to the rightof the origin.

14. Write the ordered pair that is 8 units to the left of the origin and lies on the x-axis.

x

y

O

P

QR

N

S

TU

V

W

Z

AB


Graph Points To graph an ordered pair means to draw a dot at the point on thecoordinate plane that corresponds to the ordered pair. To graph an ordered pair (x, y), beginat the origin. Move left or right x units. From there, move up or down y units. Draw a dot atthat point.

Plot each point on a coordinate plane.

a. R(�3, 2)Start at the origin. Move left 3 units since the x-coordinate is �3. Move up 2 units since the y-coordinate is 2. Draw a dot and label it R.

b. S(0, �3)Start at the origin. Since the x-coordinate is 0, the point will be located on the y-axis. Move down 3 units since the y-coordinate is �3. Draw a dot and label it S.

Plot each point on the coordinate plane at the right.

1. A(2, 4) 2. B(0, �3)

3. C(�4, �4) 4. D(�2, 0)

5. E(1, �4) 6. F(0, 0)

7. G(5, 0) 8. H(�3, 4)

9. I(4, �5) 10. J(�2, �2)

11. K(2, �1) 12. L(�1, �2)

13. M(0, 3) 14. N(5, �3)

15. P(4, 5) 16. Q(�5, 2)

x

y

O

AM

N

P

Q

BC

D

E

FG

H

I

JKL

x

y

O

R

S

Study Guide and Intervention (continued)

The Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

ExampleExample

ExercisesExercises

Skills PracticeThe Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Write the ordered pair for each point shown at the right.Name the quadrant in which the point is located.

1. A 2. B

3. C 4. D

5. E 6. F

Write the ordered pair for each point shown at the right.Name the quadrant in which the point is located.

7. G 8. H

9. J 10. K

11. L 12. M


13. M(2, 4) 14. N(�3, �3)

15. P(2, �2) 16. Q(0, 3)

17. R(4, 1) 18. S(�4, 1)


19. T(4, 0) 20. U(�3, 2)

21. W(�2, �3) 22. X(2, 2)

23. Y(�3, �2) 24. Z(3, �3)W

x

y

OT

XU

Y

Z

M

x

y

O

Q

S

N

R

P

x

y

O

GJ

KH

L

M

x

y

O

D

F

EC

B

A


Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located.

1. A 2. B

3. C 4. D

5. E 6. F

7. G 8. H

9. I 10. J

11. K 12. L


13. M(�3, 3) 14. N(3, �2) 15. P(5, 1)

16. Q(�4, �3) 17. R(0, 5) 18. S(�1, �2)

19. T(�5, 1) 20. V(1, �5) 21. W(2, 0)

22. X(�2, �4) 23. Y(4, 4) 24. Z(�1, 2)

25. CHESS Letters and numbers are used to show the positions of chess pieces and to describe their moves.For example, in the diagram at the right, a white pawn is located at f5. Name the positions of each of theremaining chess pieces.

ARCHAEOLOGY For Exercises 26 and 27, use the grid at the right that shows the location of arrowheads excavated at a midden—a place where people in the past dumped trash, food remains, and other discarded items.

26. Write the coordinates of each arrowhead.

27. Suppose an archaeologist discovers two other arrowheads located at (1, 2) and (3, 3). Draw an arrowhead at each of these locations on the grid.

0 1 2Meters

Met

ers

3 4

4

3

2

1

0

87654321

KingPawn

a b c d e f g h

Wx

y

O

R

N

Y

T

Q

Z

V

P

M

X

S

x

y

O

C

JB

AK

D

L

E

F

I

GH

Practice The Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

Reading to Learn MathematicsThe Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Pre-Activity How do archaeologists use coordinate systems?

Read the introduction to Lesson 4-1 at the top of page 192 in your textbook.

What do the terms grid system, grid, and coordinate system mean to you?

Reading the Lesson

1. Use the coordinate plane shown at the right.

a. Label the origin O.

b. Label the y-axis y.

c. Label the x-axis x.

2. Explain why the coordinates of the origin are (0, 0).

3. Use the ordered pair (�2, 3).

a. Explain how to identify the x- and y-coordinates.

b. Name the x- and y-coordinates.

c. Describe the steps you would use to locate the point for (�2, 3) on the coordinateplane.

4. What does the term quadrant mean?

Helping You Remember

5. Explain how the way the axes are labeled on the coordinate plane can help youremember how to plot the point for an ordered pair.


MidpointThe midpoint of a line segment is the point that lies exactly halfwaybetween the two endpoints of the segment. The coordinates of themidpoint of a line segment whose endpoints are (x1, y1) and (x2, y2)

are given by � , �.

Find the midpoint of each line segment with the given endpoints.

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

3. (�4, 4) and (4, �4) 4. (�3, �6) and (�10, �15)

Plot each segment in the coordinate plane. Then find the coordinates of themidpoint.

5. J�K� with J(5, 2) and K(�2, �4) 6. P�Q� with P(�1, 4) and Q(3, �1)

You are given the coordinates of one endpoint of a line segment and the midpoint M. Find the coordinates of the other endpoint.

7. A(�10, 3) and M(�6, 7) 8. D(�1, 4) and M(3, �6)

(3, –1)

(–1, 4)

(1, 1.5)

x

y

O

P

Q

(5, 2)

(–2, –4)

(1.5, –1)x

y

O

K

J

y1 � y2�2

x1 � x2�2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

Study Guide and InterventionTransformations on the Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Transform Figures Transformations are movements of geometric figures. Thepreimage is the position of the figure before the transformation, and the image is theposition of the figure after the transformation.

Reflection A figure is flipped over a line.

Translation A figure is slid horizontally, vertically, or both.

Dilation A figure is enlarged or reduced.

Rotation A figure is turned around a point.

Determine whether each transformation is a reflection, translation,dilation, or rotation.

a. The figure has been flipped over a line, so this is a reflection.

b. The figure has been turned around a point, so this is a rotation.

c. The figure has been reduced in size, so this is a dilation.

d. The figure has been shifted horizontally to the right, so this is atranslation.

Determine whether each transformation is a reflection, translation, dilation, orrotation.

1. 2.

3. 4.

5. 6.

ExampleExample

ExercisesExercises


Transform Figures on the Coordinate Plane You can perform transformationson a coordinate plane by changing the coordinates of each vertex. The vertices of the imageof the transformed figure are indicated by the symbol �, which is read prime.

Reflection over x-axis (x, y ) → (x, �y )

Reflection over y-axis (x, y ) → (�x, y )

Translation (x, y ) → (x � a, y � b)

Dilation (x, y ) → (kx, ky )

Rotation 90� counterclockwise (x, y ) → (�y, x )

Rotation 180� (x, y ) → (�x, �y )

A triangle has vertices A(�1, 1), B(2, 4), and C(3, 0). Find thecoordinates of the vertices of each image below.


Transformations on the Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

ExampleExample

a. reflection over the x-axisTo reflect a point over the x-axis,multiply the y-coordinate by �1.

A(�1, 1) → A�(�1, �1)B(2, 4) → B�(2, �4)C(3, 0) → C�(3, 0)

The coordinates of the image vertices areA�(�1, �1), B�(2, �4), and C�(3, 0).

b. dilation with a scale factor of 2Find the coordinates of the dilated figureby multiplying the coordinates by 2.

A(�1, 1) → A�(�2, 2)B(2, 4) → B�(4, 8)C(3, 0) → C�(6, 0)

The coordinates of the image vertices areA�(�2, 2), B�(4, 8), and C�(6, 0).

ExercisesExercises

Find the coordinates of the vertices of each figure after the given transformationis performed.

1. triangle RST with R(�2, 4), S(2, 0), T(�1, �1) reflected over the y-axis

2. triangle ABC with A(0, 0), B(2, 4), C(3, 0) rotated about the origin 180°

3. parallelogram ABCD with A(�3, 0), B(�2, 3), C(3, 3), D(2, 0) translated 3 units down

4. quadrilateral RSTU with R(�2, 2), S(2, 4), T(4, 4), U(4, 0) dilated by a factor of

5. triangle ABC with A(�4, 0), B(�2, 3), C(0, 0) rotated counterclockwise 90°

6. hexagon ABCDEF with A(0, 0), B(�2, 3), C(0, 4), D(3, 4), E(4, 2), F(3, 0) translated 2 units up and 1 unit to the left

1�2

Skills PracticeTransformations on the Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Identify each transformation as a reflection, translation, dilation, or rotation.

1. 2. 3.

4. 5. 6.

For Exercises 7–10, complete parts a and b.a. Find the coordinates of the vertices of each figure after the given

transformation is performed.b. Graph the preimage and its image.

7. triangle ABC with A(1, 2), B(4, �1), and 8. parallelogram PQRS with P(�2, �1),C(1, �1) reflected over the y-axis Q(3, �1), R(2, �3), and S(�3, �3)

translated 3 units up

9. trapezoid JKLM with J(�2, 1), K(2, 1), 10. triangle STU with S(3, 3), T(5, 1), and L(1, �1), and M(�1, �1) dilated by a U(1, 1) rotated 90° counterclockwise scale factor of 2 about the origin

S�

T�

U�

S

TUx

y

O

J�J

K�K

L�

L

M�

Mx

y

O

P� Q�

R�S�

PQ

RS

x

y

O

A�

B� C�

A

BCx

y

O


Identify each transformation as a reflection, translation, dilation, or rotation.

1. 2. 3.

For Exercises 4–6, complete parts a and b.a. Find the coordinates of the vertices of each figure after the given

transformation is performed.b. Graph the preimage and its image.

4. triangle DEF with D(2, 3), 5. trapezoid EFGH with 6. triangle XYZ with X(3, 1),E(4, 1), and F(1, �1) E(3, 2), F(3, �3), Y(4, �2), and Z(1, �3) translated 4 units left G(1, �2), and H(1, 1) rotated 90° counterclockwiseand 3 units down reflected over the y-axis about the origin

GRAPHICS For Exercises 7–9, use the diagram at the right and the following information.

A designer wants to dilate the rocket by a scale factor of , and then translate it 5 units up.

7. Write the coordinates for the vertices of the rocket.

8. Find the coordinates of the final position of the rocket.

9. Graph the image on the coordinate plane.

10. DESIGN Ramona transformed figure ABCDEF to design a pattern for a quilt. Name two different sets of transformations she could have used to design the pattern.

AB

CD

E

F

x

y

O

1�2

1�2

AB

C

DE

F

G

x

y

O

X�Y�

Z�X

YZ

E�

F�G�

H�E

FG

H

x

y

OD�

E�

F�

D

E

Fx

y

O

Practice Transformations on the Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

Reading to Learn MathematicsTransformations on the Coordinate Plane

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Pre-Activity How are transformations used in computer graphics?


In the sentence, “Computer graphic designers can create movement thatmimics real-life situations,” what phrase indicates the use oftransformations?

Reading the Lesson

1. Suppose you look at a diagram that shows two figures ABCDE and A�B�C�D�E�. If onefigure was obtained from the other by using a transformation, how do you tell which wasthe original figure?

2. Write the letter of the term and the Roman numeral of the figure that best matches eachstatement.

a. A figure is flipped over a line. A. dilation I.

b. A figure is turned around a point. B. translation II.

c. A figure is enlarged or reduced. C. reflection III.

d. A figure is slid horizontally, vertically, D. rotation IV.or both.


3. Give examples of things in everyday life that can help you remember what reflections,dilations, and rotations are.


The Legendary City of Ur The city of Ur was founded more than five thousand years ago inMesopotamia (modern-day Iraq). It was one of the world’s first cities.Between 1922 and 1934, archeologists discovered many treasures fromthis ancient city. A large cemetery from the 26th century B.C. wasfound to contain large quantities of gold, silver, bronze, and jewels. Themany cultural artifacts that were found, such as musical instruments,weapons, mosaics, and statues, have provided historians with valuableclues about the civilization that existed in early Mesopotamia.

1. Suppose that the ordered pairs below represent the volume (cm3) and mass (grams) of ten artifacts from the city of Ur. Plot each point on the graph.A(10, 150) B(150, 1350) C(200, 1760) D(50, 525) E(100, 1500) F(10, 88) G(200, 2100) H(150, 1675) I(100, 900) J(50, 440)

2. The equation relating mass, density, and volume for silver is m � 10.5V. Which of the points in Exercise 1 are solutions for this equation?

3. Suppose that the equation m � 8.8V relates mass, density, andvolume for the kind of bronze used in the ancient city of Ur. Whichof the points in Exercise 1 are solutions for this equation?

4. Explain why the graph in Exercise 1 shows only quadrant 1.

AF

JD

I

EB

H C

G

Volume (cm3)

Mas

s (g

ram

s)

0 50 100 150 200

2400

2200

2000

1800

1600

1400

1200

1000

800

600

400

200

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

Study Guide and InterventionRelations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

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4-3

Represent Relations A relation is a set of ordered pairs. A relation can berepresented by a set of ordered pairs, a table, a graph, or a mapping. A mapping illustrateshow each element of the domain is paired with an element in the range.

Express the relation{(1, 1), (0, 2), (3, �2)} as a table, agraph, and a mapping. State thedomain and range of the relation.

The domain for this relation is {0, 1, 3}.The range for this relation is {�2, 1, 2}.

X Y

103

12

�2

x

y

O

x y

1 1

0 2

3 �2

A person playingracquetball uses 4 calories per hour forevery pound he or she weighs.

a. Make a table to show the relation between weight and calories burned in onehour for people weighing 100, 110, 120, and 130 pounds.Source: The Math Teacher’s Book of Lists

b. Give the domain and range.domain: {100, 110, 120, 130}range: {400, 440, 480, 520}

c. Graph the relation.

Weight (pounds)

Cal

ori

es

100 1200

520

480

440

400

x

y

x y

100 400

110 440

120 480

130 520


ExercisesExercises

1. Express the relation{(�2, �1), (3, 3), (4, 3)} as a table, a graph, and amapping. Then determinethe domain and range.

2. The temperature in a house drops 2° for every hour the air conditioner is on between the hours of 6 A.M. and 11 A.M.Make a graph to show the relationship between time andtemperature if the temperature at 6 A.M. was 82°F.

Time (A.M.)

Tem

per

atu

re (

�F)

6 87 9 10 11 120

86

84

82

80

78

76

74

72

x

y

O

X Y

�234

�13

x y


Inverse Relations The inverse of any relation is obtained by switching the coordinatesin each ordered pair.

Express the relation shown in the mapping as a set of orderedpairs. Then write the inverse of the relation.

Relation: {(6, 5), (2, 3), (1, 4), (0, 3)}Inverse: {(5, 6), (3, 2), (4, 1), (3, 0)}

Express the relation shown in each table, mapping, or graph as a set of orderedpairs. Then write the inverse of each relation.

1. 2.

3. 4.

5. 6.

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Relations

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ExampleExample

ExercisesExercises

Skills PracticeRelations

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Express each relation as a table, a graph, and a mapping. Then determine thedomain and range.

1. {(�1, �1), (1, 1), (2, 1), (3, 2)}

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

3. {(3, �2), (1, 0), (�2, 4), (3, 1)}

Express the relation shown in each table, mapping, or graph as a set of orderedpairs. Then write the inverse of the relation.

4. 5. 6.

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Express each relation as a table, a graph, and a mapping. Then determine thedomain and range.

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

Express the relation shown in each table, mapping, or graph as a set of orderedpairs. Then write the inverse of the relation.

2. 3. 4.

BASEBALL For Exercises 5 and 6, use the graph that shows the batting average for Barry Bonds of the SanFrancisco Giants. Source: www.sportsillustrated.cnn.com

5. Find the domain and estimate the range.

6. Which seasons did Bonds have the lowest and highest batting averages?

METEORS For Exercises 7 and 8, use the table that shows the number of meteorsAnn observed each hour during a meteor shower.

7. What are the domain and range?

8. Graph the relation. Time (A.M.)

Meteor Shower

Nu

mb

er o

f M

eteo

rs

12–1 1–2 2–3 3–4 4–50

30

25

20

15

10

Time Number of (A.M.) Meteors

12 15

1 26

2 28

3 28

4 15

Year

Barry Bonds’ Batting

Ave

rag

e

’96 ’98’97 ’99 ’00 ’01 ’020

.34

.32

.30

.28

.26

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x y

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Practice Relations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

Reading to Learn MathematicsRelations

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How can relations be used to represent baseball statistics?


In 1997, Ken Griffey, Jr. had home runs and strikeouts.

This can be represented with the ordered pair ( , ).

Reading the Lesson

1. Look at page 205 in your textbook. There you see the same relation represented by a setof ordered pairs, a table, a graph, and a mapping.

a. In the list of ordered pairs, where do you see the numbers for the domain? thenumbers for the range?

b. What parts of the table show the domain and the range?

c. How do the table, the graph, and the mapping show that there are three orderedpairs in the relation?

2. Which tells you more about a relation, a list of the ordered pairs in the relation or thedomain and range of the relation? Explain.

3. Describe how you would find the inverse of the relation {(1, 2), (2, 4), (3, 6), (4, 8)}.


4. The first letters in two words and their order in the alphabet can sometimes help youremember their mathematical meaning. Two key terms in this lesson are domain andrange. Describe how the alphabet method could help you remember their meaning.


Inverse RelationsOn each grid below, plot the points in Sets A and B. Then connect thepoints in Set A with the corresponding points in Set B. Then find theinverses of Set A and Set B, plot the two sets, and connect those points.

Set A Set B

(�4, 0) (0, 1)(�3, 0) (0, 2)(�2, 0) (0, 3)(�1, 0) (0, 4)

Set A Set B

(�3, �3) (�2, 1)(�2, �2) (�1, 2)(�1, �1) (0, 3)(0, 0) (1, 4)

Set A Set B

(�4, 1) (3, 2)(�3, 2) (3, 2)(�2, 3) (3, 2)(�1, 4) (3, 2)

13. What is the graphical relationship between the line segments you drewconnecting points in Sets A and B and the line segments connecting pointsin the inverses of those two sets?

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Enrichment

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InverseSet A Set B

1.

2.

3.

4.

InverseSet A Set B

5.

6.

7.

8.

InverseSet A Set B

9.

10.

11.

12.

Study Guide and InterventionEquations as Relations

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Solve Equations The equation y � 3x � 4 is an example of an equation in twovariables because it contains two variables, x and y. The solution of an equation in twovariables is an ordered pair of replacements for the variables that results in a truestatement when substituted into the equation.

Find the solution set for y � �2x � 1, given the replacement set {(�2, 3), (0, �1), (1, �2), (3, 1)}.

Make a table. Substitute the x and y-values of eachordered pair into the equation.

x y y � �2x � 1 True or False

�2 33 � �2(�2) � 1

True3 � 3

0 �1�1 � �2(0) � 1

True�1 � �1

1 �2�2 � �2(1) � 1

False�2 � �3

3 11 � �2(3) � 1

False1 � �7

The ordered pairs (�2, 3), and (0, �1) result in truestatements. The solution set is {(�2, 3), (0, �1)}.

Solve b � 2a � 1 ifthe domain is {�2, �1, 0, 2, 4}.

Make a table. The values of a comefrom the domain. Substitute eachvalue of a into the equation todetermine the corresponding valuesof b in the range.

a 2a � 1 b (a, b)

�2 2(�2) � 1 �5 (�2, �5)

�1 2(�1) � 1 �3 (�1, �3)

0 2(0) � 1 �1 (0, �1)

2 2(2) � 1 3 (2, 3)

4 2(4) � 1 7 (4, 7)

The solution set is {(�2, �5),(�1, �3), (0, �1), (2, 3), (4, 7)}.


ExercisesExercises

Find the solution set of each equation, given the replacement set.

1. y � 3x � 1; {(0, 1), � , 2�, ��1, � �, (�1, �2)}

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

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

Solve each equation if the domain is (�4, �2, 0, 2, 4}.

4. x � y � 4

5. y � �4x � 6

6. 5a � 2b � 10

7. 3x � 2y � 12

8. 6x � 3y � 18

9. 4x � 8 � 2y

10. x � y � 8

11. 2x � y � 10

2�3

1�3


Graph Solution Sets You can graph the ordered pairs in the solution set of anequation in two variables. The domain contains values represented by the independentvariable. The range contains the corresponding values represented by the dependentvariable, which are determined by the given equation.

Solve 4x � 2y � 12 if the domain is (�1, 0, 2, 4}. Graph the solution set.

First solve the equation for y in terms of x.4x � 2y � 12 Original equation

4x � 2y � 4x � 12 � 4x Subtract 4x from each side.

2y � 12 � 4x Simplify.

� Divide each side by 2.

y � 6 � 2x Simplify.

12 � 4x�2

2y�2


Equations as Relations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4

ExampleExample

Substitute each value of x from the domain todetermine the corresponding value of y in the range.

x 6 � 2x y (x, y )

�1 6 � 2(�1) 8 (�1, 8)

0 6 � 2(0) 6 (0, 6)

2 6 � 2(2) 2 (2, 2)

4 6 � 2(4) �2 (4, �2)

Graph the solution set.

x

y

O

ExercisesExercises

Solve each equation for the given domain. Graph the solution set.

1. x � 2y � 4 for x � {�2, 0, 2, 4} 2. y � �2x � 3 for x � {�2, �1, 0, 1}

3. x � 3y � 6 for x � {�3, 0, 3, 6} 4. 2x � 4y � 8 for x � {�4, �2, 0, 2}

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Skills PracticeEquations as Relations

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Find the solution set for each equation, given the replacement set.

1. y � 3x � 1; {(2, 5), (�2, 7), (0, �1), (1, 1)}

2. y � 2x � 4; {(�1, �2), (�3, 2), (1, 6), (�2, 8)}

3. y � 7 � 2x; {(3, 1), (4, �1), (5, �3), (�1, 5)}

4. �3x � y � 2; {(�3, 7), (�2, �4), (�1, �1), (3, 11)}

Solve each equation if the domain is {�2, �1, 0, 2, 5}.

5. y � x � 4

6. y � 3x � 2

7. y � 2x � 1

8. x � y � 2

9. x � 3 � y

10. 2x � y � 4

11. 2x � y � 7

12. 4x � 2y � 6


13. y � 2x � 5 for x � {�5, �4, �2, �1, 0} 14. y � 2x � 3 for x � {�1, 1, 2, 3, 4}

15. 2x � y � 1 for x � {�2, �1, 0, 2, 3} 16. 2x � 2y � 6 for x � {�3, �1, 3, 4, 6}

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Find the solution set for each equation, given the replacement set.

1. y � 2 � 5x; {(3, 12), (�3, �17), (2, �8), (�1, 7)}

2. 3x � 2y � �1; {(�1, 1), (�2, �2.5), (�1, �1.5), (0, 0.5)}

Solve each equation if the domain is {�2, �1, 2, 3, 5}.

3. y � 4 � 2x

4. x � 8 � y

5. 4x � 2y � 10

6. 3x � 6y � 12

7. 2x � 4y � 16

8. x � y � 6


9. 2x � 4y � 8 for x � {�4, �3, �2, 2, 5} 10. x � y � 1 for x � {�4, �3, �2, 0, 4}

EARTH SCIENCE For Exercises 11 and 12, use the following information.Earth moves at a rate of 30 kilometers per second around the Sun. The equation d � 30t relates the distance d inkilometers Earth moves to time t in seconds.

11. Find the set of ordered pairs when t � {10, 20, 30, 45, 70}.

12. Graph the set of ordered pairs.

GEOMETRY For Exercises 13–15, use the following information.

The equation for the area of a triangle is A � bh. Suppose the area of triangle DEF is 30 square inches.

13. Solve the equation for h.

14. State the independent and dependent variables.

15. Choose 5 values for b and find the corresponding values for h.

1�2

Time (s)

Distance Earth Travels

Dis

tan

ce (

km)

0 10 20 30 40 50 60 70 80

2400

2100

1800

1500

1200

900

600

300

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NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsEquations as Relations

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Pre-Activity Why are equations of relations important in traveling?


• In the equation p � 0.69d, p represents and d

represents .

• How many variables are in the equation p � 0.69d?

Reading the Lesson

1. Suppose you make the following table to solve an equation that uses the domain {�3, �2, �1, 0, 1}.

x x � 4 y (x, y)

�3 �3 � 4 �7 (�3, �7)

�2 �2 � 4 �6 (�2, �6)

�1 �1 � 4 �5 (�1, �5)

0 0 � 4 �4 (0, �4)

1 1 � 4 �3 (1, �3)

a. What is the equation?

b. Which column shows the domain?

c. Which column shows the range?

d. Which column shows the solution set?

2. The solution set of the equation y � 2x for a given domain is {(�2, �4), (0, 0), (2, 4), (7, 14)}. Tell whether each sentence is true or false. If false,replace the underlined word(s) to make a true sentence.

a. The domain contains the values represented by the independent variable.

b. The domain contains the numbers �4, 0, 4, and 14.

c. For each number in the domain, the range contains a corresponding number that is avalue of the dependent variable.

3. What is meant by “solving an equation for y in terms of x”?


4. Remember, when you solve an equation for a given variable, that variable becomes thedependent variable. Write an equation and describe how you would identify thedependent variable.


Coordinate Geometry and AreaHow would you find the area of a triangle whose vertices have the coordinates A(-1, 2), B(1, 4), and C(3, 0)?

When a figure has no sides parallel to either axis, the height and base are difficult to find.

One method of finding the area is to enclose the figure in a rectangle and subtract the area of the surrounding triangles from the area of the rectangle.

Area of rectangle DEFC� 4 � 4 � 16 square units

Area of triangle I � (2)(4) � 4

Area of triangle II � (2)(4) � 4

Area of triangle III � (2)(2) � 2

Total � 10 square units

Area of triangle ABC � 16 � 10, or 6 square units

Find the areas of the figures with the following vertices.

1. A(-4, -6), B(0, 4), 2. A(6, -2), B(8, -10), 3. A(0, 2), B(2, 7),C(4, 2) C(12, -6) C(6, 10), D(9, -2)

24 square units 20 square units 55 square units

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II

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Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

4-44-4

Study Guide and InterventionGraphing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


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Identify Linear Equations A linear equation is an equation that can be written inthe form Ax � By � C. This is called the standard form of a linear equation.

Standard Form of a Linear EquationAx � By � C, where A � 0, A and B are not both zero, and A, B, and C are integers whose GCF is 1.

Determine whether y � 6 � 3xis a linear equation. If so, write the equationin standard form.

First rewrite the equation so both variables are onthe same side of the equation.

y � 6 � 3x Original equation

y � 3x � 6 � 3x � 3x Add 3x to each side.

3x � y � 6 Simplify.

The equation is now in standard form, with A � 3,B � 1 and C � 6. This is a linear equation.

Determinewhether 3xy � y � 4 � 2x is alinear equation. If so, write theequation in standard form.

Since the term 3xy has two variables,the equation cannot be written in theform Ax � By � C. Therefore, this isnot a linear equation.


ExercisesExercises

Determine whether each equation is a linear equation. If so, write the equation instandard form.

1. 2x � 4y 2. 6 � y � 8 3. 4x � 2y � �1

4. 3xy � 8 � 4y 5. 3x � 4 � 12 6. y � x2 � 7

7. y � 4x � 9 8. x � 8 � 0 9. �2x � 3 � 4y

10. 2 � x � y 11. y � 12 � 4x 12. 3xy � y � 8

13. 6x � 4y � 3 � 0 14. yx � 2 � 8 15. 6a � 2b � 8 � b

16. x � 12y � 1 17. 3 � x � x2 � 0 18. x2 � 2xy1�4

1�4

1�2


Graph Linear Equations The graph of a linear equation is a line. The line represents allsolutions to the linear equation. Also, every ordered pair on this line satisfies the equation.

Graph the equation y � 2x � 1.

Solve the equation for y.y � 2x � 1 Original equation

y � 2x � 2x � 1 � 2x Add 2x to each side.

y � 2x � 1 Simplify.

Select five values for the domain and make a table. Thengraph the ordered pairs and draw a line through the points.

x 2x � 1 y (x, y)

�2 2(�2) � 1 �3 (�2, �3)

�1 2(�1) � 1 �1 (�1, �1)

0 2(0) � 1 1 (0, 1)

1 2(1) � 1 3 (1, 3)

2 2(2) � 1 5 (2, 5)

Graph each equation.

1. y � 4 2. y � 2x 3. x � y � �1

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

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

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8

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Graphing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExampleExample

ExercisesExercises

Skills PracticeGraphing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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1. xy � 6 2. y � 2 � 3x 3. 5x � y � 4

4. y � 2x � 5 5. y � �7 � 6x 6. y � 3x2 � 1

7. y � 4 � 0 8. 5x � 6y � 3x � 2 9. y � 1


10. y � 4 11. y � 3x 12. y � x � 4

13. y � x � 2 14. y � 4 � x 15. y � 4 � 2x

16. x � y � 3 17. 10x � �5y 18. 4x � 2y � 6

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1. 4xy � 2y � 9 2. 8x � 3y � 6 � 4x 3. 7x � y � 3 � y

4. 5 � 2y � 3x 5. 4y � x � 9x 6. a � b � 2

7. 6x � 2y 8. � � 1 9. � � 7


10. x � y � 2 11. 5x � 2y � 7 12. 1.5x � 3y � 9

COMMUNICATIONS For Exercises 13–15, use the following information.A telephone company charges $4.95 per month for longdistance calls plus $0.05 per minute. The monthly cost cof long distance calls can be described by the equation c � 0.05m � 4.95, where m is the number of minutes.

13. Find the y-intercept of the graph of the equation.

14. Graph the equation.

15. If you talk 140 minutes, what is the monthly cost for long distance?

MARINE BIOLOGY For Exercises 16 and 17, use the following information.Killer whales usually swim at a rate of 3.2–9.7 kilometers per hour, though they can travel up to 48.4 kilometers perhour. Suppose a migrating killer whale is swimming at anaverage rate of 4.5 kilometers per hour. The distance d thewhale has traveled in t hours can be predicted by theequation d � 4.5t.

16. Graph the equation.

17. Use the graph to predict the time it takes the killer whale to travel 30 kilometers.

Time (minutes)

Long Distance

Co

st (

$)

0 40 80 120 160

14

12

10

8

6

4

2

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Practice Graphing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5

Reading to Learn MathematicsGraphing Linear Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


Less

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Pre-Activity How can linear equations be used in nutrition?


In the equation f � 0.3� �, what are the independent and dependent

variables?

Reading the Lesson

1. Describe the graph of a linear equation.

2. Determine whether each equation is a linear equation. Explain.

Equation Linear or non-linear? Explanation

a. 2x � 3y � 1

b. 4xy � 2y � 7

c. 2x2 � 4y � 3

d. � � 2

3. What do the terms x-intercept and y-intercept mean?


4. Describe the method you would use to graph 4x � 2y � 8.

4y�3

x�5

C�9


Taxicab GraphsYou have used a rectangular coordinate system to graph equations such as y � x � 1 on a coordinate plane. In a coordinate plane, the numbers in an ordered pair (x, y) can be any two real numbers.

A taxicab plane is different from the usual coordinate plane.The only points allowed are those that exist along the horizontal and vertical grid lines. You may think of the points as taxicabs that must stay on the streets.

The taxicab graph shows the equations y � �2 and y � x � 1.Notice that one of the graphs is no longer a straight line. It is now a collection of separate points.

Graph these equations on the taxicab plane at the right.

1. y � x � 1 2. y � �2x � 3

3. y � 2.5 4. x � �4

Use your graphs for these problems.

5. Which of the equations has the same graph in both the usual coordinate plane and the taxicab plane? x � �4

6. Describe the form of equations that have the same graph in both the usual coordinate plane and the taxicab plane.x � A and y � B, where A and B are integers

In the taxicab plane, distances are not measured diagonally, but along the streets.Write the taxi-distance between each pair of points.

7. (0, 0) and (5, 2) 8. (0, 0) and (-3, 2) 9. (0, 0) and (2, 1.5)7 units 5 units 3.5 units

10. (1, 2) and (4, 3) 11. (2, 4) and (-1, 3) 12. (0, 4) and (-2, 0)4 units 4 units 6 units

Draw these graphs on the taxicab grid at the right.

13. The set of points whose taxi-distance from (0, 0) is 2 units. indicated by crosses

14. The set of points whose taxi-distance from (2, 1) is 3 units. indicated by dots

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y � x � 1

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Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

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Study Guide and InterventionFunctions

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6


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Identify Functions Relations in which each element of the domain is paired withexactly one element of the range are called functions.

Determinewhether the relation {(6, �3),(4, 1), (7, �2), (�3, 1)} is afunction. Explain.

Since each element of the domain ispaired with exactly one element ofthe range, this relation is afunction.

Determine whether 3x � y � 6 is a function.

Since the equation is in the form Ax � By � C, the graph of theequation will be a line, as shownat the right.

If you draw a vertical line through each value of x, thevertical line passes through justone point of the graph. Thus, the line represents a function.

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ExercisesExercises

Determine whether each relation is a function.

1. 2. 3.

4. 5. 6.

7. {(4, 2), (2, 3), (6, 1)} 8. {(�3, �3), (�3, 4), (�2, 4)} 9. {(�1, 0), (1, 0)}

10. �2x � 4y � 0 11. x2 � y2 � 8 12. x � �4

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Function Values Equations that are functions can be written in a form called functionnotation. For example, y � 2x �1 can be written as f(x) � 2x � 1. In the function, xrepresents the elements of the domain, and f(x) represents the elements of the range.Suppose you want to find the value in the range that corresponds to the element 2 in thedomain. This is written f(2) and is read “f of 2.” The value of f(2) is found by substituting 2 for x in the equation.

If f(x) � 3x � 4, find each value.

a. f(3)f(3) � 3(3) � 4 Replace x with 3.

� 9 � 4 Multiply.

� 5 Simplify.

b. f(�2)f(�2) � 3(�2) � 4 Replace x with �2.

� �6 � 4 Multiply.

� �10 Simplify.

If f(x) � 2x � 4 and g(x) � x2 � 4x, find each value.

1. f(4) 2. g(2) 3. f(�5)

4. g(�3) 5. f(0) 6. g(0)

7. f(3) � 1 8. f � � 9. g� �

10. f(a2) 11. f(k � 1) 12. g(2c)

13. f(3x) 14. f(2) � 3 15. g(�4)

1�4

1�4


Functions

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ExampleExample

ExercisesExercises

Skills PracticeFunctions

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1. 2. 3.

4. 5. 6.

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

9. y � 2x � 5 10. y � 11

11. 12. 13.

If f(x) � 3x � 2 and g(x) � x2 � x, find each value.

14. f(4) 15. f(8)

16. f(�2) 17. g(2)

18. g(�3) 19. g(�6)

20. f(2) � 1 21. f(1) � 1

22. g(2) � 2 23. g(�1) � 4

24. f(x � 1) 25. g(3b)

x

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3 7

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x y

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13



1. 2. 3.

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

6. x � �2 7. y � 2

If f(x) � 2x � 6 and g(x) � x � 2x2, find each value.

8. f(2) 9. f �� 10. g(�1)

11. g�� 12. f(7) � 9 13. g(�3) � 13

14. f(h � 9) 15. g(3y) 16. 2[g(b) � 1]

WAGES For Exercises 17 and 18, use the following information.

Martin earns $7.50 per hour proofreading ads at a local newspaper. His weekly wage w canbe described by the equation w � 7.5h, where h is the number of hours worked.

17. Write the equation in functional notation.

18. Find f(15), f(20), and f(25).

ELECTRICITY For Exercises 19–21, use the following information.

The table shows the relationship between resistance R and current I in a circuit.

19. Is the relationship a function? Explain.

20. If the relation can be represented by the equation IR � 12, rewrite the equation in functional notation so that the resistance R is a function of the current I.

21. What is the resistance in a circuit when the current is 0.5 ampere?

Resistance (ohms) 120 80 48 6 4

Current (amperes) 0.1 0.15 0.25 2 3

1�3

1�2

x

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x y

1 �5

�4 3

7 6

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15

Practice Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsFunctions

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Pre-Activity How are functions used in meteorology?


If pressure is the independent variable and temperature is the dependentvariable, what are the ordered pairs for this set of data?

Reading the Lesson

1. The statement, “Relations in which each element of the is paired with exactly one element of the are called functions,” is false. How can you change the underlined words to make the statement true?

2. Describe how each method shows that the relation represented is a function.

a. mapping

b. vertical line test


3. A student who was trying to help a friend remember how functions are different fromrelations that are not functions gave the following advice: Just remember that functionsare very strict and never give you a choice. Explain how this might help you rememberwhat a function is.

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X Y

1237

�2�1

04

domainrange


Composite FunctionsThree things are needed to have a function—a set called the domain,a set called the range, and a rule that matches each element in thedomain with only one element in the range. Here is an example.

Rule: f(x) � 2x � 1f(x) � 2x � 1

f(1) � 2(1) � 1 � 2 � 1 � 3

f(2) � 2(2) � 1 � 4 � 1 � 5

f(-3) � 2(�3) � 1 � �6 � 1 � �5

Suppose we have three sets A, B, and C and two functions describedas shown below.

Rule: f(x) � 2x � 1 Rule: g( y) � 3y � 4g( y) � 3y � 4

g(3) � 3(3) � 4 � 5

Let’s find a rule that will match elements of set A with elements ofset C without finding any elements in set B. In other words, let’s finda rule for the composite function g[f(x)].

Since f(x) � 2x � 1, g[ f(x)] � g(2x � 1).Since g( y) � 3y � 4, g(2x � 1) � 3(2x � 1) � 4, or 6x � 1.Therefore, g[ f(x)] � 6x � 1.

Find a rule for the composite function g[f(x)].

1. f(x) � 3x and g( y) � 2y � 1 2. f(x) � x2 � 1 and g( y) � 4y

g[f(x)] � 6x � 1 g[f(x)] � 4x2 � 4

3. f(x) � �2x and g( y) � y2 � 3y 4. f(x) � and g( y) � y�1

g[f(x)] � 4x2 � 6x g[f(x)] � x � 3

5. Is it always the case that g[ f(x)] � f[ g(x)]? Justify your answer.

No. For example, in Exercise 1,f [g(x)] � f(2x � 1) � 3(2x � 1) � 6x � 3, not 6x � 1.

1�x � 3

1 3

x f(x)

5

g[f(x)]

A B C

1 3

2 5

–3 –5

x f(x)

Enrichment

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Study Guide and InterventionArithmetic Sequences

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Recognize Arithmetic Sequences A sequence is a set of numbers in a specificorder. If the difference between successive terms is constant, then the sequence is called anarithmetic sequence.

Arithmetic Sequencea numerical pattern that increases or decreases at a constant rate or value called thecommon difference

Determine whether thesequence 1, 3, 5, 7, 9, 11, … is anarithmetic sequence. Justify youranswer.

If possible, find the common differencebetween the terms. Since 3 � 1 � 2,5 � 3 � 2, and so on, the common differenceis 2.

Since the difference between the terms of 1, 3, 5, 7, 9, 11, … is constant, this is anarithmetic sequence.

Determine whether thesequence 1, 2, 4, 8, 16, 32, … is anarithmetic sequence. Justify youranswer.

If possible, find the common differencebetween the terms. Since 2 � 1 � 1 and 4 � 2 � 2, there is no common difference.

Since the difference between the terms of 1, 2, 4, 8, 16, 32, … is not constant, this isnot an arithmetic sequence.


ExercisesExercises

Determine whether each sequence is an arithmetic sequence. If it is, state thecommon difference.

1. 1, 5, 9, 13, 17, … 2. 8, 4, 0, �4, �8, … 3. 1, 3, 9, 27, 81, …

4. 10, 15, 25, 40, 60, … 5. �10, �5, 0, 5, 10, … 6. 8, 6, 4, 2, 0, �2, …

7. 4, 8, 12, 16, … 8. 15, 12, 10, 9, … 9. 1.1, 2.1, 3.1, 4.1, 5.1, …

10. 8, 7, 6, 5, 4, … 11. 0.5, 1.5, 2.5, 3.5, 4.5, … 12. 1, 4, 16, 64, …

13. 10, 14, 18, 22, … 14. �3, �6, �9, �12, … 15. 7, 0, �7, �14, …


Write Arithmetic Sequences You can use the common difference of an arithmeticsequence to find the next term of the sequence. Each term after the first term is found byadding the preceding term and the common difference.

Terms of an Arithmetic SequenceIf a1 is the first term of an arithmetic sequence with common difference d, then the sequence is a1, a1 � d, a1 � 2d, a1 � 3d, … .

nth Term of an Arithmetic Sequence an � a1 � (n � 1)d


Arithmetic Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

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Find the next threeterms of the arithmetic sequence 28,32, 36, 40, ….

Find the common difference bysubtracting successive terms.

The common difference is 4.Add 4 to the last given term, 40, to getthe next term. Continue adding 4 untilthe next three terms are found.

The next three terms are 44, 48, 52.

40

� 4 � 4 � 4

44 48 52

28

� 4 � 4 � 4

32 36 40

Write an equation for thenth term of the sequence 12, 15, 18, 21, … .

In this sequence, a1 is 12. Find the commondifference.

The common difference is 3.Use the formula for the nth term to write anequation.

an � a1 � (n � 1)d Formula for the nth term

an � 12 � (n � 1)3 a1 � 12, d � 3

an � 12 � 3n � 3 Distributive Property

an � 3n � 9 Simplify.

The equation for the nth term is an � 3n � 9.

12

� 3 � 3 � 3

15 18 21


ExercisesExercises

Find the next three terms of each arithmetic sequence.

1. 9, 13, 17, 21, 25, … 2. 4, 0, �4, �8, �12, … 3. 29, 35, 41, 47, …

4. �10, �5, 0, 5, … 5. 2.5, 5, 7.5, 10, … 6. 3.1, 4.1, 5.1, 6.1, …

Find the nth term of each arithmetic sequence described.

7. a1 � 6, d � 3, n � 10 8. a1 � �2, d � �3, n � 8 9. a1 � 1, d � �5, n � 20

10. a1 � �3, d � �2, n � 50 11. a1 � �12, d � 4, n � 20 12. a1 � 1, d � , n � 11

Write an equation for the nth term of the arithmetic sequence.

13. 1, 3, 5, 7, … 14. �1, �4, �7, �10, … 15. �4, �9, �14, �19, …

1�2

Skills PracticeArithmetic Sequences

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1. 4, 7, 9, 12, … 2. 15, 13, 11, 9, …

3. 7, 10, 13, 16, … 4. �6, �5, �3, �1, …

5. �5, �3, �1, 1, … 6. �9, �12, �15, �18, …


7. 3, 7, 11, 15, … 8. 22, 20, 18, 16, …

9. �13, �11, �9, �7, … 10. �2, �5, �8, �11, …

11. 19, 24, 29, 34, … 12. 16, 7, �2, �11, …


13. a1 � 6, d � 3, n � 12 14. a1 � �2, d � 5, n � 11

15. a1 � 10, d � �3, n � 15 16. a1 � �3, d � �3, n � 22

17. a1 � 24, d � 8, n � 25 18. a1 � 8, d � �6, n � 14

19. 8, 13, 18, 23, … for n � 17 20. �10, �3, 4, 11, … for n � 12

21. 12, 10, 8, 6, … for n � 16 22. 12, 7, 2, �3, … for n � 25

Write an equation for the nth term of each arithmetic sequence. Then graph thefirst five terms of the sequence.

23. 7, 13, 19, 25, … 24. 30, 26, 22, 18, … 25. �7, �4, �1, 2, …

n

an

2 4 6

4

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1. 21, 13, 5, �3, … 2. �5, 12, 29, 46, … 3. �2.2, �1.1, 0.1, 1.3, …


4. 82, 76, 70, 64, … 5. �49, �35, �21, �7, … 6. , , , 0, …


7. a1 � 7, d � 9, n � 18 8. a1 � �12, d � 4, n � 36

9. �18, �13, �8, �3, … for n � 27 10. 4.1, 4.8, 5.5, 6.2, … for n � 14

11. a1 � , d � , n � 15 12. a1 � 2 , d � 1 , n � 24

Write an equation for the nth term of each arithmetic sequence. Then graph thefirst five terms of the sequence.

13. 9, 13, 17, 21, … 14. �5, �2, 1, 4, … 15. 19, 31, 43, 55, …

BANKING For Exercises 16 and 17, use the following information.

Chem deposited $115.00 in a savings account. Each week thereafter, he deposits $35.00 intothe account.

16. Write a formula to find the total amount Chem has deposited for any particular numberof weeks after his initial deposit.

17. How much has Chem deposited 30 weeks after his initial deposit?

18. STORE DISPLAY Tamika is stacking boxes of tissue for a store display. Each row oftissues has 2 fewer boxes than the row below. The first row has 23 boxes of tissues. Howmany boxes will there be in the tenth row?

2 4 6

60

40

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30

20

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1�2

1�2

1�4

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1�2

3�4

Practice Arithmetic Sequences

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Reading to Learn MathematicsArithmetic Sequences

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Pre-Activity How are arithmetic sequences used to solve problems in science?


Describe the pattern in the data.

Reading the Lesson

1. Do the recorded altitudes in the introduction form an arithmetic sequence? Explain.

2. What is meant by successive terms?

3. Complete the table.

PatternIs the sequence increasing Is there a common difference? or decreasing? If so, what is it?

a. 2, 5, 8, 11, 14, …

b. 55, 50, 45, 40, …

c. 1, 2, 4, 9, 16, …

d. , 0, � , �1, …

e. 2.6, 2.9, 3.2, 3.5, …


4. Use the pattern 3, 7, 11, 15, … to explain how you would help someone else learn how tofind the 10th term of an arithmetic sequence.

1�2

1�2


Arithmetic SeriesAn arithmetic series is a series in which each term after the first may befound by adding the same number to the preceding term. Let S stand forthe following series in which each term is 3 more than the preceding one.

S � 2 � 5 � 8 � 11 � 14 � 17 � 20

The series remains the same if we S � 2 � 5 � 8 � 11 � 14 � 17 � 20reverse the order of all the terms. So S � 20 � 17 � 14 � 11 � 8 � 5 � 2let us reverse the order of the terms 2S � 22 � 22 � 22 � 22 � 22 � 22 � 22and add one series to the other, term 2S � 7(22)by term. This is shown at the right.

S � � 7(11) � 77Let a represent the first term of the series.Let � represent the last term of the series.Let n represent the number of terms in the series.

In the preceding example, a � 2, l � 20, and n � 7. Notice that whenyou add the two series, term by term, the sum of each pair of terms is 22.That sum can be found by adding the first and last terms, 2 � 20 or a � �. Notice also that there are 7, or n, such sums. Therefore, the valueof 2S is 7(22), or n(a � �) in the general case. Since this is twice the sumof the series, you can use the following formula to find the sum of anyarithmetic series.

S �

Find the sum: 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9

a � 1, � � 9, n � 9, so S � � � 45

Find the sum: �9 � (�5) � (�1) � 3 � 7 � 11 � 15

a � 29, � � 15, n � 7, so S � � � 21

Find the sum of each arithmetic series.

1. 3 � 6 � 9 � 12 � 15 � 18 � 21 � 24

2. 10 � 15 � 20 � 25 � 30 � 35 � 40 � 45 � 50

3. �21 � (�16) � (�11) � (�6) � (�1) � 4 � 9 � 14

4. even whole numbers from 2 through 100

5. odd whole numbers between 0 and 100

7 6�2

7(�9 � 15)��2

9 10�2

9(1 � 9)��2

n(a � �)��2

7(22)�2

Enrichment

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Example 1Example 1

Example 2Example 2

Study Guide and InterventionWriting Equations from Patterns

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Look for Patterns A very common problem-solving strategy is to look for a pattern.Arithmetic sequences follow a pattern, and other sequences can follow a pattern.

Find the next threeterms in the sequence 3, 9, 27, 81, … .

Study the pattern in the sequence.

Successive terms are found bymultiplying the last given term by 3.


81

� 3 � 3 � 3

243 729 2187

3

� 3 � 3 � 3

9 27 81

Find the next three termsin the sequence 10, 6, 11, 7, 12, 8, … .

Study the pattern in the sequence.

Assume that the pattern continues.


8

� 5 � 4 � 5

13 9 14

� 4 � 5

10 6 11

� 4 � 5

7 12

� 4

8


ExercisesExercises

1. Give the next two items for the pattern below.

Give the next three numbers in each sequence.

2. 2, 12, 72, 432, … 3. 7, �14, 28, �56, …

4. 0, 10, 5, 15, 10, … 5. 0, 1, 3, 6, 10, …

6. x � 1, x � 2, x � 3, … 7. x, , , , …x�4

x�3

x�2


Write Equations Sometimes a pattern can lead to a general rule that can be written asan equation.

Suppose you purchased a number of packages of blank compactdisks. If each package contains 3 compact disks, you could make a chart to showthe relationship between the number of packages of compact disks and thenumber of disks purchased. Use x for the number of packages and y for thenumber of compact disks.

Make a table of ordered pairs for several points of the graph.

The difference in the x values is 1, and the difference in the y values is 3. This patternshows that y is always three times x. This suggests the relation y � 3x. Since the relation isalso a function, we can write the equation in functional notation as f(x) � 3x.

1. Write an equation for the function in 2. Write an equation for the function infunctional notation. Then complete functional notation. Then complete the table. the table.

3. Write an equation in functional notation. 4. Write an equation in functional notation.

x

y

Ox

y

O

x �2 �1 0 1 2 3

y 10 7 4

x �1 0 1 2 3 4

y �2 2 6

Number of Packages 1 2 3 4 5

Number of CDs 3 6 9 12 15


Writing Equations from Patterns

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ExampleExample

ExercisesExercises

Skills PracticeWriting Equations from Patterns

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Find the next two items for each pattern. Then find the 19th figure in the pattern.

1.

2.

Find the next three terms in each sequence.

3. 1, 4, 10, 19, 31, … 4. 15, 14, 16, 15, 17, 16, …

5. 29, 28, 26, 23, 19, … 6. 2, 3, 2, 4, 2, 5, …

7. x, x � 1, x � 2, … 8. y, 4y, 9y, 16y, …

Write an equation in function notation for each relation.

9. 10. 11.

12. 13. 14.

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1. Give the next two items for the pattern. Then find the 21st figure in the pattern.

Find the next three terms in each sequence

2. �5, �2, �3, 0, �1, 2, 1, 4, … 3. 0, 1, 3, 6, 10, 15, …

4. 0, 1, 8, 27, … 5. 3, 2, 4, 3, 5, 4, …

6. a � 1, a � 4, a � 9, … 7. 3d � 1, 4d � 2, 5d � 3, …

Write an equation in function notation for each relation.

8. 9. 10.

BIOLOGY For Exercises 11 and 12, use the following information.

Male fireflies flash in various patterns to signal location and perhaps to ward off predators.Different species of fireflies have different flash characteristics, such as the intensity of theflash, its rate, and its shape. The table below shows the rate at which a male firefly is flashing.

11. Write an equation in function notation for the relation.

12. How many times will the firefly flash in 20 seconds?

13. GEOMETRY The table shows the number of diagonals that can be drawn from one vertex in a polygon. Write anequation in function notation for the relation and find thenumber of diagonals that can be drawn from one vertex in a 12-sided polygon.

Sides 3 4 5 6

Diagonals 0 1 2 3

Time (seconds) 1 2 3 4 5

Number of Flashes 2 4 6 8 10

x

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x

y

O

;

Practice Writing Equations from Patterns

NAME ______________________________________________ DATE ____________ PERIOD _____

4-84-8

Reading to Learn MathematicsWriting Equations From Patterns

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity Why is writing equations from patterns important in science?


• What is meant by the term linear pattern?

• Describe any arithmetic sequences in the data.

Reading the Lesson

1. What is meant by the term inductive reasoning?

2. For the figures below, explain why Figure 5 does not follow the pattern.

3. Describe the steps you would use to find the pattern in the sequence 1, 5, 25, 125, … .


4. What are some basic things to remember when you are trying to discover whether thereis a pattern in a sequence of numbers?

1 2 3 4 5


Traceable FiguresTry to trace over each of the figures below without tracing the samesegment twice.

The figure at the left cannot be traced, but the one at the right can.The rule is that a figure is traceable if it has no points, or exactly twopoints where an odd number of segments meet. The figure at the lefthas three segments meeting at each of the four corners. However, thefigure at the right has exactly two points, L and Q, where an oddnumber of segments meet.

Determine whether each figure can be traced. If it can, thenname the starting point and number the sides in the order inwhich they should be traced.

1. 2.

3. 4.

E

A B C

D KF

HJG

611

10 5

4

13 2781213

9

E

W XY

F

H G

6

11

5

4

1

3

278 10

9

P V

T

S

U

R

A B

E C

D

65

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2

A B

CD

K

Q

M

J P L

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Chapter 4 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

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

For Questions 1 and 2, use the graph to answer each question.

1. Write the ordered pair for point J.A. (1, �2) B. (2, 1)C. (1, 2) D. (2, �1) 1.

2. Name the quadrant in which point J is located.A. I B. II C. III D. IV 2.

3. To plot the point (1, �2), start at the origin and moveA. left 1 unit and down 2 units. B. left 2 units and down 1 unit.C. left 2 units and up 1 unit. D. right 1 unit and down 2 units. 3.

4. Which best describes a dilation?A. a flip B. an enlargement or reductionC. a slide D. a turn around a point 4.

5. Determine what type or transformation is shown.A. translation B. rotationC. reflection D. dilation 5.

6. A line segment with end points at point M(�2, 0) and N(�1, 2) is translated 4 units left. What are the coordinates of the endpoints after the transformation is performed?A. M�(�6, �4), N�(�5, �2) B. M�(�6, 0), N�(�5, 2)C. M�(�2, �4), N�(�1, �2) D. M�(2, 0), N�(3, 2) 6.

7. Which table, mapping, or graph does not show the relation {(�1, 1), (1, 2), (2, �2), (4, 3)}?A. B.

C. D. 7.

8. What is the inverse of the relation {(0, 1), (1, 0), (3, �4)}?A. {(0, 0), (1, 1), (3, �4)} B. {(0, 1), (1, 0), (3, �4)}C. {(1, 0), (0, 1), (4, �3)} D. {(1, 0), (0, 1), (�4, 3)} 8.

9. Solve the equation y � 3x if the domain is {�1, 0, 1}.A. {(�3, �1), (0, 0), (3, 1)} B. {(�1, �3), (0, 0), (1, 3)}C. {(�1, 2), (0, 3), (1, 4)} D. {(�1, �1), (0, 0), (1, 1)} 9.

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xO

J

x �1 1 2 4

y 1 2 �2 3


10. The distance d a car travels in t hours is given by the function d � 55t.Find d when t � 5.A. 275 B. 11 C. 60 D. 50 10.

11. Which equation is a linear equation?

A. 4m2 � 6 B. 3a � 5b � �3 C. �23�xy � �

34�y � 0 D. x2 � y2 � 0 11.

12. Which line shown at the right is the graph of y � 2x � 4?A. l B. the x-axisC. p D. q 12.

13. Determine which relation is a function.A. B.

C. y � �15�x � 2 D. {(3, 0), (�2, �2), (7, �2), (�2, 0)} 13.

14. Determine which relation is a function.A. {(1, 1), (1, 2)} B. x � 5 � 1 C. y � 9 D. x � 2 14.

15. If h(r) � �23�r � 6, what is the value of h(�9)?

A. 12 B. 0 C. �12 D. �6�23� 15.

16. Determine which sequence is an arithmetic sequence.

A. 3, 6, 12, 24, … B. �15�, �

17�, �

19�, �1

11�

, …

C. �7, �3, 1, 5, … D. �10, 5, ��52�, �

54�, … 16.

17. Find the next three terms of the arithmetic sequence 5, 9, 13, 17, … .A. 21 B. 21, 25, 29 C. 41, 45, 49 D. 21, 41, 61 17.

18. Find the next two numbers of the sequence 1, 2, 4, 8, 16, … .A. 32, 64 B. 24, 32 C. 20, 22 D. 18, 20 18.

19. Find the next item in the pattern, .

A. B. C. D. 19.

20. Write an equation in function notation for the relation at the right.A. f(x) � 2x B. f(x) � �xC. f(x) � x � 1 D. f(x) � 1 � x 20.

Bonus Solve y � x � 3 if the domain is {0, 1, 2, 3}.Graph the equation y � x � 3. B:

X

12

34

5�2

Y

NAME DATE PERIOD

44 Chapter 4 Test, Form 1 (continued)

y

xO

p

q

�

y

xO

x 3 4 4 5

y �1 2 3 6

Chapter 4 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent



1. Write the ordered pair for point I.A. (1, �4) B. (�1, 4)C. (�4, 1) D. (4, �1) 1.

2. Name the quadrant in which point K is located.A. I B. II C. III D. IV 2.

3. To plot the point (�7, 4), start at the origin and moveA. left 7 units and down 4 units. B. left 4 units and down 7 units.C. left 7 units and up 4 units. D. left 4 units and up 7 units. 3.

4. What type of transformation is shown at the right?A. reflection B. translationC. dilation D. rotation 4.

5. The letter P is flipped over a line. Which transformation is this? A. reflection B. translation C. dilation D. rotation 5.

6. Triangle ABC with A(�4, �1), B(�2, �1), and C(�4, 4) is rotated 180�about the origin. What are the coordinates of the vertices of the figure after the triangle is rotated?A. A�(�1, 4), B�(�1, 2), C�(4, 4) B. A�(�1, �4), B�(�1, �2), C�(4, �4)C. A�(4, 1), B�(2, 1), C�(4, �4) D. A�(4, �1), B�(2, �1), C�(4, 4) 6.

7. State the range of the relation {(�2, 5), (0, �1), (�1, 4), (�1, 5)}.A. {�2, 0, �1} B. {�1, 4, 5} C. {�2, 0} D. {�1, 4} 7.

8. What is the inverse of the relation {(3, �1), (2, 5), (�3, 4), (�2, 0)}?A. {(�3, 1), (�2, �5), (3, �4), (2, 0)} B. {(�2, 0), (�3, 4), (2, 5), (3, �1)} C. {(�1, 3), (5, 2), (4, �3), (0, �2)} D. {(1, �3), (�5, �2), (�4, 3), (0, 2)} 8.

9. Solve y � 2x � 7 if the domain is {�3, 1, 5}.A. {(�3, �1), (1, �9), (5, �17)} B. {(�3, �10), (1, �6), (5, �2)}C. {(�3, �13), (1, �5), (5, 3)} D. {(�3, �6), (1, 2), (5, 10)} 9.

10. Which equation is not a linear equation?

A. �4v � 2w � 7 B. �4x

� � y C. x � �5 D. �2x� � �

3y� � 6 10.

11. If g(x) � 3x2�4x � 1, what is the value of g(�2)?A. 21 B. 29 C. 45 D. 5 11.

12. Find the next two numbers of the sequence 7, 11, 8, 12, 9, 13, 10, … .A. 11, 15 B. 14, 15 C. 14, 11 D. 15, 12 12.

44

y

xOE

F

L

K

J

HI


Chapter 4 Test, Form 2A (continued)

13. Which is the graph of 2x � y � 1 if the domain is {�2, �1, 0, 1, 2}?A. B. C. D. 13.

14. Which line shown at the right is the graph of x � 2y � 4?A. l B. mC. p D. q 14.

15. Which equation has a graph that is a vertical line?A. 2x � y B. 3x � 2 � 0C. y � 5 � 3 D. x � y � 0 15.

16. Determine which relation is not a function.A. B.

C. D. 16.

17. Determine which sequence is not an arithmetic sequence.

A. �7, 0, 7, 14, … B. 0, �12�, 1, �

32�, … C. 10, 6, 2, �2, … D. 2, 4, 8, 16, … 17.

18. Which equation describes the nth term of the arithmetic sequence 7, 10, 13, 16, …?A. an � 3n � 4 B. an � 7 � 3n C. an � �4n � 3 D. an � 3n�4 18.

19. Find the next two items in the pattern .

A. B. C. D. 19.

20. Write an equation in function notation for the relation shown at the right.A. f(x)��2x B. f(x) � x�2C. f(x) � 2x � 2 D. f(x) � �x � 2 20.

Bonus Find the solution set of 2x � y2 � 8, given the replacement set {(2, �2), (6, 2), (6, 22), (4, 0), (12, �4)}. B:

X

0

2

9

�4�3

56

Y

X

0�4

2�3

95

Yy

xO

y

xO

y

xO

y

xO

y

xO

NAME DATE PERIOD

44

y

xOpq

�

m

y

xO

x �2 0 1 3

y 0 0 2 1

Chapter 4 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent



1. Write the ordered pair for point L.A. (1, �4) B. (�4, 1)C. (�1, �4) D. (�4, �1) 1.

2. Name the quadrant in which point E is located.A. I B. II C. III D. IV 2.

3. To plot the point (�5, 2), start at the origin and moveA. right 2 units and down 5 units. B. left 5 units and up 2 units.C. left 5 units and down 2 units. D. right 2 units and up 5 units. 3.

4. What type of transformation is shown at the right?A. reflection B. translationC. dilation D. rotation 4.

5. A letter R is turned around a point. Which type of transformation is this?A. reflection B. translation C. dilation D. rotation 5.

6. Triangle HIJ with H(�2, 1), I(1, 4), and J(3, 1) is reflected over the x-axis.What are the coordinates of the vertices of the figure after the triangle is reflected?A. H�(�2, �1), I�(1, �4), J�(3, �1) B. H�(2, 1), I�(�1, 4), J�(�3, 1)C. H�(2, �1), I�(�1,�4), J�(�3, �1) D. H�(1, �2), I�(4, 1), J�(1, 3) 6.

7. What is the range of the relation {(5, 3), (2, 8), (�1, 1), (6, 1)}?A. {(�1, 1), (6, 1)} B. {�1, 2, 5, 6}C. {1, 3, 8} D. {3, 8} 7.

8. What is the inverse of the relation {(�2, �1), (2, 1), (�2, 4), (�4, 0)}?A. {(�4, 0), (�2, 4), (2, 1), (�2, �1)} B. {(2, 1), (�2, �1), (2, �4), (4, 0)} C. {(1, 2), (�1, �2), (�4, 2), (0, 4)} D. {(�1, �2), (1, 2), (4, �2), (0, �4)} 8.

9. Solve y � �2x � 8 if the domain is {�2, 0, 4}.A. {(�2, 12), (0, 8), (4, 0)} B. {(�2, 4), (0, 8), (4, 0)}C. {(�2, 6), (0, 8), (4, 12)} D. {(�2, 4), (0, 0), (4, �8)} 9.

10. Which equation is not a linear equation?

A. 2x � 5y � 3 B. y � �10 C. 5 � 3xy D. y � �7x

� � 4 10.

11. If f(x) � 2x2�3x � 5, what is the value of f(�3)?A. 14 B. 50 C. 2 D. 32 11.

12. Find the next two numbers of the sequence 1, 8, 5, 12, 9, 16, 13, … .A. 24, 21 B. 20, 17 C. 20, 24 D. 17, 21 12.

44

y

xOE

F

JG

I H

K

L


Chapter 4 Test, Form 2B (continued)

13. Which is the graph of 2x � y � �5 if the domain is {�4, �3, �2, �1, 0}?A. B. C. D. 13.

14. Which line shown at the right is the graph of x � 2y � 6?A. r B. sC. t D. v 14.

15. Which equation has a graph that is a horizontal line?A. x � 7 � 0 B. x � yC. 2y � 3 � 4 D. x � y � 0 15.

16. Determine which relation is a function.A. B. C. D. 16.

17. Determine which sequence is an arithmetic sequence.A. �16, �12, �8, �4, … B. 1, 4, 2, 5, 3, …C. 4, 8, 16, 32, … D. 1, 1, 2, 3, 5, … 17.

18. Which equation describes the nth term of the arithmetic sequence �12, �14, �16, �18, …?A. an � �2n�10 B. an � �12�2nC. an � 10n�2 D. an � �2n � 10 18.

19. Find the next two items in the pattern, .

A. B. C. D. 19.

20. Write an equation in function notation for the relation.A. f(x) � �2x B. f(x) � �x � 2C. f(x) � x�2 D. f(x) � 2x � 2 20.

Bonus If f(x) � x � x2, find f(2m � 3). B:

X

�32

�41

3

�1

25

Yy

xO

y

xO

y

xO

y

xO

y

xO

y

xO

NAME DATE PERIOD

44

r s

xOt v

y

y

xO

x y�2 7

0 0

1 �2

1 3

Chapter 4 Test, Form 2C


Write the ordered pair for each point 1.shown at the right. Name the quadrant in which the point is located. 2.

1. S 2. T 3.3. U 4–6.


4. A(3, 2) 5. B(0, �2) 6. C(�3, 4)

For Questions 7 and 8, determine whether each transformation is a reflection, translation, dilation, or rotation.

7. 8.

9. Express the relation {(1, 4), (�2, 3), (�5, 0), (7, 4), (3, 2)} asa mapping. Then determine the domain and range.

10. Express the relation shown in the graph as a set of ordered pairs.Then write the inverse of the relation.

11. Find the solution set for x � 2y � 9,given the replacement set {(�1, �5), (3, �3), (5, 2), (7, 1)}.

12. Solve y � �2x � 1 if the domain is {�2, �1, 0, 1, 2}.Graph the solution set.

For Questions 13 and 14, determine whether each equation is a linear equation. If so, write the equation in standard form.

13. xy � 6 14. 2x � 3y � 7 � 3

15. Graph the equation x � 4y � 2.

y

xO

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

y

xOU

S

T

y

xO

7.

8.

9.

10.

11.

12.

13.

14.

15. y

xO

y

xO

X Y


Chapter 4 Test, Form 2C (continued)


16. 17.

For Questions 18 and 19, if f(x) � x2 � 3x � 2, find each value.

18. f(2) 19. f(5v)

20. Determine whether the sequence �10, �7, �4, �1, … is an arithmetic sequence. If so, state the common difference.

21. Find the next three terms of the arithmetic sequence 8, 15, 22, 29, ….}.

22. Write an equation for the nth term of the sequence 12, 5, �2, �9, … . Then graph the first five terms of the sequence.

23. Find the next two items in the pattern,

24. Use the table below that shows the amount of gasoline a car consumes for different distances driven.

Write an equation in function notation for the relationship between distance and gasoline used.

25. Maya wants to enlarge a rectangular cartoon that is 2 inches wide and 3 inches tall by a scale factor of 1.5.What will be the dimensions of the new cartoon? Use a coordinate system to draw a picture of the original cartoon and the enlarged cartoon. Place one corner of each cartoon at the origin and for each cartoon write the coordinates of the other three vertices.

Bonus Graph x � 3, y � �1, and 2x � 2y � 0 on a coordinate plane. Give the vertices of the figure formed by the three lines.

.

y

xO

X

7�3

42

3�2

4

�11

Y

NAME DATE PERIOD

44

distance (mi) 1 2 3 4 5

gasoline (gal) 0.04 0.08 0.12 0.16 0.20

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B: y

xO

y

xO

an

nO 1

51015

2 4 5

�10�15

Chapter 4 Test, Form 2D


Write the ordered pair for each point 1.shown at the right. Name the quadrant in which the point is located. 2.

1. M 2. N 3.

3. P 4–6.


4. A(�3, 0) 5. B(2, �2) 6. C(1, 3)

For Questions 7 and 8, determine whether each transformation is a reflection, translation, dilation, or rotation.

7. 8.

9. Express the relation {(8, 7), (�2, 0), (1, 0), (4, �5), (�1, 3)} as a mapping. Then determine the domain and range.


11. Find the solution set for 3x � y � 7,given the replacement set {(2, 1), (�3, �16), (5, 8), (4, �19)}.

12. Solve y � �2x � 1 if the domain is {�3, �2, �1, 0, 1}.Graph the solution set.

For Questions 13 and 14, determine whether each equation is a linear equation. If so, write the equation in standard form.

13. �1x� � �

1y� � �

23�

14. 4x � 2y

15. Graph the equation x � 4y � 3 � 0.

y

xO

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

y

xO

P

MN

y

xO

7.

8.

9.

10.

11.

12.

13.

14.15. y

xO

y

xO

X Y


Chapter 4 Test, Form 2D (continued)


16. 17.

For Questions 18 and 19, if g(x) � x2 � 2x � 4, find each value.

18. g(�5) 19. g(t � 1)

20. Determine whether the sequence 7, 11, 15, 19, … is an arithmetic sequence. If so, state the common difference.

21. Find the next three terms of the arithmetic sequence �25, �22, �19, �16, … .

22. Write an equation for the nth term of the sequence 3, 12, 21, 30, … . Then graph the first five terms of the sequence.

23. Find the next two items in the pattern.

24. Use the table below that shows the amount of gasoline a car consumes for different distances driven.

Write an equation in function notation for the relationship between distance and gasoline used.

25. Joseph wants to reduce a rectangular photograph that is 3 inches wide and 5 inches tall by a scale factor of 2.5 so it will fit on his report. What will be the dimensions of the new photograph? Use a coordinate system to draw a photograph of the original rectangle and the reduced photograph. Place one corner of each photograph at the origin and for each photograph write the coordinates of the other three vertices.

Bonus Graph the points (4, �2), (4, 3), (�2, 3). Find a fourth point that completes a rectangle with the given three points, then graph the rectangle on the same coordinate plane.

...

y

xO

NAME DATE PERIOD

44

x y

�2 7

0 0

1 �2

1 3

Distance (mi) 1 2 3 4 5

Gasoline (gal) 0.05 0.10 0.15 0.20 0.25

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:y

xO

y

xO

an

nO 1

5101520253035

2 3 4 5 6 7

Chapter 4 Test, Form 3


1. Write the ordered pair that describes a point 8 units below the origin and on the y-axis.

2. Plot the points (�3, 1.5) and ��32�, �4� on a coordinate plane,

and label each point with the ordered pairs.

3. Determine whether a reflection, translation, dilation, or rotation is being shown.

4. Triangle MNP with M(1, 2), N(3, 4), and P(4, �1) is reflected over the y-axis, then translated 2 units right and 2 units down. Find the coordinates of the vertices after the given transformations are performed. Then, graph the preimage and its image.

5. Express the relation {(4, �1), (�2, 3), (�1, �2), (6, �2), (3, �1)} as a mapping. Then determine the domain and range.


7. Find the solution set for 4x � 5y � 15,given the replacement set {(�10, �12), (�3, �5.4), (5, 1.2), (6, 3), (15, 9)}.

8. Solve the equation 7x � 2y � 8 if the domain is {�4, 3, 5, 6}. Graph the solution set.

9. Determine whether 3x � 4y � 7 � 3y � 1 is a linear equation. If so, write the equation in standard form.

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

y

xO

1.

2.

3.

4.

5.

6.

7.

8.

9.

y

xO

X Y

y

xO

y

xO


Chapter 4 Test, Form 3 (continued)

10. Graph �23x� � �2

y� � 1.


11. 3x � 11 12.

13. If f(x) � 3x2 � 4x, find �2[f(2p)].

14. Determine whether the sequence 0, �12�, 1, �

32�, … is an

arithmetic sequence. If it is, state the common difference.

15. Find the value of y that makes 9, 4, y, �6, … an arithmetic sequence.

16. Write an equation for the nth term of the arithmetic sequence �15, �11, �7, �3, … . Then graph the first five terms of the sequence.

17. Find the tenth figure in the pattern below.

For Questions 18 and 19, use the table below that shows the value of a vending machine over the first five years of use.

18. Write an equation in function notation for the relationship between years of use t and value v(t).

19. When will the value of the vending machine reach 0?

20. Brian collects baseball cards. His father gave him 20 cards to start his collection on his tenth birthday. Each year Brian adds about 15 cards to his collection. About how many years will it take to fill his collection binder if it holds 200 cards?

Bonus Solve the equation �2x

� � �3y

� � 10 if the domain is

{�4, �1, 2, 3, 6}. Then state the inverse of the solution set.

y

xO

NAME DATE PERIOD

44

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

an

n

y

xO

Number of years 1 2 3 4 5

Value (dollars) 1810 1620 1430 1240 1050

0

2000

Chapter 4 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 solution in more thanone way or investigate beyond the requirements of the problem.

1. a. Give the coordinates of a point on the coordinate plane.Name the quadrant in which the point is located.

b. Reflect the point in part a across the y-axis and find thecoordinates of the image. Graph the preimage and its image.

c. Explain how the coordinates of a point can be determined tobe positive or negative by knowing in which quadrant thepoint is located.

2. a. List any two transformations and describe the differencesbetween the two transformations chosen.

b. Describe any similarities between the two transformationschosen for part a.

3. Use the set {�1, 0, 1, 2} as a domain and the set {�3, �1, 4, 5} asa range.a. Create a relation. Express the relation as a set of ordered

pairs. Then state the inverse of the relation.b. Create a relation that is not a function. Express the relation

as a table, a graph, and a mapping.c. Explain why the relation created for part b is not a function.

4. Write a linear equation in standard form. Then explain how tograph the equation.

5. An equation is written using the function notation f(x).Explain what f(2) represents and describe how to find f(2).

6. a. Write a sequence that is an arithmetic sequence. State thecommon difference, and find a6.

b. Write a sequence that is not an arithmetic sequence.Determine whether the sequence has a pattern, and if sodescribe the pattern.

c. Determine if the sequence 1, 1, 1, 1, … is an arithmeticsequence. Explain your reasoning.

NAME DATE PERIOD

SCORE 44

Ass

essm

ent


Chapter 4 Vocabulary Test/Review

Choose from the terms above to complete each sentence.

1. In mathematics, points are located in reference to two

perpendicular number lines called .

2. The axes in the coordinate plane intersect at a common zero

point called the .

3. The x-axis and y-axis separate the coordinate plane into four

regions called .

4. are movements of geometric figures.

5. In a(n) , a figure is slid horizontally,vertically, or both.

6. The of any relation is obtained byswitching the coordinates of each ordered pair of the relation.

7. For a(n) Ax � By � C, the numbersA, B, and C cannot all be zero.

8. A(n) is a relation in which eachelement of the domain is paired with exactly one element of therange.

9. The can be used to determine if agraph represents a function.

In your own words—Define each term.

10. coordinate plane

11. mapping

arithmetic sequenceaxescommon differencecoordinate planedilationequation in two variablesfunctionfunction notationgraph

imageinductive reasoninginverselinear equationlook for a patternmappingoriginpreimagequadrant

reflectionrotationsequencesolutionstandard formtermstransformationtranslationvertical line test

x-axisx-coordinatex-intercepty-axisy-coordinatey-intercept

NAME DATE PERIOD

SCORE 44

Chapter 4 Quiz (Lessons 4–1 and 4–2)

44


Write the ordered pair for each point shown. Name the quadrant in which the point is located.

1. P 2. T

For Questions 3 and 4, identify each transformation as a reflection,translation, dilation, or rotation.

3. 4.

5. Find the coordinates of the vertices of triangle ABC with A(1, 2), B(3, 3), and C(3, �1), after the triangle is reflected over the y-axis. Then graph the preimage and its image.

NAME DATE PERIOD

SCORE


1. Express the relation {(3, 5), (�4, 6), (3, 8), (2, 4), (1, 3)} as a mapping. Then determine the domain and range.


3. Find the solution set for y � 4x � 3,given the replacement set {(�3, 9), (�2, �11), (0, �2), (2, 5)}.

4. Solve 3x � 2y � �6 if the domain is {�2, �1, 0, 2, 3}.

5. Solve the equation y � 2x � 4 if the domain is {�3, �2, �1, 0, 1}.

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

y

xOP

T

y

xO

1.

2.

3.

4.

5.

y

xO

1.

2.

3.

4.

5.

X Y


1. Determine whether y � 2x � 1 is a linear equation.If so, write the equation in standard form.

2. Graph 3x � y � 3.

3. Determine whether the relation is a function.

4. If g(x) � x2 � 3x � 2, find g(�4). 4.

5. STANDARDIZED TEST PRACTICE If � x � x2 � 3x � 4,then � 6 �

A. �8. B. 22. C. 58. D. 14. 5.


1. Determine whether 2, 5, 9, 14, … is an arithmetic sequence. 1.If so, state the common difference.

2. Find the next three terms of the arithmetic sequence 2.5, 9, 13, 17, … .

3. Find the 15th term of the arithmetic sequence if 3.a1 � �3 and d � 2.

4. Find the next three terms in the sequence 4.3125, 625, 125, 25, … .

5. Write an equation in function 5.notation for the relation graphed.

y

xO

NAME DATE PERIOD

SCORE


44

44

1.

2.

3.

y

xO

NAME DATE PERIOD

SCORE

y

xO

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



1. Write the ordered pair for point P and name the quadrant in which it is located.A. (2, �3); II B. (2, �3); IIIC. (�2, �3); III D. (�2, �3); II 1.

2. Determine what type of transformation is shown.A. reflection B. translationC. dilation D. rotation 2.

3. What is the inverse of the relation {(�1, 2), (3, �4), (0, 5)}?A. {(2, �1), (�4, 3), (5, 0)} B. {(�2, 1), (4, �3), (�5, 0)}C. {(1, �2), (�3, 4), (0, �5)} D. {(1, 2), (�3, �4), (0, 5)} 3.

For Questions 4 and 5 use the following information.The total T charged by a phone company can be represented by the equation T � 30 � 0.10m where m is the number of minutes.

4. Solve the equation for m.

A. m � �0.T10� � 30 B. m � 0.1(T � 30) C. m � �0.

T10� � 30 D. m � �

T0�.10

30� 4.

5. Find the number of minutes if the total charge is $31.00.A. 100 B. 10 C. 1 D. 340 5.

6. Express the relation {(�3, �2), (3, �1), (5, �2), (7, 0)} 6.as a mapping.

7. For the triangle described below, find the coordinates of the 7.vertices of each figure after the given transformation is performed. Then graph the preimage and its image.Triangle DEF with D(�1, 1), E(�4, 1), and F(�2, 4) reflected over the y-axis

Solve each equation for the given domain.

8. y � 2x � 3 for x � {�2, �1, 0, 2, 3} 8.

9. 3y � x � 6 for x � {�3, 0, 3, 6} 9.

X Y

Part II

Part I

NAME DATE PERIOD

SCORE 44

Ass

essm

ent

y

xO

P


Chapter 4 Cumulative Review (Chapters 1–4)

1. Find the solution of y � �23� � �

2125�

if the replacement set is

��25�, �

35�, �

45�, 1, 1�

15��. (Lesson 1–3)

2. Simplify 5m � 8n � 3m � n. (Lesson 1–6)

3. The average longevity in years of 10 different farm animals is listed below. Make a stem-and-leaf plot. (Lesson 2–5)

12 12 12 15 12 8 20 10 5 12

4. Find �38�. Round to the nearest hundredth. (Lesson 2–7)

5. Translate the following equation into a verbal sentence.

�4x

� � y � �2��xy��

(Lesson 3–1)

6. Find the discounted price. clock: $15.00discount: 15% (Lesson 3–7)

7. Solve �7x � 23 � 37. (Lesson 3–4)

8. Use cross products to determine whether the ratios �47� and �1

151�

form a proportion. Write yes or no. (Lesson 3–6)

For Questions 9–11, use the graph.

9. Write the ordered pair for point A, and name the quadrant in which the point is located.(Lesson 4–1)

10. Express the relation as a set of ordered pairs. Then determine the domain and range. (Lesson 4–3)

11. Determine whether the relation is a function. (Lesson 4–6)

12. Find the coordinates of the vertices of triangle ABC withA(�1, 0), B(2, 4), and C(3, 1) reflected over the x-axis. Thengraph the preimage and its image. (Lesson 4–2)

13. Solve 4x � 5 � y if the domain is {�3, �1, 1, 2, 5}. (Lesson 4–4)

14. Graph 2x � 3y � 6. (Lesson 4–5)

y

xO

BE

C D

A

G

F

NAME DATE PERIOD

44

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14. y

xO

y

x

Standardized Test Practice (Chapters 1–4)


1. Which vehicle is a counterexample for the statement.If an automobile is a sports car, then it is red. (Lesson 1–7)

A. a red pick-up truck B. a white motor homeC. a red sports car D. a yellow sports car 1.

2. Dion owns a delivery service. He charges his customers $15.00 for each delivery. His expenses include $7000 for the motorcycle he drives and $0.42 for gasoline per trip. Which equation could Dion use to calculate his profit p for d deliveries? (Lesson 1–8)

E. p � 15 � 0.42d F. p � 14.58d � 7000G. p � 7000 � 15d H. p � 0.42d � 7000 2.

3. Evaluate �4xy if x � 5.3 and y � �0.4. (Lesson 2–3)

A. 9.7 B. �9.7 C. 8.48 D. �8.48 3.

4. Which measure of central tendency best represents the data? new car prices: $14,500, $12,400, $18,500, $12,400, $28,900,$19,100, $17,300, $22,900, $17,800, $21,600 (Lesson 2–5)

E. median F. mean G. mode H. none 4.

5. Translate the sentence into an equation. (Lesson 3–1)

Five times the sum of s and t is as much as four times r.A. 5s � t � 4 B. 5s � t � r C. 5(s � t) � 4r D. s � t � 5(4r) 5.

6. Solve 8(x � 5) � 12(4x � 1) � 12. (Lesson 3–5)

E. ��170�

F. ��57� G. �2 H. �1 6.

7. Paul and Charlene are 420 miles apart. They start toward each other with Paul driving 16 miles per hour faster than Charlene.They meet in 5 hours. Find Charlene’s speed. (Lesson 3–9)

A. 34 mph B. 50 mph C. 40.4 mph D. 68 mph 7.

8. Determine which equation is a linear equation. (Lesson 4–5)

E. x2 � y � 4 F. x � y � 4 G. xy � 4 H. �1x� � y � 4 8.

9. If f(x) � 7 � 2x, find f(3) � 6. (Lesson 4–6)

A. 11 B. 7 C. 14 D. –11 9.

10. Chapa is beginning an exercise program that calls for 30 push-ups each day for the first week. Each week thereafter, she has to increase her push-ups by 2. Which week of her program will be the first one in which she will do 50 push-ups a day? (Lesson 4–7)

E. 9th week F. 10th week G. 11th week H. 12th week 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

44

Ass

essm

ent

Part 1: Multiple Choice

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


Standardized Test Practice (continued)

NAME DATE PERIOD

44

NAME DATE PERIOD

11. Evaluate 27 � � y � z � if y � �4.9 and z � �5.1. (Lesson 2–1)

12. The ratio of a to b is �47�. If a is 16, find the

value of b. (Lesson 3–6)

13. Tariq is enlarging a map by a scale factor of

3�12�. On his original map the distance from

his home to his school is 4 inches. How far in inches is this distance on the enlargement?(Lesson 4–2)

14. The equation C � �F

1�.8

32� relates the

temperature in degrees Fahrenheit F to degrees Celsius C. If the temperature is 25°C in Rome, Italy, what is the temperature in degrees Fahrenheit?(Lesson 4–4)

Column A Column B

15. 15.

(Lessons 2–3 and 2–4)

16. Given: x � 4 � 10 16.

(Lesson 3–2)

17. 17.

(Lesson 4–1)

DCBAthe quadrant in whichpoint (�2, 4) is located

the quadrant in whichpoint (�3, �1) is located

15x

DCBA

DCBA3(a � 2)�14 �

77a

�

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.

11. 12.

13. 14.

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

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 252–253 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 1

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.

Also enter your answer by writing each number or symbol in a box.Then fill in the corresponding oval for that number or symbol.

10 (grid in) 10 13 14

11

12

13 (grid in)

14 (grid in)

15 (grid in) 15 16

16 (grid in)

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

17 18 19 20

Record your answers for Question 21 on the back of this paper.

DCBADCBADCBADCBA

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

NAME DATE PERIOD

44

An

swer

s

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

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison

Part 4 Open-EndedPart 4 Open-Ended

Part 1 Multiple ChoicePart 1 Multiple Choice


Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

e C

oo

rdin

ate

Pla

ne

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

©G

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ill21

3G

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ra 1

Lesson 4-1

Iden

tify

Po

ints

In t

he d

iagr

am a

t th

e ri

ght,

poin

ts a

re

loca

ted

in r

efer

ence

to

two

perp

endi

cula

r nu

mbe

r li

nes

call

ed

axes

.The

hor

izon

tal n

umbe

r li

ne is

the

x-a

xis,

and

the

vert

ical

num

ber

line

is t

he y

-axi

s.T

he p

lane

con

tain

ing

the

x- a

nd y

-axe

s is

cal

led

the

coor

din

ate

pla

ne.

Poi

nts

in t

he c

oord

inat

e pl

ane

are

nam

ed b

y or

dere

d pa

irs

of t

he f

orm

(x,

y).T

he f

irst

num

ber,

or

x-co

ord

inat

eco

rres

pond

s to

a n

umbe

r on

the

x-a

xis.

The

sec

ond

num

ber,

or y

-coo

rdin

ate,

corr

espo

nds

to a

num

ber

on t

he y

-axi

s.

Th

e ax

es d

ivid

e th

e co

ordi

nat

e pl

ane

into

Qu

adra

nts

I,I

I,II

I,an

d IV

,as

show

n.T

he

poin

t w

her

e th

e ax

es i

nte

rsec

t is

cal

led

the

orig

in.T

he

orig

in h

as c

oord

inat

es (

0,0)

.

x

y

O

PQ

R

Quad

rant

III

Quad

rant

IIQu

adra

nt I

Quad

rant

IV

Wri

te a

n o

rder

edp

air

for

poi

nt

Rab

ove.

Th

e x-

coor

din

ate

is 0

an

d th

e y-

coor

din

ate

is 4

.Th

us

the

orde

red

pair

for

Ris

(0,

4).

Wri

te o

rder

ed p

airs

for

poi

nts

Pan

d Q

abov

e.T

hen

nam

e th

e q

uad

ran

t in

wh

ich

eac

h p

oin

t is

loc

ated

.

Th

e x-

coor

din

ate

of P

is �

3 an

d th

e y-

coor

din

ate

is�

2.T

hu

s th

e or

dere

d pa

ir f

or P

is (

�3,

�2)

.Pis

in

Qu

adra

nt

III.

Th

e x-

coor

din

ate

of Q

is 4

an

d th

e y-

coor

din

ate

is�

1.T

hu

s th

e or

dere

d pa

ir f

or Q

is (

4,�

1).Q

is i

nQ

uad

ran

t IV

.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te t

he

ord

ered

pai

r fo

r ea

ch p

oin

t sh

own

at

the

righ

t.N

ame

the

qu

adra

nt

in w

hic

h t

he

poi

nt

is l

ocat

ed.

1.N

(3,0

),n

on

e2.

P(�

2,�

3),I

II

3.Q

(0,5

),n

on

e4.

R(�

3,4)

,II

5.S

(5,�

2),I

V6.

T(�

5,0)

,no

ne

7.U

(1,1

),I

8.V

(5,4

),I

9.W

(�2,

1),I

I10

.Z(�

1,0)

,no

ne

11.A

(3,�

3),I

V12

.B(0

,�4)

,no

ne

13.W

rite

th

e or

dere

d pa

ir t

hat

des

crib

es a

poi

nt

4 u

nit

s do

wn

fro

m a

nd

3 u

nit

s to

th

e ri

ght

of t

he

orig

in.

(3,�

4)

14.W

rite

th

e or

dere

d pa

ir t

hat

is

8 u

nit

s to

th

e le

ft o

f th

e or

igin

an

d li

es o

n t

he

x-ax

is.

(�8,

0)

x

y O

P

QR

N

S

TU

V

W

Z

AB

©G

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Gra

ph

Po

ints

To

grap

han

ord

ered

pai

r m

ean

s to

dra

w a

dot

at

the

poin

t on

th

eco

ordi

nat

e pl

ane

that

cor

resp

onds

to

the

orde

red

pair

.To

grap

h a

n o

rder

ed p

air

(x,y

),be

gin

at t

he

orig

in.M

ove

left

or

righ

t x

un

its.

Fro

m t

her

e,m

ove

up

or d

own

yu

nit

s.D

raw

a d

ot a

tth

at p

oin

t.

Plo

t ea

ch p

oin

t on

a c

oord

inat

e p

lan

e.

a.R

(�3,

2)S

tart

at

the

orig

in.M

ove

left

3 u

nit

s si

nce

th

e x-

coor

din

ate

is �

3.M

ove

up

2 u

nit

s si

nce

th

e y-

coor

din

ate

is 2

.Dra

w a

do

t an

d la

bel

it R

.

b.

S(0

,�3)

Sta

rt a

t th

e or

igin

.Sin

ce t

he

x-co

ordi

nat

e is

0,t

he

poin

t w

ill

be l

ocat

ed o

n t

he

y-ax

is.M

ove

dow

n 3

un

its

sin

ce t

he

y-co

ordi

nat

e is

�3.

Dra

w a

dot

an

d la

bel

it S

.

Plo

t ea

ch p

oin

t on

th

e co

ord

inat

e p

lan

e at

th

e ri

ght.

1.A

(2,4

)2.

B(0

,�3)

1–16

.See

gra

ph

.

3.C

(�4,

�4)

4.D

(�2,

0)

5.E

(1,�

4)6.

F(0

,0)

7.G

(5,0

)8.

H(�

3,4)

9.I(

4,�

5)10

.J(�

2,�

2)

11.K

(2,�

1)12

.L(�

1,�

2)

13.M

(0,3

)14

.N(5

,�3)

15.P

(4,5

)16

.Q(�

5,2)

x

y

O

AM

N

P

Q

BC

D

E

FG

H

I

JK

L

x

y O

R

S

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Th

e C

oo

rdin

ate

Pla

ne

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 4-1)


An

swer

s

Skil

ls P

ract

ice

Th

e C

oo

rdin

ate

Pla

ne

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

©G

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5G

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Lesson 4-1

Wri

te t

he

ord

ered

pai

r fo

r ea

ch p

oin

t sh

own

at

the

righ

t.N

ame

the

qu

adra

nt

in w

hic

h t

he

poi

nt

is l

ocat

ed.

1.A

(�4,

�4)

;III

2.B

(3,2

);I

3.C

(2,�

1);

IV4.

D(�

3,4)

;II

5.E

(�2,

0);

no

ne

6.F

(4,�

3);

IV

Wri

te t

he

ord

ered

pai

r fo

r ea

ch p

oin

t sh

own

at

the

righ

t.N

ame

the

qu

adra

nt

in w

hic

h t

he

poi

nt

is l

ocat

ed.

7.G

(�2,

2);

II8.

H(1

,�1)

;IV

9.J

(3,4

);I

10.K

(�2,

�2)

;III

11.L

(1,3

);I

12.M

(0,�

4);

no

ne

Plo

t ea

ch p

oin

t on

th

e co

ord

inat

e p

lan

e at

th

e ri

ght.

13.M

(2,4

)14

.N(�

3,�

3)

15.P

(2,�

2)16

.Q(0

,3)

17.R

(4,1

)18

.S(�

4,1)

Plo

t ea

ch p

oin

t on

th

e co

ord

inat

e p

lan

e at

th

e ri

ght.

19.T

(4,0

)20

.U(�

3,2)

21.W

(�2,

�3)

22.X

(2,2

)

23.Y

(�3,

�2)

24.Z

(3,�

3)W

x

y

OT

XU

Y

Z

M

x

y

O

Q

S N

R

P

x

y

O

GJ

KHL

M

x

y

O

D

F

EC

B

A

©G

lenc

oe/M

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ill21

6G

lenc

oe A

lgeb

ra 1

Wri

te t

he

ord

ered

pai

r fo

r ea

ch p

oin

t sh

own

at

the

righ

t.N

ame

the

qu

adra

nt

in w

hic

h t

he

poi

nt

is l

ocat

ed.

1.A

(0,�

1);

no

ne

2.B

(�5,

5);

II

3.C

(�5,

�2)

;III

4.D

(5,�

5);

IV

5.E

(3,3

);I

6.F

(�1,

1);

II

7.G

(2,�

3);

IV8.

H(�

1,�

4);

III

9.I

(�3,

0);

no

ne

10.J

(2,5

);I

11.K

(5,�

2);

IV12

.L(4

,2);

I

Plo

t ea

ch p

oin

t on

th

e co

ord

inat

e p

lan

e at

th

e ri

ght.

13.M

(�3,

3)14

.N(3

,�2)

15.P

(5,1

)

16.Q

(�4,

�3)

17.R

(0,5

)18

.S(�

1,�

2)

19.T

(�5,

1)20

.V(1

,�5)

21.W

(2,0

)

22.X

(�2,

�4)

23.Y

(4,4

)24

.Z(�

1,2)

25.C

HES

SL

ette

rs a

nd

nu

mbe

rs a

re u

sed

to s

how

th

e po

siti

ons

of c

hes

s pi

eces

an

d to

des

crib

e th

eir

mov

es.

For

exa

mpl

e,in

th

e di

agra

m a

t th

e ri

ght,

a w

hit

e pa

wn

is

loc

ated

at

f5.N

ame

the

posi

tion

s of

eac

h o

f th

ere

mai

nin

g ch

ess

piec

es.

wh

ite

paw

ns:

a7,c

6;b

lack

paw

ns:

b4,

d7;

wh

ite

kin

g:

g3;

bla

ck k

ing

:e8

AR

CH

AEO

LOG

YF

or E

xerc

ises

26

and

27,

use

th

e gr

id a

t th

e ri

ght

that

sh

ows

the

loca

tion

of

arro

wh

ead

s ex

cava

ted

at

a m

idd

en—

a p

lace

wh

ere

peo

ple

in

th

e p

ast

du

mp

ed

tras

h,f

ood

rem

ain

s,an

d o

ther

dis

card

ed i

tem

s.

26.W

rite

th

e co

ordi

nat

es o

f ea

ch a

rrow

hea

d.(1

,1),

(3,2

),(1

,4)

27.S

upp

ose

an a

rch

aeol

ogis

t di

scov

ers

two

oth

er a

rrow

hea

ds

loca

ted

at (

1,2)

an

d (3

,3).

Dra

w a

n a

rrow

hea

d at

eac

h o

f th

ese

loca

tion

s on

th

e gr

id.

01

2M

eter

s

Meters

34

4 3 2 1 0

8 7 6 5 4 3 2 1

King

Paw

n

ab

cd

ef

ghW

x

y

O

R

N

Y

T Q

Z

V

P

M

XS

x

y

O

C

JB

AK

DL

E

F

I

GH

Pra

ctic

e (

Ave

rag

e)

Th

e C

oo

rdin

ate

Pla

ne

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1



Readin

g t

o L

earn

Math

em

ati

csT

he

Co

ord

inat

e P

lan

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-H

ill21

7G

lenc

oe A

lgeb

ra 1

Lesson 4-1

Pre-

Act

ivit

yH

ow d

o ar

chae

olog

ists

use

coo

rdin

ate

syst

ems?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-1

at

the

top

of p

age

192

in y

our

text

book

.

Wh

at d

o th

e te

rms

grid

sys

tem

,gri

d,a

nd

coor

din

ate

syst

emm

ean

to

you

?S

ee s

tud

ents

’wo

rk.

Rea

din

g t

he

Less

on

1.U

se t

he

coor

din

ate

plan

e sh

own

at

the

righ

t.

a.L

abel

th

e or

igin

O.

b.

Lab

el t

he

y-ax

is y

.

c.L

abel

th

e x-

axis

x.

2.E

xpla

in w

hy

the

coor

din

ates

of

the

orig

in a

re (

0,0)

.S

amp

le a

nsw

er:T

he

ori

gin

is t

he

inte

rsec

tio

n o

f tw

o n

um

ber

lin

es a

t th

eir

com

mo

n z

ero

po

int.

3.U

se t

he

orde

red

pair

(�

2,3)

.

a.E

xpla

in h

ow t

o id

enti

fy t

he

x- a

nd

y-co

ordi

nat

es.

Th

e x-

coo

rdin

ate

is t

he

firs

tn

um

ber

,th

e y-

coo

rdin

ate

is t

he

seco

nd

nu

mb

er.

b.

Nam

e th

e x-

an

d y-

coor

din

ates

.T

he

x-co

ord

inat

e is

�2,

the

y-co

ord

inat

e is

3.

c.D

escr

ibe

the

step

s yo

u w

ould

use

to

loca

te t

he

poin

t fo

r (�

2,3)

on

th

e co

ordi

nat

epl

ane.

Sta

rt a

t th

e o

rig

in.M

ove

left

2 u

nit

s.T

hen

mov

e u

p 3

un

its.

4.W

hat

doe

s th

e te

rm q

uad

ran

tm

ean

?S

amp

le a

nsw

er:

on

e o

f th

e fo

ur

reg

ion

s in

the

coo

rdin

ate

pla

ne

Hel

pin

g Y

ou

Rem

emb

er

5.E

xpla

in h

ow t

he

way

th

e ax

es a

re l

abel

ed o

n t

he

coor

din

ate

plan

e ca

n h

elp

you

rem

embe

r h

ow t

o pl

ot t

he

poin

t fo

r an

ord

ered

pai

r.S

amp

le a

nsw

er:T

he

rig

ht

sid

e o

f th

e h

ori

zon

tal a

xis

is la

bel

ed w

ith

th

e le

tter

x.T

his

is t

he

sid

e o

fth

e h

ori

zon

tal n

um

ber

lin

e w

her

e yo

u f

ind

po

siti

ve n

um

ber

s.T

he

top

en

do

f th

e ve

rtic

al n

um

ber

lin

e is

lab

eled

wit

h t

he

lett

er y

.Th

is is

th

e p

art

of

the

vert

ical

nu

mb

er li

ne

wh

ere

you

fin

d p

osi

tive

nu

mb

ers.

x

y

O

©G

lenc

oe/M

cGra

w-H

ill21

8G

lenc

oe A

lgeb

ra 1

Mid

po

int

Th

e m

idpo

int

of a

lin

e se

gmen

t is

th

e po

int

that

lie

s ex

actl

y h

alfw

aybe

twee

n t

he

two

endp

oin

ts o

f th

e se

gmen

t.T

he

coor

din

ates

of

the

mid

poin

t of

a l

ine

segm

ent

wh

ose

endp

oin

ts a

re (

x 1,y

1) a

nd

(x2,

y 2)

are

give

n b

y �

,�.

Fin

d t

he

mid

poi

nt

of e

ach

lin

e se

gmen

t w

ith

th

e gi

ven

en

dp

oin

ts.

1.(7

,1)

and

(�3,

1)2.

(5,�

2) a

nd

(9,�

8)

(2,1

)(7

,�5)

3.(�

4,4)

an

d (4

,�4)

4.(�

3,�

6) a

nd

(�10

,�15

)

(0,0

)(�

6.5,

�10

.5)

Plo

t ea

ch s

egm

ent

in t

he

coor

din

ate

pla

ne.

Th

en f

ind

th

e co

ord

inat

es o

f th

em

idp

oin

t.

5.J �K�

wit

h J

(5,2

) an

d K

(�2,

�4)

6.P�

Q�w

ith

P(�

1,4)

an

d Q

(3,�

1)

(1.5

,�1)

(1,1

.5)

You

are

giv

en t

he

coor

din

ates

of

one

end

poi

nt

of a

lin

e se

gmen

t an

d t

he

mid

poi

nt

M.F

ind

th

e co

ord

inat

es o

f th

e ot

her

en

dp

oin

t.

7.A

(�10

,3)

and

M(�

6,7)

8.D

(�1,

4) a

nd

M(3

,�6)

(�2,

11)

(7,�

16)

(3, –

1)

(–1,

4)

(1, 1

.5)

x

y

O

P

Q

(5, 2

)

(–2,

–4)

(1.5

, –1)

x

y

O

K

J

y 1�

y 2�

2x 1

�x 2

�2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Tran

sfo

rmat

ion

s o

n t

he

Co

ord

inat

e P

lan

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-H

ill21

9G

lenc

oe A

lgeb

ra 1

Lesson 4-2

Tran

sfo

rm F

igu

res

Tra

nsf

orm

atio

ns

are

mov

emen

ts o

f ge

omet

ric

figu

res.

Th

ep

reim

age

is t

he

posi

tion

of

the

figu

re b

efor

e th

e tr

ansf

orm

atio

n,a

nd

the

imag

eis

th

epo

siti

on o

f th

e fi

gure

aft

er t

he

tran

sfor

mat

ion

.

Ref

lect

ion

Afig

ure

is f

lippe

d ov

er a

line

.

Tran

slat

ion

Afig

ure

is s

lid h

oriz

onta

lly,

vert

ical

ly,

or b

oth.

Dila

tio

nA

figur

e is

enl

arge

d or

red

uced

.

Ro

tati

on

Afig

ure

is t

urne

d ar

ound

a p

oint

.

Det

erm

ine

wh

eth

er e

ach

tra

nsf

orm

atio

n i

s a

refl

ecti

on,t

ran

sla

tion

,d

ila

tion

,or

rota

tion

.

a.T

he

figu

re h

as b

een

fli

pped

ove

r a

lin

e,so

th

is i

s a

refl

ecti

on.

b.

Th

e fi

gure

has

bee

n t

urn

ed a

rou

nd

a po

int,

so t

his

is

a ro

tati

on.

c.T

he

figu

re h

as b

een

red

uce

d in

siz

e,so

th

is i

s a

dila

tion

.

d.

Th

e fi

gure

has

bee

n s

hif

ted

hor

izon

tall

y to

th

e ri

ght,

so t

his

is

atr

ansl

atio

n.

Det

erm

ine

wh

eth

er e

ach

tra

nsf

orm

atio

n i

s a

refl

ecti

on,t

ran

sla

tion

,dil

ati

on,o

rro

tati

on.

1.re

flec

tio

n2.

dila

tio

n

3.ro

tati

on

4.tr

ansl

atio

n

5.d

ilati

on

6.ro

tati

on

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill22

0G

lenc

oe A

lgeb

ra 1

Tran

sfo

rm F

igu

res

on

th

e C

oo

rdin

ate

Plan

eYo

u c

an p

erfo

rm t

ran

sfor

mat

ion

son

a c

oord

inat

e pl

ane

by c

han

gin

g th

e co

ordi

nat

es o

f ea

ch v

erte

x.T

he

vert

ices

of

the

imag

eof

th

e tr

ansf

orm

ed f

igu

re a

re i

ndi

cate

d by

th

e sy

mbo

l �,

wh

ich

is

read

pri

me.

Ref

lect

ion

ove

r x-

axis

(x,

y)

→(x

, �

y)

Ref

lect

ion

ove

r y-

axis

(x,

y)

→(�

x, y

)

Tran

slat

ion

(x,

y)

→(x

�a,

y�

b)

Dila

tio

n(x

, y

) →

(kx,

ky

)

Ro

tati

on

90�

cou

nte

rclo

ckw

ise

(x,

y)

→(�

y, x

)

Ro

tati

on

180

�(x

, y

) →

(�x,

�y

)

A t

rian

gle

has

ver

tice

s A

(�1,

1),B

(2,4

),an

d C

(3,0

).F

ind

th

eco

ord

inat

es o

f th

e ve

rtic

es o

f ea

ch i

mag

e b

elow

.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Tran

sfo

rmat

ion

s o

n t

he

Co

ord

inat

e P

lan

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

Exam

ple

Exam

ple

a.re

flec

tion

ove

r th

e x-

axis

To

refl

ect

a po

int

over

th

e x-

axis

,m

ult

iply

th

e y-

coor

din

ate

by �

1.

A(�

1,1)

→A

�(�

1,�

1)B

(2,4

) →

B�(

2,�

4)C

(3,0

) →

C�(

3,0)

Th

e co

ordi

nat

es o

f th

e im

age

vert

ices

are

A�(

�1,

�1)

,B�(

2,�

4),a

nd

C�(

3,0)

.

b.

dil

atio

n w

ith

a s

cale

fac

tor

of 2

Fin

d th

e co

ordi

nat

es o

f th

e di

late

d fi

gure

by m

ult

iply

ing

the

coor

din

ates

by

2.

A(�

1,1)

→A

�(�

2,2)

B(2

,4)

→B

�(4,

8)C

(3,0

) →

C�(

6,0)

Th

e co

ordi

nat

es o

f th

e im

age

vert

ices

are

A�(

�2,

2),B

�(4,

8),a

nd

C�(

6,0)

.

Exer

cises

Exer

cises

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

each

fig

ure

aft

er t

he

give

n t

ran

sfor

mat

ion

is p

erfo

rmed

.

1.tr

ian

gle

RS

Tw

ith

R(�

2,4)

,S(2

,0),

T(�

1,�

1) r

efle

cted

ove

r th

e y-

axis

R�(

2,4)

,S�(

�2,

0),T

�(1,

�1)

2.tr

ian

gle

AB

Cw

ith

A(0

,0),

B(2

,4),

C(3

,0)

rota

ted

abou

t th

e or

igin

180

°A

�(0,

0),B

�(�

2,�

4),C

�(�

3,0)

3.pa

rall

elog

ram

AB

CD

wit

h A

(�3,

0),B

(�2,

3),C

(3,3

),D

(2,0

) tr

ansl

ated

3 u

nit

s do

wn

A�(

�3,

�3)

,B�(

�2,

0),C

�(3,

0),D

�(2,

�3)

4.qu

adri

late

ral

RS

TU

wit

h R

(�2,

2),S

(2,4

),T

(4,4

),U

(4,0

) di

late

d by

a f

acto

r of

R

�(�

1,1)

,S�(

1,2)

,T�(

2,2)

,U�(

2,0)

5.tr

ian

gle

AB

Cw

ith

A(�

4,0)

,B(�

2,3)

,C(0

,0)

rota

ted

cou

nte

rclo

ckw

ise

90°

A�(

0,�

4),B

�(�

3,�

2),C

�(0,

0)

6.h

exag

on A

BC

DE

Fw

ith

A(0

,0),

B(�

2,3)

,C(0

,4),

D(3

,4),

E(4

,2),

F(3

,0)

tran

slat

ed

2 u

nit

s u

p an

d 1

un

it t

o th

e le

ftA

�(�

1,2)

,B�(

�3,

5),C

�(�

1,6)

,D�(

2,6)

,E�(

3,4)

,F�(

2,2)

1 � 2



Skil

ls P

ract

ice

Tran

sfo

rmat

ion

s o

n t

he

Co

ord

inat

e P

lan

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-H

ill22

1G

lenc

oe A

lgeb

ra 1

Lesson 4-2

Iden

tify

eac

h t

ran

sfor

mat

ion

as

a re

flec

tion

,tra

nsl

ati

on,d

ila

tion

,or

rota

tion

.

1.2.

3.

rota

tio

nre

flec

tio

ntr

ansl

atio

n

4.5.

6.

dila

tio

nre

flec

tio

nro

tati

on

For

Exe

rcis

es 7

–10,

com

ple

te p

arts

a a

nd

b.

a.F

ind

th

e co

ord

inat

es o

f th

e ve

rtic

es o

f ea

ch f

igu

re a

fter

th

e gi

ven

tran

sfor

mat

ion

is

per

form

ed.

b.

Gra

ph

th

e p

reim

age

and

its

im

age.

7.tr

ian

gle

AB

Cw

ith

A(1

,2),

B(4

,�1)

,an

d 8.

para

llel

ogra

m P

QR

Sw

ith

P(�

2,�

1),

C(1

,�1)

ref

lect

ed o

ver

the

y-ax

isQ

(3,�

1),R

(2,�

3),a

nd

S(�

3,�

3)

A�(

�1,

2),B

�(�

4,�

1),a

nd

tr

ansl

ated

3 u

nit

s u

pP

�(�

2,2)

,C

�(�

1,�

1)Q

�(3,

2),R

�(2,

0),a

nd

S�(

�3,

0)

9.tr

apez

oid

JK

LM

wit

h J

(�2,

1),K

(2,1

),10

.tri

angl

e S

TU

wit

h S

(3,3

),T

(5,1

),an

d L

(1,�

1),a

nd

M(�

1,�

1) d

ilat

ed b

y a

U(1

,1)

rota

ted

90°

cou

nte

rclo

ckw

ise

scal

e fa

ctor

of

2J

�(�

4,2)

,K�(

4,2)

,ab

out

the

orig

inL

�(2,

�2)

,an

d M

�(�

2,�

2)S

�(�

3,3)

,T�(

�1,

5),U

�(�

1,1)

S�

T�

U�

S

TU

x

y

O

J�J

K�

K L �L

M�M

x

y

O

P�

Q�

R�

S�

PQ

RS

x

y

O

A�

B�

C�

A

BC

x

y

O

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lenc

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cGra

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2G

lenc

oe A

lgeb

ra 1

Iden

tify

eac

h t

ran

sfor

mat

ion

as

a re

flec

tion

,tra

nsl

ati

on,d

ila

tion

,or

rota

tion

.

1.2.

3.

refl

ecti

on

tran

slat

ion

rota

tio

n

For

Exe

rcis

es 4

–6,c

omp

lete

par

ts a

an

d b

.a.

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

each

fig

ure

aft

er t

he

give

ntr

ansf

orm

atio

n i

s p

erfo

rmed

.b

.G

rap

h t

he

pre

imag

e an

d i

ts i

mag

e.

4.tr

ian

gle

DE

Fw

ith

D(2

,3),

5.tr

apez

oid

EF

GH

wit

h

6.tr

ian

gle

XY

Zw

ith

X(3

,1),

E(4

,1),

and

F(1

,�1)

E

(3,2

),F

(3,�

3),

Y(4

,�2)

,an

d Z

(1,�

3)

tran

slat

ed 4

un

its

left

G

(1,�

2),a

nd

H(1

,1)

ro

tate

d 90

°co

unte

rclo

ckw

ise

and

3 u

nit

s do

wn

refl

ecte

d ov

er t

he

y-ax

isab

out

the

orig

in

D�(

�2,

0),E

�(0,

�2)

,E

�(�

3,2)

,F�(

�3,

�3)

,X

�(�

1,3)

,F

�(�

3,�

4)G

�(�

1,�

2),H

�(�

1,1)

Y�(

2,4)

,Z�(

3,1)

GR

APH

ICS

For

Exe

rcis

es 7

–9,u

se t

he

dia

gram

at

the

righ

t

and

th

e fo

llow

ing

info

rmat

ion

.

A d

esig

ner

wan

ts t

o di

late

th

e ro

cket

by

a sc

ale

fact

or o

f ,a

nd

then

tra

nsl

ate

it 5

un

its

up.

7.W

rite

th

e co

ordi

nat

es f

or t

he

vert

ices

of

the

rock

et.

A(0

,�2)

,B(1

,�3)

,C(1

,�5)

,D(2

,�6)

,E(�

2,�

6),

F(�

1,�

5),G

(�1,

�3)

8.F

ind

the

coor

din

ates

of

the

fin

al p

osit

ion

of

the

rock

et.

A� �0

,4�,B

� �,4

�,C� �

,3�,D

� �1,2

�,E� ��

1,2

�,F� ��

,3�,G

� ��,4

�9.

Gra

ph t

he

imag

e on

th

e co

ordi

nat

e pl

ane.

10.D

ESIG

NR

amon

a tr

ansf

orm

ed f

igu

re A

BC

DE

Fto

des

ign

a

patt

ern

for

a q

uil

t.N

ame

two

diff

eren

t se

ts o

f tr

ansf

orm

atio

ns

she

cou

ld h

ave

use

d to

des

ign

th

e pa

tter

n. S

amp

le a

nsw

er:

refl

ecti

on

ove

r th

e x-

axis

,90�

cou

nte

rclo

ckw

ise

rota

tio

n,a

nd

th

en r

efle

ctio

n o

ver

the

y-ax

is;

thre

e 90

�co

un

terc

lock

wis

e ro

tati

on

s

AB C

DEF

x

y

O

1 � 21 � 2

1 � 21 � 2

1 � 21 � 2

1 � 2

1 � 2

1 � 2

AB C

DE

FG

x

y

O

y

O

X�

Y�

Z�

X

YZ

x

E�

F�

G�

H�

E FGH

x

y

OD

�

E�

F�

D

E

Fx

y

OPra

ctic

e (

Ave

rag

e)

Tran

sfo

rmat

ion

s o

n t

he

Co

ord

inat

e P

lan

e

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csTr

ansf

orm

atio

ns

on

th

e C

oo

rdin

ate

Pla

ne

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-H

ill22

3G

lenc

oe A

lgeb

ra 1

Lesson 4-2

Pre-

Act

ivit

yH

ow a

re t

ran

sfor

mat

ion

s u

sed

in

com

pu

ter

grap

hic

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-2

at

the

top

of p

age

197

in y

our

text

book

.

In t

he

sen

ten

ce,“

Com

pute

r gr

aph

ic d

esig

ner

s ca

n c

reat

e m

ovem

ent

that

mim

ics

real

-lif

e si

tuat

ion

s,”

wh

at p

hra

se i

ndi

cate

s th

e u

se o

ftr

ansf

orm

atio

ns?

crea

te m

ovem

ent

Rea

din

g t

he

Less

on

1.S

upp

ose

you

loo

k at

a d

iagr

am t

hat

sh

ows

two

figu

res

AB

CD

Ean

d A

�B�C

�D�E

�.If

on

efi

gure

was

obt

ain

ed f

rom

th

e ot

her

by

usi

ng

a tr

ansf

orm

atio

n,h

ow d

o yo

u t

ell

wh

ich

was

the

orig

inal

fig

ure

?T

he

lett

ers

that

hav

e n

o p

rim

e sy

mb

ols

are

use

d f

or

vert

ices

of

the

ori

gin

al f

igu

re.

2.W

rite

th

e le

tter

of

the

term

an

d th

e R

oman

nu

mer

al o

f th

e fi

gure

th

at b

est

mat

ches

eac

hst

atem

ent.

a.A

fig

ure

is

flip

ped

over

a l

ine.

A.

dila

tion

I.

b.

A f

igu

re i

s tu

rned

aro

un

d a

poin

t.B

.tr

ansl

atio

nII

.

c.A

fig

ure

is

enla

rged

or

redu

ced.

C.

refl

ecti

onII

I.

d.

A f

igu

re i

s sl

id h

oriz

onta

lly,

vert

ical

ly,

D.

rota

tion

IV.

or b

oth

.

Hel

pin

g Y

ou

Rem

emb

er

3.G

ive

exam

ples

of

thin

gs i

n e

very

day

life

th

at c

an h

elp

you

rem

embe

r w

hat

ref

lect

ion

s,di

lati

ons,

and

rota

tion

s ar

e.S

amp

le a

nsw

er:

Fo

r a

refl

ecti

on

,th

ink

of

loo

kin

gat

yo

urs

elf

in a

mir

ror.

Fo

r a

dila

tio

n,t

hin

k o

f h

ow

yo

ur

han

d lo

oks

if y

ou

ho

ld it

far

fro

m y

ou

r fa

ce a

nd

th

en m

ove

it s

trai

gh

t in

,ver

y cl

ose

to

yo

ur

face

.Fo

r a

rota

tio

n,t

hin

k o

f tw

isti

ng

th

e to

p o

f a

jar

to o

pen

th

e ja

r.

B,I

V

A,I

I

D,I

II

C,I

©G

lenc

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cGra

w-H

ill22

4G

lenc

oe A

lgeb

ra 1

Th

e L

egen

dar

y C

ity

of

Ur

The

cit

y of

Ur

was

fou

nded

mor

e th

an f

ive

thou

sand

yea

rs a

go i

nM

esop

otam

ia (

mod

ern-

day

Iraq

).It

was

one

of

the

wor

ld’s

fir

st c

itie

s.B

etw

een

1922

and

193

4,ar

cheo

logi

sts

disc

over

ed m

any

trea

sure

s fr

omth

is a

ncie

nt c

ity.

A l

arge

cem

eter

y fr

om t

he 2

6th

cent

ury

B.C

.was

foun

d to

con

tain

lar

ge q

uant

itie

s of

gol

d,si

lver

,bro

nze,

and

jew

els.

The

man

y cu

ltur

al a

rtif

acts

tha

t w

ere

foun

d,su

ch a

s m

usic

al i

nstr

umen

ts,

wea

pons

,mos

aics

,and

sta

tues

,hav

e pr

ovid

ed h

isto

rian

s w

ith

valu

able

clue

s ab

out

the

civi

liza

tion

tha

t ex

iste

d in

ear

ly M

esop

otam

ia.

1.S

upp

ose

that

th

e or

dere

d pa

irs

belo

w r

epre

sen

t th

e vo

lum

e (c

m3 )

an

d m

ass

(gra

ms)

of

ten

art

ifac

ts f

rom

th

e ci

ty o

f U

r.P

lot

each

poi

nt

on t

he

grap

h.

A(1

0,15

0)

B(1

50,1

350)

C

(200

,176

0)

D(5

0,52

5)

E(1

00,1

500)

F

(10,

88)

G(2

00,2

100)

H

(150

,167

5)

I(10

0,90

0)

J(5

0,44

0)

2.T

he

equ

atio

n r

elat

ing

mas

s,de

nsi

ty,a

nd

volu

me

for

silv

er i

s m

�10

.5V

.Wh

ich

of

the

poin

ts i

n E

xerc

ise

1 ar

e so

luti

ons

for

this

equ

atio

n?

Dan

d G

3.S

upp

ose

that

th

e eq

uat

ion

m�

8.8V

rela

tes

mas

s,de

nsi

ty,a

nd

volu

me

for

the

kin

d of

bro

nze

use

d in

th

e an

cien

t ci

ty o

f U

r.W

hic

hof

th

e po

ints

in

Exe

rcis

e 1

are

solu

tion

s fo

r th

is e

quat

ion

?

C,F

,J

4.E

xpla

in w

hy

the

grap

h i

n E

xerc

ise

1 sh

ows

only

qu

adra

nt

1.

Nei

ther

th

e m

ass

no

r th

e vo

lum

e ca

n b

e a

neg

ativ

e n

um

ber

.

AF

JD

I

EB

HC

G

Vo

lum

e (c

m3 )

Mass (grams)

050

100

150

200

2400

2200

2000

1800

1600

1400

1200

1000 800

600

400

200

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill22

5G

lenc

oe A

lgeb

ra 1

Lesson 4-3

Rep

rese

nt

Rel

atio

ns

A r

elat

ion

is a

set

of

orde

red

pair

s.A

rel

atio

n c

an b

ere

pres

ente

d by

a s

et o

f or

dere

d pa

irs,

a ta

ble,

a gr

aph

,or

a m

app

ing.

A m

appi

ng

illu

stra

tes

how

eac

h e

lem

ent

of t

he

dom

ain

is

pair

ed w

ith

an

ele

men

t in

th

e ra

nge

.

Exp

ress

th

e re

lati

on{(

1,1)

,(0,

2),(

3,�

2)}

as a

tab

le,a

grap

h,a

nd

a m

app

ing.

Sta

te t

he

dom

ain

an

d r

ange

of

the

rela

tion

.

Th

e do

mai

n f

or t

his

rel

atio

n i

s {0

,1,3

}.T

he

ran

ge f

or t

his

rel

atio

n i

s {�

2,1,

2}.

XY

1 0 3

1 2�

2

x

y O

xy

11

02

3�

2

A p

erso

n p

layi

ng

racq

uet

bal

l u

ses

4 ca

lori

es p

er h

our

for

ever

y p

oun

d h

e or

sh

e w

eigh

s.

a.M

ake

a ta

ble

to

show

th

e re

lati

on b

etw

een

wei

ght

and

cal

orie

s b

urn

ed i

n o

ne

hou

r fo

r p

eop

le w

eigh

ing

100,

110,

120,

and

130

pou

nd

s.So

urce

: The

Mat

h Te

ache

r’s B

ook

of L

ists

b.

Giv

e th

e d

omai

n a

nd

ran

ge.

dom

ain

:{10

0,11

0,12

0,13

0}ra

nge

:{40

0,44

0,48

0,52

0}c.

Gra

ph

th

e re

lati

on.

Wei

gh

t (p

ou

nd

s)Calories

100

120

0

520

480

440

400

x

y

xy

100

400

110

440

120

480

130

520

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

1.E

xpre

ss t

he

rela

tion

{(�

2,�

1),(

3,3)

,(4,

3)}

as

a ta

ble,

a gr

aph

,an

d a

map

pin

g.T

hen

det

erm

ine

the

dom

ain

an

d ra

nge

.d

om

ain

:{�

2,3,

4};

ran

ge:

{�1,

3}

2.T

he

tem

pera

ture

in

a h

ouse

dro

ps 2

°fo

r ev

ery

hou

r th

e ai

r co

ndi

tion

er i

s on

bet

wee

n t

he

hou

rs o

f 6

A.M

.an

d 11

A.M

.M

ake

a gr

aph

to

show

th

e re

lati

onsh

ip b

etw

een

tim

e an

dte

mpe

ratu

re i

f th

e te

mpe

ratu

re a

t 6

A.M

.was

82°

F.

Tim

e (A

.M.)

Temperature (�F)

68

79

1011

120

86 84 82 80 78 76 74 72

x

y

O

XY

�2 3 4

�1 3

xy

�2

�1

33

43

©G

lenc

oe/M

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w-H

ill22

6G

lenc

oe A

lgeb

ra 1

Inve

rse

Rel

atio

ns

Th

e in

vers

eof

an

y re

lati

on i

s ob

tain

ed b

y sw

itch

ing

the

coor

din

ates

in e

ach

ord

ered

pai

r.

Exp

ress

th

e re

lati

on s

how

n i

n t

he

map

pin

g as

a s

et o

f or

der

edp

airs

.Th

en w

rite

th

e in

vers

e of

th

e re

lati

on.

Rel

atio

n:{

(6,5

),(2

,3),

(1,4

),(0

,3)}

Inve

rse:

{(5,

6),(

3,2)

,(4,

1),(

3,0)

}

Exp

ress

th

e re

lati

on s

how

n i

n e

ach

tab

le,m

app

ing,

or g

rap

h a

s a

set

of o

rder

edp

airs

.Th

en w

rite

th

e in

vers

e of

eac

h r

elat

ion

.

1.2.

{(�

2,4)

,(�

1,3)

,(2,

1),(

4,5)

};{(

�1,

0),(

2,�

8),(

4,�

1),(

5,2)

};{(

4,�

2),(

3,�

1),(

1,2)

,(5,

4)}

{(0,

�1)

,(�

8,2)

,(�

1,4)

,(2,

5)}

3.4.

{(�

3,5)

,(�

2,�

1),(

1,0)

,(2,

4)};

{(�

2,�

1),(

0,4)

,(1,

6),(

4,7)

};{(

5,�

3),(

�1,

�2)

,(0,

1),(

4,2)

}{(

�1,

�2)

,(4,

0),(

6,1)

,(7,

4)}

5.6.

{(�

2,2)

,(1,

�2)

,(2,

4),(

3,0)

};{(

�4,

0),(

�1,

�1)

,(0,

�2)

,(2,

1),(

4,3)

};{(

2,�

2),(

�2,

1),(

4,2)

,(0,

3)}

{(0,

�4)

,(�

1,�

1),(

�2,

0),(

1,2)

,(3,

4)}

x

y

Ox

y

O

XY

�2 0 1 4

�1 4 6 7

xy

�3

5

�2

�1

10

24

XY

�1 2 4 5

�8

�1 0 2

xy

�2

4

�1

3

21

45

XY

6 2 1 0

3 4 5

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill22

7G

lenc

oe A

lgeb

ra 1

Lesson 4-3

Exp

ress

eac

h r

elat

ion

as

a ta

ble

,a g

rap

h,a

nd

a m

app

ing.

Th

en d

eter

min

e th

ed

omai

n a

nd

ran

ge.

1.{(

�1,

�1)

,(1,

1),(

2,1)

,(3,

2)}

D �

{�1,

1,2,

3};

R �

{�1,

1,2}

2.{(

0,4)

,(�

4,�

4),(

�2,

3),(

4,0)

}

D �

{�4,

�2,

0,4}

;R

�{�

4,0,

3,4}

3.{(

3,�

2),(

1,0)

,(�

2,4)

,(3,

1)}

D �

{�2,

1,3}

;R

�{�

2,0,

1,4}

Exp

ress

th

e re

lati

on s

how

n i

n e

ach

tab

le,m

app

ing,

or g

rap

h a

s a

set

of o

rder

edp

airs

.Th

en w

rite

th

e in

vers

e of

th

e re

lati

on.

4.5.

6.

{(3,

�5)

,(�

4,3)

,(7,

6),

{(9,

7),(

�5,

6),(

4,8)

,{(

�3,

1),(

�3,

2),(

3,1)

,(1

,�2)

};{(

�5,

3),

(4,2

)};

{(7,

9),

(3,�

3)};

{(1,

�3)

,(3

,�4)

,(6,

7),(

�2,

1)}

(6,�

5),(

8,4)

,(2,

4)}

(2,�

3),(

1,3)

,(�

3,3)

}

x

y

O

XY

9�

5 4

7 6 8 2

xy

3�

5

�4

3

76

1�

2

XY

3 1�

2

�2 0 4 1

x

y

O

xy

3�

2

10

�2

4

3 1

XY

0�

4�

2 4

4�

4 3 0x

y

O

xy

04

�4

�4

�2

3

40

XY

�1 1 2 3

�1 1 2

x

y

O

xy

�1

�1

11

21

3 2

©G

lenc

oe/M

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ill22

8G

lenc

oe A

lgeb

ra 1

Exp

ress

eac

h r

elat

ion

as

a ta

ble

,a g

rap

h,a

nd

a m

app

ing.

Th

en d

eter

min

e th

ed

omai

n a

nd

ran

ge.

1.{(

4,3)

,(�

1,4)

,(3,

�2)

,(2,

3),(

�2,

1)}

D �

{�2,

�1,

2,3,

4};

R �

{�2,

1,3,

4}

Exp

ress

th

e re

lati

on s

how

n i

n e

ach

tab

le,m

app

ing,

or g

rap

h a

s a

set

of o

rder

edp

airs

.Th

en w

rite

th

e in

vers

e of

th

e re

lati

on.

2.3.

4.

{(0,

9),(

�8,

3),

{(9,

5),(

9,3)

,(�

6,�

5),

{(�

3,�

1),(

�2,

�2)

,(2

,�6)

,(1,

4)};

(4,3

),(8

,�5)

,(8,

7)};

(�1,

�3)

,(1,

1),(

2,1)

,{(

9,0)

,(3,

�8)

,{(

5,9)

,(3,

9),(

�5,

�6)

,(3

,1)}

;{(�

1,�

3),(

�2,

�2)

,(�

6,2)

,(4,

1)}

(3,4

),(�

5,8)

,(7,

8)}

(�3,

�1)

,(1,

1),(

1,2)

,(1,

3)}

BA

SEB

ALL

For

Exe

rcis

es 5

an

d 6

,use

th

e gr

aph

th

at

show

s th

e b

atti

ng

aver

age

for

Bar

ry B

ond

s of

th

e S

anF

ran

cisc

o G

ian

ts.

Sour

ce: w

ww.sp

ortsi

llustr

ated

.cnn.

com

5.F

ind

the

dom

ain

an

d es

tim

ate

the

ran

ge.

D �

{199

6,19

97,1

998,

1999

,200

0,20

01};

R �

{.26

2,.2

91,.

303,

.306

,.30

8,.3

28}

6.W

hic

h s

easo

ns

did

Bon

ds h

ave

the

low

est

and

hig

hes

t ba

ttin

g av

erag

es?

low

est:

1999

,hig

hes

t:20

01

MET

EOR

SF

or E

xerc

ises

7 a

nd

8,u

se t

he

ta

ble

th

at s

how

s th

e n

um

ber

of

met

eors

An

n o

bse

rved

eac

h h

our

du

rin

g a

met

eor

show

er.

7.W

hat

are

th

e do

mai

n a

nd

ran

ge?

D �

{12,

1,2,

3,4}

;R

�{1

5,26

,28}

8.G

raph

th

e re

lati

on.

Tim

e (A

.M.)

Mete

or

Sh

ow

er

Number of Meteors

12–1

1–2

2–3

3–4

4–5

0

30 25 20 15 10

Tim

eN

um

ber

of

(A.M

.)M

eteo

rs

1215

126

228

328

415

Year

Barr

y B

on

ds’

Batt

ing

Average

’96

’98

’97

’99

’00

’01

’02

0

.34

.32

.30

.28

.26

x

y

O

XY 5

�5 3 7

9�

6 4 8

xy

09

�8

3

2�

6

1 4

XY 3 4

�2 1

x

y

O

xy

43

�1

4

3�

2

23

�2

1

Pra

ctic

e (

Ave

rag

e)

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3



Readin

g t

o L

earn

Math

em

ati

csR

elat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill22

9G

lenc

oe A

lgeb

ra 1

Lesson 4-3

Pre-

Act

ivit

yH

ow c

an r

elat

ion

s b

e u

sed

to

rep

rese

nt

bas

ebal

l st

atis

tics

?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-3

at

the

top

of p

age

205

in y

our

text

book

.

In 1

997,

Ken

Gri

ffey

,Jr.

had

h

ome

run

s an

d st

rike

outs

.

Th

is c

an b

e re

pres

ente

d w

ith

th

e or

dere

d pa

ir (

,).

Rea

din

g t

he

Less

on

1.L

ook

at p

age

205

in y

our

text

book

.Th

ere

you

see

th

e sa

me

rela

tion

rep

rese

nte

d by

a s

etof

ord

ered

pai

rs,a

tab

le,a

gra

ph,a

nd

a m

appi

ng.

a.In

th

e li

st o

f or

dere

d pa

irs,

wh

ere

do y

ou s

ee t

he

nu

mbe

rs f

or t

he

dom

ain

? th

en

um

bers

for

th

e ra

nge

?b

efo

re t

he

com

mas

;af

ter

the

com

mas

b.

Wh

at p

arts

of

the

tabl

e sh

ow t

he

dom

ain

an

d th

e ra

nge

?T

he

colu

mn

of

nu

mb

ers

un

der

th

e le

tter

xsh

ow

s th

e d

om

ain

,an

d t

he

colu

mn

of

nu

mb

ers

un

der

th

e le

tter

ysh

ow

s th

e ra

ng

e.

c.H

ow d

o th

e ta

ble,

the

grap

h,a

nd

the

map

pin

g sh

ow t

hat

th

ere

are

thre

e or

dere

dpa

irs

in t

he

rela

tion

?T

he

tab

le h

as t

hre

e ro

ws

of

nu

mb

ers,

the

gra

ph

sho

ws

thre

e p

oin

ts m

arke

d w

ith

do

ts,a

nd

th

e m

app

ing

use

s th

ree

arro

ws.

2.W

hic

h t

ells

you

mor

e ab

out

a re

lati

on,a

lis

t of

th

e or

dere

d pa

irs

in t

he

rela

tion

or

the

dom

ain

an

d ra

nge

of

the

rela

tion

? E

xpla

in.

Sam

ple

an

swer

:A

list

of

the

ord

ered

pai

rs t

ells

yo

u m

ore

.Yo

u c

an u

se it

to

fin

d t

he

do

mai

n a

nd

th

e ra

ng

e,an

dyo

u k

no

w e

xact

ly h

ow

th

e n

um

ber

s ar

e p

aire

d.I

f yo

u o

nly

kn

ow

th

ed

om

ain

an

d t

he

ran

ge,

you

can

no

t b

e su

re h

ow

th

e n

um

ber

s ar

e p

aire

d.

3.D

escr

ibe

how

you

wou

ld f

ind

the

inve

rse

of t

he

rela

tion

{(1

,2),

(2,4

),(3

,6),

(4,8

)}.

Sw

itch

th

e co

ord

inat

es in

eac

h o

rder

ed p

air

to g

et

{(2,

1),(

4,2)

,(6,

3),(

8,4)

}.

Hel

pin

g Y

ou

Rem

emb

er

4.T

he

firs

t le

tter

s in

tw

o w

ords

an

d th

eir

orde

r in

th

e al

phab

et c

an s

omet

imes

hel

p yo

ure

mem

ber

thei

r m

ath

emat

ical

mea

nin

g.T

wo

key

term

s in

th

is l

esso

n a

re d

omai

nan

dra

nge

.Des

crib

e h

ow t

he

alph

abet

met

hod

cou

ld h

elp

you

rem

embe

r th

eir

mea

nin

g.S

amp

le a

nsw

er:

dco

mes

fir

st f

or

the

firs

t co

ord

inat

e,r

com

es s

eco

nd

fo

rth

e se

con

d c

oo

rdin

ate.

121

5612

156

©G

lenc

oe/M

cGra

w-H

ill23

0G

lenc

oe A

lgeb

ra 1

Inve

rse

Rel

atio

ns

On

eac

h g

rid

bel

ow,p

lot

the

poi

nts

in

Set

s A

an

d B

.Th

en c

onn

ect

the

poi

nts

in

Set

A w

ith

th

e co

rres

pon

din

g p

oin

ts i

n S

et B

.Th

en f

ind

th

ein

vers

es o

f S

et A

an

d S

et B

,plo

t th

e tw

o se

ts,a

nd

con

nec

t th

ose

poi

nts

.

Set

AS

et B

(�4,

0)(0

,1)

(�3,

0)(0

,2)

(�2,

0)(0

,3)

(�1,

0)(0

,4)

Set

AS

et B

(�3,

�3)

(�2,

1)(�

2,�

2)(�

1,2)

(�1,

�1)

(0,3

)(0

,0)

(1,4

)

Set

AS

et B

(�4,

1)(3

,2)

(�3,

2)(3

,2)

(�2,

3)(3

,2)

(�1,

4)(3

,2)

13.W

hat

is

the

grap

hic

al r

elat

ion

ship

bet

wee

n t

he

lin

e se

gmen

ts y

ou d

rew

con

nec

tin

g po

ints

in

Set

s A

an

d B

an

d th

e li

ne

segm

ents

con

nec

tin

g po

ints

in t

he

inve

rses

of

thos

e tw

o se

ts?

An

swer

s w

ill v

ary.

A p

oss

ible

an

swer

is t

hat

th

e g

rap

hs

are

refl

ecte

d a

cro

ss t

he

line

y�

xby

th

eir

inve

rses

.

x

y

O

x

y

O

x

y

OEn

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

Inve

rse

Set

AS

et B

1.(0

,�4)

(1,0

)

2.(0

,�3)

(2,0

)

3.(0

,�2)

(3,0

)

4.(0

,�1)

(4,0

)

Inve

rse

Set

AS

et B

5.(�

3,�

3)(1

,�2)

6.(�

2,�

2)(2

,�1)

7.(�

1,�

1)(3

,0)

8.(0

,0)

(4,1

)

Inve

rse

Set

AS

et B

9.(1

,�4)

(2,3

)

10.

(2,�

3)(2

,3)

11.

(3,�

2)(2

,3)

12.

(4,�

1)(2

,3)



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Eq

uat

ion

s as

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill23

1G

lenc

oe A

lgeb

ra 1

Lesson 4-4

Solv

e Eq

uat

ion

sT

he

equ

atio

n y

�3x

�4

is a

n e

xam

ple

of a

n e

qu

atio

n i

n t

wo

vari

able

sbe

cau

se i

t co

nta

ins

two

vari

able

s,x

and

y.T

he

solu

tion

of a

n e

quat

ion

in

tw

ova

riab

les

is a

n o

rder

ed p

air

of r

epla

cem

ents

for

th

e va

riab

les

that

res

ult

s in

a t

rue

stat

emen

t w

hen

su

bsti

tute

d in

to t

he

equ

atio

n.

Fin

d t

he

solu

tion

set

for

y

��

2x�

1,gi

ven

th

e re

pla

cem

ent

set

{(�

2,3)

,(0,

�1)

,(1,

�2)

,(3,

1)}.

Mak

e a

tabl

e.S

ubs

titu

te t

he

xan

d y-

valu

es o

f ea

chor

dere

d pa

ir i

nto

th

e eq

uat

ion

.

xy

y�

�2x

�1

Tru

e o

r Fa

lse

�2

33

��

2(�

2) �

1Tr

ue3

�3

0�

1�

1 �

�2(

0) �

1Tr

ue�

1 �

�1

1�

2�

2 �

�2(

1) �

1F

alse

�2

��

3

31

1 �

�2(

3) �

1F

alse

1 �

�7

Th

e or

dere

d pa

irs

(�2,

3),a

nd

(0,�

1) r

esu

lt i

n t

rue

stat

emen

ts.T

he

solu

tion

set

is

{(�

2,3)

,(0,

�1)

}.

Sol

ve b

�2a

�1

ifth

e d

omai

n i

s {�

2,�

1,0,

2,4}

.

Mak

e a

tabl

e.T

he

valu

es o

f a

com

efr

om t

he

dom

ain

.Su

bsti

tute

eac

hva

lue

of a

into

th

e eq

uat

ion

to

dete

rmin

e th

e co

rres

pon

din

g va

lues

of b

in t

he

ran

ge.

a2a

�1

b(a

,b)

�2

2(�

2) �

1�

5(�

2, �

5)

�1

2(�

1) �

1�

3(�

1, �

3)

02(

0) �

1�

1(0

, �

1)

22(

2) �

13

(2,

3)

42(

4) �

17

(4,

7)

Th

e so

luti

on s

et i

s {(

�2,

�5)

,(�

1,�

3),(

0,�

1),(

2,3)

,(4,

7)}.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

solu

tion

set

of

each

eq

uat

ion

,giv

en t

he

rep

lace

men

t se

t.

1.y

�3x

�1;

{(0,

1), �

,2�, �

�1,

��,(

�1,

�2)

}{(

0,1)

, �,2

�,(�

1,�

2)}

2.3x

�2y

�6;

{(�

2,3)

,(0,

1),(

0,�

3),(

2,0)

}{(

0,�

3),(

2,0)

}

3.2x

�5

�y;

{(1,

3),(

2,1)

,(3,

2),(

4,3)

}{(

1,3)

,(2,

1)}

Sol

ve e

ach

eq

uat

ion

if

the

dom

ain

is

(�4,

�2,

0,2,

4}.

4.x

�y

�4

{(�

4,8)

,(�

2,6)

,(0,

4),(

2,2)

(4,

0)}

5.y

��

4x�

6{(

�4,

10),

(�2,

2),(

0,�

6),(

2,�

14),

(4,�

22)}

6.5a

�2b

�10

{(�

4,�

15),

(�2,

�10

),(0

,�5)

,(2,

0),(

4,5)

}

7.3x

�2y

�12

{(�

4,�

12),

(�2,

�9)

,(0,

�6)

,(2,

�3)

(4,

0)}

8.6x

�3y

�18

{(�

4,14

),(�

2,10

),(0

,6),

(2,2

),(4

,�2)

}

9.4x

�8

�2y

{(�

4,�

4),(

�2,

0),(

0,4)

,(2,

8),(

4,12

)}

10.x

�y

�8

{(�

4,�

12),

(�2,

�10

),(0

,�8)

,(2,

�6)

(4,

�4)

}

11.2

x�

y�

10{(

�4,

18),

(�2,

14),

(0,1

0),(

2,6)

,(4,

2)}1 � 3

2 � 31 � 3

©G

lenc

oe/M

cGra

w-H

ill23

2G

lenc

oe A

lgeb

ra 1

Gra

ph

So

luti

on

Set

sYo

u c

an g

raph

th

e or

dere

d pa

irs

in t

he

solu

tion

set

of

aneq

uat

ion

in

tw

o va

riab

les.

Th

e do

mai

n c

onta

ins

valu

es r

epre

sen

ted

by t

he

ind

epen

den

tva

riab

le.T

he

ran

ge c

onta

ins

the

corr

espo

ndi

ng

valu

es r

epre

sen

ted

by t

he

dep

end

ent

vari

able

,wh

ich

are

det

erm

ined

by

the

give

n e

quat

ion

.

Sol

ve 4

x�

2y�

12 i

f th

e d

omai

n i

s (�

1,0,

2,4}

.Gra

ph

th

e so

luti

on s

et.

Fir

st s

olve

th

e eq

uat

ion

for

yin

ter

ms

of x

.4x

�2y

�12

Orig

inal

equ

atio

n

4x�

2y�

4x�

12 �

4xS

ubtr

act

4xfr

om e

ach

side

.

2y�

12 �

4xS

impl

ify.

�D

ivid

e ea

ch s

ide

by 2

.

y�

6 �

2xS

impl

ify.

12 �

4x�

22y � 2S

tudy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Eq

uat

ion

s as

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

Exam

ple

Exam

ple

Su

bsti

tute

eac

h v

alu

e of

xfr

om t

he

dom

ain

to

dete

rmin

e th

e co

rres

pon

din

g va

lue

of y

in t

he

ran

ge.

x6

�2x

y(x

,y)

�1

6 �

2(�

1)8

(�1,

8)

06

�2(

0)6

(0,

6)

26

�2(

2)2

(2,

2)

46

�2(

4)�

2(4

, �

2)

Gra

ph t

he

solu

tion

set

.

x

y

O

Exer

cises

Exer

cises

Sol

ve e

ach

eq

uat

ion

for

th

e gi

ven

dom

ain

.Gra

ph

th

e so

luti

on s

et.

1.x

�2y

�4

for

x�

{�2,

0,2,

4}2.

y�

�2x

�3

for

x�

{�2,

�1,

0,1}

{(�

2,3)

,(0,

2),(

2,1)

,(4,

0)}

{(�

2,1)

,(�

1,�

1),(

0,�

3),(

1,�

5)}

3.x

�3y

�6

for

x�

{�3,

0,3,

6}4.

2x�

4y�

8 fo

r x

�{�

4,�

2,0,

2}

{(�

3,�

3),(

0,�

2),(

3,�

1),(

6,0)

}{(

�4,

�4)

,(�

2,�

3),(

0,�

2),(

2,�

1)}

x

y

Ox

y

O

x

y

O

x

y

O



Skil

ls P

ract

ice

Eq

uat

ion

s as

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill23

3G

lenc

oe A

lgeb

ra 1

Lesson 4-4

Fin

d t

he

solu

tion

set

for

eac

h e

qu

atio

n,g

iven

th

e re

pla

cem

ent

set.

1.y

�3x

�1;

{(2,

5),(

�2,

7),(

0,�

1),(

1,1)

}{(

2,5)

,(0,

�1)

}

2.y

�2x

�4;

{(�

1,�

2),(

�3,

2),(

1,6)

,(�

2,8)

}{(

1,6)

}

3.y

�7

�2x

;{(3

,1),

(4,�

1),(

5,�

3),(

�1,

5)}

{(3,

1),(

4,�

1),(

5,�

3)}

4.�

3x�

y�

2;{(

�3,

7),(

�2,

�4)

,(�

1,�

1),(

3,11

)}{(

�2,

�4)

,(�

1,�

1),(

3,11

)}

Sol

ve e

ach

eq

uat

ion

if

the

dom

ain

is

{�2,

�1,

0,2,

5}.

5.y

�x

�4

{(�

2,2)

,(�

1,3)

,(0,

4),(

2,6)

,(5,

9)}

6.y

�3x

�2

{(�

2,�

8),(

�1,

�5)

,(0,

�2)

,(2,

4),(

5,13

)}

7.y

�2x

�1

{(�

2,�

3),(

�1,

�1)

,(0,

1),(

2,5)

,(5,

11)}

8.x

�y

�2

{(�

2,�

4),(

�1,

�3)

,(0,

�2)

,(2,

0),(

5,3)

}

9.x

�3

�y

{(�

2,5)

,(�

1,4)

,(0,

3),(

2,1)

,(5,

�2)

}

10.2

x�

y�

4{(

�2,

8),(

�1,

6),(

0,4)

,(2,

0),(

5,�

6)}

11.2

x�

y�

7{(

�2,

�11

),(�

1,�

9),(

0,�

7),(

2,�

3),(

5,3)

}

12.4

x�

2y�

6{(

�2,

7),(

�1,

5),(

0,3)

,(2,

�1)

,(5,

�7)

}

Sol

ve e

ach

eq

uat

ion

for

th

e gi

ven

dom

ain

.Gra

ph

th

e so

luti

on s

et.

13.y

�2x

�5

for

x�

{�5,

�4,

�2,

�1,

0}14

.y�

2x�

3 fo

r x

�{�

1,1,

2,3,

4}

15.2

x�

y�

1 fo

r x

�{�

2,�

1,0,

2,3}

16.2

x�

2y�

6 fo

r x

�{�

3,�

1,3,

4,6}

{(�

3,�

6),

(�1,

�4)

,(3,

0),

(4,1

),(6

,3)}

x

y

O

{(�

2,5)

,(�

1,3)

,(0

,1),

(2,�

3),

(3,�

5)}

x

y

O

{(�

1,�

5),

(1,�

1),(

2,1)

,(3

,3),

(4,5

)}

x

y

O

{(�

5,�

5),

(�4,

�3)

,(�

2,1)

,(�

1,3)

,(0

,5)}

x

y

O

©G

lenc

oe/M

cGra

w-H

ill23

4G

lenc

oe A

lgeb

ra 1

Fin

d t

he

solu

tion

set

for

eac

h e

qu

atio

n,g

iven

th

e re

pla

cem

ent

set.

1.y

�2

�5x

;{(3

,12)

,(�

3,�

17),

(2,�

8),(

�1,

7)}

{(2,

�8)

,(�

1,7)

}

2.3x

�2y

��

1;{(

�1,

1),(

�2,

�2.

5),(

�1,

�1.

5),(

0,0.

5)}

{(�

2,�

2.5)

,(0,

0.5)

}

Sol

ve e

ach

eq

uat

ion

if

the

dom

ain

is

{�2,

�1,

2,3,

5}.

3.y

�4

�2x

{(�

2,8)

,(�

1,6)

,(2,

0),(

3,�

2),(

5,�

6)}

4.x

�8

�y

{(�

2,10

),(�

1,9)

,(2,

6),(

3,5)

,(5,

3)}

5.4x

�2y

�10

{(�

2,9)

,(�

1,7)

,(2,

1),(

3,�

1),(

5,�

5)}

6.3x

�6y

�12

{(�

2,�

3),(

�1,

�2.

5),(

2,�

1),(

3,�

0.5)

,(5,

0.5)

}

7.2x

�4y

�16

{(�

2,5)

,(�

1,4.

5),(

2,3)

,(3,

2.5)

,(5,

1.5)

}

8.x

�y

�6

{(�

2,�

16),

(�1,

�14

),(2

,�8)

,(3,

�6)

,(5,

�2)

}

Sol

ve e

ach

eq

uat

ion

for

th

e gi

ven

dom

ain

.Gra

ph

th

e so

luti

on s

et.

9.2x

�4y

�8

for

x�

{�4,

�3,

�2,

2,5}

10.

x�

y�

1 fo

r x

�{�

4,�

3,�

2,0,

4}

EAR

TH S

CIE

NC

EF

or E

xerc

ises

11

and

12,

use

th

e fo

llow

ing

info

rmat

ion

.E

arth

mov

es a

t a

rate

of

30 k

ilom

eter

s pe

r se

con

d ar

oun

d th

e S

un

.Th

e eq

uat

ion

d�

30t

rela

tes

the

dist

ance

din

kilo

met

ers

Ear

th m

oves

to

tim

e t

in s

econ

ds.

11.F

ind

the

set

of o

rder

ed p

airs

wh

en

t�

{10,

20,3

0,45

,70}

.{(

10,3

00),

(20,

600)

,(3

0,90

0),(

45,1

350)

,(70

,210

0)}

12.G

raph

th

e se

t of

ord

ered

pai

rs.

GEO

MET

RYF

or E

xerc

ises

13–

15,u

se t

he

foll

owin

g in

form

atio

n.

Th

e eq

uat

ion

for

th

e ar

ea o

f a

tria

ngl

e is

A�

bh.S

upp

ose

the

area

of

tria

ngl

e D

EF

is

30 s

quar

e in

ches

.

13.S

olve

th

e eq

uat

ion

for

h.

h�

14.S

tate

the

ind

epen

dent

and

dep

ende

nt v

aria

bles

.b

is in

dep

end

ent;

his

dep

end

ent.

15.C

hoo

se 5

val

ues

for

ban

d fi

nd

the

corr

espo

ndi

ng

valu

es f

or h

.S

amp

le a

nsw

er:

{(5,

12),

(6,1

0),(

10,6

),(1

2,5)

,(15

,4)}

60 � b

1 � 2

Tim

e (s

)

Dis

tan

ce E

art

h T

ravels

Distance (km)

010

2030

4050

6070

80

2400

2100

1800

1500

1200 900

600

300

{(�

4,3)

, ��3,

2�1 2� �,

(�2,

2),(

0,1)

,(4

,�1)

}x

y

O

{(�

4,�

4),

(�3,

�3.

5),

(�2,

�3)

,(2

,�1)

,(5,

0.5)

}

x

y

O

1 � 2

1 � 2

Pra

ctic

e (

Ave

rag

e)

Eq

uat

ion

s as

Rel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csE

qu

atio

ns

as R

elat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill23

5G

lenc

oe A

lgeb

ra 1

Lesson 4-4

Pre-

Act

ivit

yW

hy

are

equ

atio

ns

of r

elat

ion

s im

por

tan

t in

tra

veli

ng?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-4

at

the

top

of p

age

212

in y

our

text

book

.

•In

th

e eq

uat

ion

p�

0.69

d,p

repr

esen

ts

and

d

repr

esen

ts

.

•H

ow m

any

vari

able

s ar

e in

th

e eq

uat

ion

p�

0.69

d?

two

Rea

din

g t

he

Less

on

1.S

upp

ose

you

mak

e th

e fo

llow

ing

tabl

e to

sol

ve a

n e

quat

ion

th

at u

ses

the

dom

ain

{�

3,�

2,�

1,0,

1}.

xx

�4

y(x

,y)

�3

�3

�4

�7

(�3,

�7)

�2

�2

�4

�6

(�2,

�6)

�1

�1

�4

�5

(�1,

�5)

00

�4

�4

(0,

�4)

11

�4

�3

(1,

�3)

a.W

hat

is

the

equ

atio

n?

y�

x�

4

b.

Wh

ich

col

um

n s

how

s th

e d

omai

n?

the

firs

t co

lum

n

c.W

hic

h c

olu

mn

sh

ows

the

ran

ge?

the

thir

d c

olu

mn

d.

Wh

ich

col

um

n s

how

s th

e so

luti

on s

et?

the

fou

rth

co

lum

n

2.T

he

solu

tion

set

of

the

equ

atio

n y

�2x

for

a gi

ven

dom

ain

is

{(�

2,�

4),(

0,0)

,(2,

4),(

7,14

)}.T

ell

wh

eth

er e

ach

sen

ten

ce i

s tr

ue

or f

alse

.If

fals

e,re

plac

e th

e u

nde

rlin

ed w

ord(

s) t

o m

ake

a tr

ue

sen

ten

ce.

a.T

he

dom

ain

con

tain

s th

e va

lues

rep

rese

nte

d by

th

e in

depe

nde

nt

vari

able

.tr

ue

b.

Th

e do

mai

n c

onta

ins

the

nu

mbe

rs �

4,0,

4,an

d 14

.fa

lse;

ran

ge

c.F

or e

ach

nu

mbe

r in

th

e do

mai

n,t

he

ran

ge c

onta

ins

a co

rres

pon

din

g n

um

ber

that

is

ava

lue

of t

he

depe

nde

nt

vari

able

.tr

ue

3.W

hat

is

mea

nt

by “

solv

ing

an e

quat

ion

for

yin

ter

ms

of x

”?is

ola

tin

g y

on

on

e si

de

of

the

equ

atio

n

Hel

pin

g Y

ou

Rem

emb

er

4.R

emem

ber,

wh

en y

ou s

olve

an

equ

atio

n f

or a

giv

en v

aria

ble,

that

var

iabl

e be

com

es t

he

dep

end

ent

vari

able

.Wri

te a

n e

quat

ion

an

d de

scri

be h

ow y

ou w

ould

ide

nti

fy t

he

depe

nde

nt

vari

able

.S

amp

le a

nsw

er:

y�

3xsh

ow

s th

at y

ou

hav

e so

lved

fo

ry,

so y

is t

he

dep

end

ent

vari

able

.

do

llars

po

un

ds

©G

lenc

oe/M

cGra

w-H

ill23

6G

lenc

oe A

lgeb

ra 1

Co

ord

inat

e G

eom

etry

an

d A

rea

How

wou

ld y

ou f

ind

the

area

of

a tr

ian

gle

wh

ose

vert

ices

hav

e

the

coor

din

ates

A(-

1,2)

,B(1

,4),

and

C(3

,0)?

Wh

en a

fig

ure

has

no

side

s pa

rall

el t

o ei

ther

axi

s,th

e h

eigh

t an

d ba

se a

re d

iffi

cult

to

fin

d.

On

e m

eth

od o

f fi

ndi

ng

the

area

is

to e

ncl

ose

the

figu

re i

n a

re

ctan

gle

and

subt

ract

th

e ar

ea o

f th

e su

rrou

ndi

ng

tria

ngl

es

from

th

e ar

ea o

f th

e re

ctan

gle.

Are

a of

rec

tan

gle

DE

FC

�4

�4

�16

squ

are

un

its

Are

a of

tri

angl

e I

�(2

)(4)

�4

Are

a of

tri

angl

e II

�

(2)(

4) �

4

Are

a of

tri

angl

e II

I �

(2)(

2) �

2

Tot

al �

10 s

quar

e u

nit

s

Are

a of

tri

angl

e A

BC

�16

�10

,or

6 sq

uar

e u

nit

s

Fin

d t

he

area

s of

th

e fi

gure

s w

ith

th

e fo

llow

ing

vert

ices

.

1.A

(-4,

-6),

B(0

,4),

2.A

(6,-

2),B

(8,-

10),

3.A

(0,2

),B

(2,7

),C

(4,2

)C

(12,

-6)

C(6

,10)

,D(9

,-2)

24 s

qu

are

un

its

20 s

qu

are

un

its

55 s

qu

are

un

its

A

Dx

y

O

B

C

2

2

x

y

OA

B

C

2

2x

y

O

A

B

C2

2

1 � 21 � 21 � 2x

y

O

BE

III

II

I

D

F C

A

x

y

O

B

C

A

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Gra

ph

ing

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-H

ill23

7G

lenc

oe A

lgeb

ra 1

Lesson 4-5

Iden

tify

Lin

ear

Equ

atio

ns

A l

inea

r eq

uat

ion

is a

n e

quat

ion

th

at c

an b

e w

ritt

en i

nth

e fo

rm A

x�

By

�C

.Th

is i

s ca

lled

th

e st

and

ard

for

mof

a l

inea

r eq

uat

ion

.

Sta

nd

ard

Fo

rm o

f a

Lin

ear

Eq

uat

ion

Ax

�B

y�

C,

whe

re A

�0,

Aan

d B

are

not

both

zer

o, a

nd A

, B

, an

d C

are

inte

gers

who

se G

CF

is 1

.

Det

erm

ine

wh

eth

er y

�6

�3x

is a

lin

ear

equ

atio

n.I

f so

,wri

te t

he

equ

atio

nin

sta

nd

ard

for

m.

Fir

st r

ewri

te t

he

equ

atio

n s

o bo

th v

aria

bles

are

on

the

sam

e si

de o

f th

e eq

uat

ion

.y

�6

�3x

Orig

inal

equ

atio

n

y�

3x�

6 �

3x�

3xA

dd 3

xto

eac

h si

de.

3x�

y�

6S

impl

ify.

Th

e eq

uat

ion

is

now

in

sta

nda

rd f

orm

,wit

h A

�3,

B�

1 an

d C

�6.

Th

is i

s a

lin

ear

equ

atio

n.

Det

erm

ine

wh

eth

er 3

xy�

y�

4 �

2xis

ali

nea

r eq

uat

ion

.If

so,w

rite

th

eeq

uat

ion

in

sta

nd

ard

for

m.

Sin

ce t

he

term

3xy

has

tw

o va

riab

les,

the

equ

atio

n c

ann

ot b

e w

ritt

en i

n t

he

form

Ax

�B

y�

C.T

her

efor

e,th

is i

sn

ot a

lin

ear

equ

atio

n.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Det

erm

ine

wh

eth

er e

ach

eq

uat

ion

is

a li

nea

r eq

uat

ion

.If

so,w

rite

th

e eq

uat

ion

in

stan

dar

d f

orm

.

1.2x

�4y

2.6

�y

�8

3.4x

�2y

��

1

yes;

2x�

4y�

0ye

s;y

�2

yes;

4x�

2y�

�1

4.3x

y�

8 �

4y5.

3x�

4 �

126.

y�

x2�

7

no

yes;

3x�

16n

o

7.y

�4x

�9

8.x

�8

�0

9.�

2x�

3 �

4y

yes;

4x�

y�

�9

yes;

x�

�8

yes;

2x�

4y�

3

10.2

�x

�y

11.

y�

12 �

4x12

.3xy

�y

�8

yes;

x�

2y�

�4

yes;

16x

�y

�48

no

13.6

x�

4y�

3 �

014

.yx

�2

�8

15.6

a�

2b�

8 �

b

yes;

6x�

4y�

3n

oye

s;6a

�3b

�8

16.

x�

12y

�1

17.3

�x

�x2

�0

18.x

2�

2xy

yes;

x�

48y

�4

no

no

1 � 4

1 � 41 � 2

©G

lenc

oe/M

cGra

w-H

ill23

8G

lenc

oe A

lgeb

ra 1

Gra

ph

Lin

ear

Equ

atio

ns

The

gra

ph o

f a

line

ar e

quat

ion

is a

line

.The

line

rep

rese

nts

all

solu

tion

s to

the

line

ar e

quat

ion.

Als

o,ev

ery

orde

red

pair

on

this

line

sat

isfi

es t

he e

quat

ion.

Gra

ph

th

e eq

uat

ion

y�

2x�

1.

Sol

ve t

he

equ

atio

n f

or y

.y

�2x

�1

Orig

inal

equ

atio

n

y�

2x�

2x�

1 �

2xA

dd 2

xto

eac

h si

de.

y�

2x�

1S

impl

ify.

Sel

ect

five

val

ues

for

th

e do

mai

n a

nd

mak

e a

tabl

e.T

hen

grap

h t

he

orde

red

pair

s an

d dr

aw a

lin

e th

rou

gh t

he

poin

ts.

x2x

�1

y(x

,y)

�2

2(�

2) �

1�

3(�

2, �

3)

�1

2(�

1) �

1�

1(�

1, �

1)

02(

0) �

11

(0,

1)

12(

1) �

13

(1,

3)

22(

2) �

15

(2,

5)

Gra

ph

eac

h e

qu

atio

n.

1.y

�4

2.y

�2x

3.x

�y

��

1

4.3x

�2y

�6

5.x

�2y

�4

6.2x

�y

��

2

7.3x

�6y

��

38.

�2x

�y

��

29.

x�

y�

6

x

y

O4

812

8 4x

y

Ox

y O

3 � 41 � 4

x

y Ox

y

Ox

y

O

x

y

Ox

y O

x

y

O

x

y O

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Gra

ph

ing

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Gra

ph

ing

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-H

ill23

9G

lenc

oe A

lgeb

ra 1

Lesson 4-5

Det

erm

ine

wh

eth

er e

ach

eq

uat

ion

is

a li

nea

r eq

uat

ion

.If

so,w

rite

th

e eq

uat

ion

in

stan

dar

d f

orm

.

1.xy

�6

2.y

�2

�3x

3.5x

�y

�4

no

yes;

3x�

y�

2ye

s;5x

�y

��

4

4.y

�2x

�5

5.y

��

7 �

6x6.

y�

3x2

�1

yes;

2x�

y�

�5

yes;

6x�

y�

7 n

o

7.y

�4

�0

8.5x

�6y

�3x

�2

9.y

�1

yes;

y�

4ye

s;x

�3y

�1

yes;

y�

2

Gra

ph

eac

h e

qu

atio

n.

10.y

�4

11.y

�3x

12.y

�x

�4

13.y

�x

�2

14.y

�4

�x

15.y

�4

�2x

16.x

�y

�3

17.1

0x�

�5y

18.4

x�

2y�

6

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

1 � 2

©G

lenc

oe/M

cGra

w-H

ill24

0G

lenc

oe A

lgeb

ra 1

Det

erm

ine

wh

eth

er e

ach

eq

uat

ion

is

a li

nea

r eq

uat

ion

.If

so,w

rite

th

e eq

uat

ion

in

stan

dar

d f

orm

.

1.4x

y�

2y�

92.

8x�

3y�

6 �

4x3.

7x�

y�

3 �

yn

oye

s;4x

�y

�2

yes;

7x�

�3

4.5

�2y

�3x

5.4y

�x

�9x

6.a

�b

�2

yes;

3x�

2y�

5ye

s;8x

�4y

�0

yes;

5a�

b�

10

7.6x

�2y

8.�

�1

9.�

�7

yes;

6x�

2y�

0ye

s;3x

�4y

�12

n

o

Gra

ph

eac

h e

qu

atio

n.

10.

x�

y�

211

.5x

�2y

�7

12.1

.5x

�3y

�9

CO

MM

UN

ICA

TIO

NS

For

Exe

rcis

es 1

3–15

,use

th

e fo

llow

ing

info

rmat

ion

.A

tel

eph

one

com

pan

y ch

arge

s $4

.95

per

mon

th f

or l

ong

dist

ance

cal

ls p

lus

$0.0

5 pe

r m

inu

te.T

he

mon

thly

cos

t c

of l

ong

dist

ance

cal

ls c

an b

e de

scri

bed

by t

he

equ

atio

n

c�

0.05

m�

4.95

,wh

ere

mis

th

e n

um

ber

of m

inu

tes.

13.F

ind

the

y-in

terc

ept

of t

he

grap

h o

f th

e eq

uat

ion

.(0

,4.9

5)

14.G

raph

th

e eq

uat

ion

.

15.I

f yo

u t

alk

140

min

ute

s,w

hat

is

the

mon

thly

cos

t fo

r lo

ng

dist

ance

?$1

1.95

MA

RIN

E B

IOLO

GY

For

Exe

rcis

es 1

6 an

d 1

7,u

se t

he

foll

owin

g in

form

atio

n.

Kil

ler

wh

ales

usu

ally

sw

im a

t a

rate

of

3.2–

9.7

kilo

met

ers

per

hou

r,th

ough

th

ey c

an t

rave

l u

p to

48.

4 ki

lom

eter

s pe

rh

our.

Su

ppos

e a

mig

rati

ng

kill

er w

hal

e is

sw

imm

ing

at a

nav

erag

e ra

te o

f 4.

5 ki

lom

eter

s pe

r h

our.

Th

e di

stan

ce d

the

wh

ale

has

tra

vele

d in

th

ours

can

be

pred

icte

d by

th

eeq

uat

ion

d�

4.5t

.

16.G

raph

th

e eq

uat

ion

.

17.U

se t

he

grap

h t

o pr

edic

t th

e ti

me

it t

akes

th

e ki

ller

w

hal

e to

tra

vel

30 k

ilom

eter

s.b

etw

een

6 h

an

d 7

hTi

me

(ho

urs

)

Kil

ler

Wh

ale

Tra

vels

Distance (km)

02

46

89

13

57

40 35 30 25 20 15 10 5

Tim

e (m

inu

tes)

Lon

g D

ista

nce

Cost ($)

040

8012

016

0

14 12 10 8 6 4 2

x

y

O

x

y

Ox

y

O

1 � 2

2 � y5 � x

y � 3x � 4

1 � 5

Pra

ctic

e (

Ave

rag

e)

Gra

ph

ing

Lin

ear

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5



Readin

g t

o L

earn

Math

em

ati

csG

rap

hin

g L

inea

r E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-H

ill24

1G

lenc

oe A

lgeb

ra 1

Lesson 4-5

Pre-

Act

ivit

yH

ow c

an l

inea

r eq

uat

ion

s b

e u

sed

in

nu

trit

ion

?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-5

at

the

top

of p

age

218

in y

our

text

book

.

In t

he

equ

atio

n f

�0.

3 ��,w

hat

are

th

e in

depe

nde

nt

and

depe

nde

nt

vari

able

s?C

is in

dep

end

ent,

fis

dep

end

ent.

Rea

din

g t

he

Less

on

1.D

escr

ibe

the

grap

h o

f a

lin

ear

equ

atio

n.

Th

e g

rap

h is

a s

trai

gh

t lin

e.

2.D

eter

min

e w

het

her

eac

h e

quat

ion

is

a li

nea

r eq

uat

ion

.Exp

lain

.

Eq

uat

ion

Lin

ear

or

no

n-l

inea

r?E

xpla

nat

ion

a.2x

�3y

�1

linea

rT

he

equ

atio

n c

an b

e w

ritt

en a

s 2x

�3y

�1.

b.

4xy

�2y

�7

no

n-l

inea

r4x

yh

as t

wo

var

iab

les.

c.2x

2�

4y�

3n

on

-lin

ear

Th

e va

riab

le x

has

an

exp

on

ent

of

2.

d.

��

2lin

ear

Th

e eq

uat

ion

can

be

wri

tten

as

3x�

20y

�30

.

3.W

hat

do

the

term

s x-

inte

rcep

tan

d y-

inte

rcep

tm

ean

?T

he

x-in

terc

ept

is t

he

x-co

ord

inat

e o

f th

e p

oin

t w

her

e th

e g

rap

h o

f an

eq

uat

ion

cro

sses

th

e x-

axis

,an

d t

he

y-in

terc

ept

is t

he

y-co

ord

inat

e o

f th

e p

oin

t w

her

e th

eg

rap

h c

ross

es t

he

y-ax

is.

Hel

pin

g Y

ou

Rem

emb

er

4.D

escr

ibe

the

met

hod

you

wou

ld u

se t

o gr

aph

4x

�2y

�8.

Sam

ple

an

swer

:F

ind

th

ex-

an

d y

-in

terc

epts

,wh

ich

are

2 a

nd

4.P

lot

the

po

ints

fo

r th

e o

rder

edp

airs

(2,

0) a

nd

(0,

4),a

nd

dra

w a

lin

e th

at c

on

nec

ts t

he

po

ints

.

4y � 3x � 5

C � 9

©G

lenc

oe/M

cGra

w-H

ill24

2G

lenc

oe A

lgeb

ra 1

Taxi

cab

Gra

ph

sYo

u h

ave

use

d a

rect

angu

lar

coor

din

ate

syst

em t

o gr

aph

eq

uat

ion

s su

ch a

s y

�x

�1

on a

coo

rdin

ate

plan

e.In

a

coor

din

ate

plan

e,th

e n

um

bers

in

an

ord

ered

pai

r (x

,y)

can

be

an

y tw

o re

al n

um

bers

.

A t

axic

ab p

lan

eis

dif

fere

nt

from

th

e u

sual

coo

rdin

ate

plan

e.T

he

only

poi

nts

all

owed

are

th

ose

that

exi

st a

lon

g th

e h

oriz

onta

l an

d ve

rtic

al g

rid

lin

es.Y

ou m

ay t

hin

k of

th

e po

ints

as

tax

icab

s th

at m

ust

sta

y on

th

e st

reet

s.

Th

e ta

xica

b gr

aph

sh

ows

the

equ

atio

ns

y�

�2

and

y�

x�

1.N

otic

e th

at o

ne

of t

he

grap

hs

is n

o lo

nge

r a

stra

igh

t li

ne.

It i

s n

ow a

col

lect

ion

of

sepa

rate

poi

nts

.

Gra

ph

th

ese

equ

atio

ns

on t

he

taxi

cab

pla

ne

at t

he

righ

t.

1.y

�x

�1

2.y

��

2x�

3

3.y

�2.

54.

x�

�4

Use

you

r gr

aph

s fo

r th

ese

pro

ble

ms.

5.W

hic

h o

f th

e eq

uat

ion

s h

as t

he

sam

e gr

aph

in

bot

h t

he

usu

al c

oord

inat

e pl

ane

and

the

taxi

cab

plan

e?x

��

4

6.D

escr

ibe

the

form

of

equ

atio

ns

that

hav

e th

e sa

me

grap

h i

n b

oth

th

e u

sual

coo

rdin

ate

plan

e an

d th

e ta

xica

b pl

ane.

x�

Aan

d y

�B

,wh

ere

Aan

d B

are

inte

ger

s

In t

he

taxi

cab

pla

ne,

dis

tan

ces

are

not

mea

sure

d d

iago

nal

ly,b

ut

alon

g th

e st

reet

s.W

rite

th

e ta

xi-d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

7.(0

,0)

and

(5,2

)8.

(0,0

) an

d (-

3,2)

9.(0

,0)

and

(2,1

.5)

7 u

nit

s5

un

its

3.5

un

its

10.(

1,2)

an

d (4

,3)

11.(

2,4)

an

d (-

1,3)

12.(

0,4)

an

d (-

2,0)

4 u

nit

s4

un

its

6 u

nit

s

Dra

w t

hes

e gr

aph

s on

th

e ta

xica

b g

rid

at

the

righ

t.

13.T

he

set

of p

oin

ts w

hos

e ta

xi-d

ista

nce

fro

m (

0,0)

is

2 u

nit

s.in

dic

ated

by

cro

sses

14.T

he

set

of p

oin

ts w

hos

e ta

xi-d

ista

nce

fro

m (

2,1)

is

3 u

nit

s.in

dic

ated

by

do

ts

x

y

O

x

y

O

1.

1.2.

2.

3.3.

4. 4.

x

y

O

y �

x �

1

y �

�2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill24

3G

lenc

oe A

lgeb

ra 1

Lesson 4-6

Iden

tify

Fu

nct

ion

sR

elat

ion

s in

wh

ich

eac

h e

lem

ent

of t

he

dom

ain

is

pair

ed w

ith

exac

tly

one

elem

ent

of t

he

ran

ge a

re c

alle

d fu

nct

ion

s.

Det

erm

ine

wh

eth

er t

he

rela

tion

{(6

,�3)

,(4

,1),

(7,�

2),(

�3,

1)}

is a

fun

ctio

n.E

xpla

in.

Sin

ce e

ach

ele

men

t of

th

e do

mai

n i

spa

ired

wit

h e

xact

ly o

ne

elem

ent

ofth

e ra

nge

,th

is r

elat

ion

is

afu

nct

ion

.

Det

erm

ine

wh

eth

er 3

x�

y�

6 is

a f

un

ctio

n.

Sin

ce t

he

equ

atio

n i

s in

th

e fo

rm

Ax

�B

y�

C,t

he

grap

h o

f th

eeq

uat

ion

wil

l be

a l

ine,

as s

how

nat

th

e ri

ght.

If y

ou d

raw

a v

erti

cal

lin

e th

rou

gh e

ach

val

ue

of x

,th

eve

rtic

al l

ine

pass

es t

hro

ugh

just

one

poin

t of

th

e gr

aph

.Th

us,

the

lin

e re

pres

ents

a f

un

ctio

n.

x

y

O

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.

1.2.

3.

yes

yes

no

4.5.

6.

no

no

yes

7.{(

4,2)

,(2,

3),(

6,1)

}8.

{(�

3,�

3),(

�3,

4),(

�2,

4)}

9.{(

�1,

0),(

1,0)

}ye

sn

oye

s

10.�

2x�

4y�

011

.x2

�y2

�8

12.x

��

4ye

sn

on

o

x

y

Ox

y

Ox

y O

XY

�1 0 1 2

4 5 6 7

x

y

Ox

y

O

©G

lenc

oe/M

cGra

w-H

ill24

4G

lenc

oe A

lgeb

ra 1

Fun

ctio

n V

alu

esE

quat

ion

s th

at a

re f

un

ctio

ns

can

be

wri

tten

in

a f

orm

cal

led

fun

ctio

nn

otat

ion

.For

exa

mpl

e,y

�2x

�1

can

be

wri

tten

as

f(x)

�2x

�1.

In t

he

fun

ctio

n,x

repr

esen

ts t

he

elem

ents

of

the

dom

ain

,an

d f(

x) r

epre

sen

ts t

he

elem

ents

of

the

ran

ge.

Su

ppos

e yo

u w

ant

to f

ind

the

valu

e in

th

e ra

nge

th

at c

orre

spon

ds t

o th

e el

emen

t 2

in t

he

dom

ain

.Th

is i

s w

ritt

en f

(2)

and

is r

ead

“fof

2.”

Th

e va

lue

of f

(2)

is f

oun

d by

su

bsti

tuti

ng

2 fo

r x

in t

he

equ

atio

n.

If f

(x)

�3x

�4,

fin

d e

ach

val

ue.

a.f(

3)f(

3) �

3(3)

�4

Rep

lace

xw

ith 3

.

�9

�4

Mul

tiply

.

�5

Sim

plify

.

b.

f(�

2)f(

�2)

�3(

�2)

�4

Rep

lace

xw

ith �

2.

��

6 �

4M

ultip

ly.

��

10S

impl

ify.

If f

(x)

�2x

�4

and

g(x

) �

x2�

4x,f

ind

eac

h v

alu

e.

1.f(

4)2.

g(2)

3.f(

�5)

4�

4�

14

4.g(

�3)

5.f(

0)6.

g(0)

21�

40

7.f(

3) �

18.

f ��

9.g �

�1

�3

�

10.f

(a2 )

11.f

(k�

1)12

.g(2

c)

2a2

�4

2k�

24c

2�

8c

13.f

(3x)

14.f

(2)

�3

15.g

(�4)

6x�

43

32

15 � 161 � 2

1 � 41 � 4

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill24

5G

lenc

oe A

lgeb

ra 1

Lesson 4-6

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.

1.ye

s2.

yes

3.n

o

4.ye

s5.

no

6.n

o

7.{(

2,5)

,(4,

�2)

,(3,

3),(

5,4)

,(�

2,5)

}ye

s8.

{(6,

�1)

,(�

4,2)

,(5,

2),(

4,6)

,(6,

5)}

no

9.y

�2x

�5

yes

10.y

�11

yes

11.

yes

12.

no

13.

no

If f

(x)

�3x

�2

and

g(x

) �

x2�

x,fi

nd

eac

h v

alu

e.

14.f

(4)

1415

.f(8

)26

16.f

(�2)

�4

17.g

(2)

2

18.g

(�3)

1219

.g(�

6)42

20.f

(2)

�1

921

.f(1

) �

14

22.g

(2)

�2

023

.g(�

1) �

46

24.f

(x�

1)3x

�5

25.g

(3b)

9b2

�3b

x

y

Ox

y

Ox

y

O

xy

37

�1

1

10

35

73

xy

27

5�

3

35

�4

�2

52

xy

4�

5

�1

�10

0�

9

1�

7

91

XY 2�

1 3 5

4 6 7

XY 4 1�

2

5 2 0�

3

XY 4 1

�3

�5

�6

�2 1 3

©G

lenc

oe/M

cGra

w-H

ill24

6G

lenc

oe A

lgeb

ra 1

Det

erm

ine

wh

eth

er e

ach

rel

atio

n i

s a

fun

ctio

n.

1.ye

s2.

no

3.ye

s

4.{(

1,4)

,(2,

�2)

,(3,

�6)

,(�

6,3)

,(�

3,6)

}5.

{(6,

�4)

,(2,

�4)

,(�

4,2)

,(4,

6),(

2,6)

}ye

sn

o

6.x

��

2 n

o7.

y�

2 ye

s

If f

(x)

�2x

�6

and

g(x

) �

x�

2x2 ,

fin

d e

ach

val

ue.

8.f(

2)�

29.

f ��

710

.g(�

1)�

3

11.g

��

12.f

(7)

�9

�1

13.g

(�3)

�13

�8

14.f

(h�

9)15

.g(3

y)16

.2[g

(b)

�1]

2h�

123y

�18

y22b

�4b

2�

2

WA

GE

SF

or E

xerc

ises

17

and

18,

use

th

e fo

llow

ing

info

rmat

ion

.

Mar

tin

ear

ns

$7.5

0 pe

r h

our

proo

frea

din

g ad

s at

a l

ocal

new

spap

er.H

is w

eekl

y w

age

wca

nbe

des

crib

ed b

y th

e eq

uat

ion

w�

7.5h

,wh

ere

his

th

e n

um

ber

of h

ours

wor

ked.

17.W

rite

th

e eq

uat

ion

in

fu

nct

ion

al n

otat

ion

.f(

h)

�7.

5h

18.F

ind

f(15

),f(

20),

and

f(25

).11

2.50

,150

,187

.50

ELEC

TRIC

ITY

For

Exe

rcis

es 1

9–21

,use

th

e fo

llow

ing

info

rmat

ion

.

Th

e ta

ble

show

s th

e re

lati

onsh

ip b

etw

een

res

ista

nce

Ran

d cu

rren

t I

in a

cir

cuit

.

19.I

s th

e re

lati

onsh

ip a

fu

nct

ion

?E

xpla

in.Y

es;

for

each

val

ue

in t

he

do

mai

n,t

her

eis

on

ly o

ne

valu

e in

th

e ra

ng

e.

20.I

f th

e re

lati

on c

an b

e re

pres

ente

d by

th

e eq

uat

ion

IR

�12

,rew

rite

th

e eq

uat

ion

in

fu

nct

ion

al n

otat

ion

so

that

th

e re

sist

ance

Ris

a f

un

ctio

n o

f th

e cu

rren

t I.

f(I)

�

21.W

hat

is

the

resi

stan

ce i

n a

cir

cuit

wh

en t

he

curr

ent

is 0

.5 a

mpe

re?

24 o

hm

s

12 � I

Res

ista

nce

(o

hm

s)12

080

486

4

Cu

rren

t (a

mp

eres

)0.

10.

150.

252

3

5 � 91 � 3

1 � 2

x

y

O

xy

1�

5

�4

3

76

1�

2

XY 0 3�

2

�3

�2 1 5

Pra

ctic

e (

Ave

rag

e)

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csF

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill24

7G

lenc

oe A

lgeb

ra 1

Lesson 4-6

Pre-

Act

ivit

yH

ow a

re f

un

ctio

ns

use

d i

n m

eteo

rolo

gy?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-6

at

the

top

of p

age

226

in y

our

text

book

.

If p

ress

ure

is

the

inde

pen

den

t va

riab

le a

nd

tem

pera

ture

is

the

depe

nde

nt

vari

able

,wh

at a

re t

he

orde

red

pair

s fo

r th

is s

et o

f da

ta?

{(10

13,3

),(1

006,

4),(

997,

10),

(995

,13)

,(99

5,8)

,(10

00,4

),(1

006,

1),(

1011

,�2)

,(10

16,�

6),(

1019

,�9)

}

Rea

din

g t

he

Less

on

1.T

he

stat

emen

t,“R

elat

ion

s in

wh

ich

eac

h e

lem

ent

of t

he

is p

aire

d w

ith

exa

ctly

on

e el

emen

t of

th

e ar

e ca

lled

fu

nct

ion

s,”

is f

alse

.How

can

you

ch

ange

th

e u

nde

rlin

ed

wor

ds t

o m

ake

the

stat

emen

t tr

ue?

Ch

ang

e ra

ng

eto

do

mai

nan

d d

om

ain

to r

ang

e.

2.D

escr

ibe

how

eac

h m

eth

od s

how

s th

at t

he

rela

tion

rep

rese

nte

d is

a f

un

ctio

n.

a.m

appi

ng

Th

e el

emen

ts o

f th

e d

om

ain

are

pai

red

wit

h c

orr

esp

on

din

gel

emen

ts in

th

e ra

ng

e.E

ach

elem

ent

of

the

do

mai

n h

as o

nly

on

e ar

row

go

ing

fro

m it

.

b.

vert

ical

lin

e te

stN

o v

erti

cal l

ine

pas

ses

thro

ug

hm

ore

th

an o

ne

po

int

of

the

gra

ph

.

Hel

pin

g Y

ou

Rem

emb

er

3.A

stu

den

t w

ho

was

try

ing

to h

elp

a fr

ien

d re

mem

ber

how

fu

nct

ion

s ar

e di

ffer

ent

from

rela

tion

s th

at a

re n

ot f

un

ctio

ns

gave

th

e fo

llow

ing

advi

ce:J

ust

rem

embe

r th

at f

un

ctio

ns

are

very

str

ict

and

nev

er g

ive

you

a c

hoi

ce.E

xpla

in h

ow t

his

mig

ht

hel

p yo

u r

emem

ber

wh

at a

fu

nct

ion

is.

Sam

ple

an

swer

:A

fu

nct

ion

alw

ays

pai

rs e

ach

ele

men

t in

the

do

mai

n w

ith

exa

ctly

on

e el

emen

t in

th

e ra

ng

e.If

tw

o p

eop

le s

tart

wit

hth

e sa

me

elem

ent

of

the

do

mai

n,t

hey

are

fo

rced

to

pai

r it

wit

h t

he

sam

eel

emen

t in

th

e ra

ng

e.T

he

seco

nd

per

son

can

no

t p

ick

a d

iffe

ren

t n

um

ber

fro

m t

he

firs

t p

erso

n.

x

y

O

XY 1 2 3 7

�2

�1 0 4

dom

ain

ran

ge

©G

lenc

oe/M

cGra

w-H

ill24

8G

lenc

oe A

lgeb

ra 1

Co

mp

osi

te F

un

ctio

ns

Th

ree

thin

gs a

re n

eede

d to

hav

e a

fun

ctio

n—

a se

t ca

lled

th

e do

mai

n,

a se

t ca

lled

th

e ra

nge

,an

d a

rule

th

at m

atch

es e

ach

ele

men

t in

th

edo

mai

n w

ith

on

ly o

ne

elem

ent

in t

he

ran

ge.H

ere

is a

n e

xam

ple.

Ru

le:f

(x)

�2x

�1

f(x)

�2x

�1

f(1)

�2(

1) �

1 �

2 �

1 �

3

f(2)

�2(

2) �

1 �

4 �

1 �

5

f(-3

) �

2(�

3) �

1 �

�6

�1

��

5

Su

ppos

e w

e h

ave

thre

e se

ts A

,B,a

nd

C a

nd

two

fun

ctio

ns

desc

ribe

das

sh

own

bel

ow.

Ru

le:f

(x)

�2x

�1

Ru

le:g

(y)

�3y

�4

g(y)

�3y

�4

g(3)

�3(

3) �

4 �

5

Let

’s f

ind

a ru

le t

hat

wil

l m

atch

ele

men

ts o

f se

t A

wit

h e

lem

ents

of

set

C w

ith

out

fin

din

g an

y el

emen

ts i

n s

et B

.In

oth

er w

ords

,let

’s f

ind

a ru

le f

or t

he

com

pos

ite

fun

ctio

n g

[f(x

)].

Sin

ce f

(x)

�2x

�1,

g[f(

x)]

�g(

2x�

1).

Sin

ce g

(y)

�3y

�4,

g(2x

�1)

�3(

2x�

1) �

4,or

6x

�1.

Th

eref

ore,

g[f(

x)]

�6x

�1.

Fin

d a

ru

le f

or t

he

com

pos

ite

fun

ctio

n g

[f(x

)].

1.f(

x) �

3xan

d g(

y) �

2y�

12.

f(x)

�x2

�1

and

g(y)

�4y

g[f

(x)]

�6x

�1

g[f

(x)]

�4x

2�

4

3.f(

x) �

�2x

and

g(y)

�y2

�3y

4.f(

x) �

and

g(y)

�y�

1

g[f

(x)]

�4x

2�

6xg

[f(x

)] �

x�

3

5.Is

it

alw

ays

the

case

th

at g

[f(x

)] �

f[g(

x)]?

Ju

stif

y yo

ur

answ

er.

No

.Fo

r ex

amp

le,i

n E

xerc

ise

1,f[

g(x

)] �

f(2x

�1)

�3(

2x�

1) �

6x�

3,n

ot

6x�

1.

1� x

�3

13

xf (

x)

5

g[f(

x)]

AB

C

13

25

–3–5

xf(

x)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Ari

thm

etic

Seq

uen

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-H

ill24

9G

lenc

oe A

lgeb

ra 1

Lesson 4-7

Rec

og

niz

e A

rith

met

ic S

equ

ence

sA

seq

uen

ceis

a s

et o

f n

um

bers

in

a s

peci

fic

orde

r.If

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Fin

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4-7

4-7



Readin

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Lesson 4-7

Pre-

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re a

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4-7

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th

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n a

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xpla

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Yes;

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dif

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8.2.

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3,7

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the

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the

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2 �

5 �

8 �

11 �

14 �

17 �

20

Th

e se

ries

rem

ain

s th

e sa

me

if w

e S

�2

�5

�8

�11

�14

�17

�20

reve

rse

the

orde

r of

all

th

e te

rms.

So

S�

20 �

17 �

14 �

11 �

8 �

5 �

2le

t u

s re

vers

e th

e or

der

of t

he

term

s 2S

�22

�22

�22

�22

�22

�22

�22

and

add

one

seri

es t

o th

e ot

her

,ter

m

2S�

7(22

)by

ter

m.T

his

is

show

n a

t th

e ri

ght.

S�

�7(

11)

�77

Let

are

pres

ent

the

firs

t te

rm o

f th

e se

ries

.L

et �

repr

esen

t th

e la

st t

erm

of

the

seri

es.

Let

nre

pres

ent

the

nu

mbe

r of

ter

ms

in t

he

seri

es.

In t

he

prec

edin

g ex

ampl

e,a

�2,

l �

20,a

nd

n�

7.N

otic

e th

at w

hen

you

add

the

two

seri

es,t

erm

by

term

,the

sum

of

each

pai

r of

ter

ms

is 2

2.T

hat

su

m c

an b

e fo

un

d by

add

ing

the

firs

t an

d la

st t

erm

s,2

�20

or

a�

�.N

otic

e al

so t

hat

th

ere

are

7,or

n,s

uch

su

ms.

Th

eref

ore,

the

valu

eof

2S

is 7

(22)

,or

n(a

��)

in

th

e ge

ner

al c

ase.

Sin

ce t

his

is

twic

e th

e su

mof

th

e se

ries

,you

can

use

th

e fo

llow

ing

form

ula

to

fin

d th

e su

m o

f an

yar

ith

met

ic s

erie

s.

S�

Fin

d t

he

sum

:1 �

2 �

3 �

4 �

5 �

6 �

7 �

8 �

9

a�

1,�

�9,

n�

9,so

S�

��

45

Fin

d t

he

sum

:�9

�(�

5) �

(�1)

�3

�7

�11

�15

a�

29,�

�15

,n�

7,so

S�

��

21

Fin

d t

he

sum

of

each

ari

thm

etic

ser

ies.

1.3

�6

�9

�12

�15

�18

�21

�24

108

2.10

�15

�20

�25

�30

�35

�40

�45

�50

270

3.�

21 �

(�16

) �

(�11

) �

(�6)

�(�

1) �

4 �

9 �

14�

28

4.ev

en w

hol

e n

um

bers

fro

m 2

th

rou

gh 1

0025

50

5.od

d w

hol

e n

um

bers

bet

wee

n 0

an

d 10

025

00

7 �

6�

27(

�9

�15

)�

� 2

9 �

10�

29(

1 �

9)�

� 2

n(a

��)

�� 2

7(22

)�

2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Wri

tin

g E

qu

atio

ns

fro

m P

atte

rns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

©G

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cGra

w-H

ill25

5G

lenc

oe A

lgeb

ra 1

Lesson 4-8

Loo

k fo

r Pa

tter

ns

A v

ery

com

mon

pro

blem

-sol

vin

g st

rate

gy i

s to

loo

k f

or a

pat

tern

.A

rith

met

ic s

equ

ence

s fo

llow

a p

atte

rn,a

nd

oth

er s

equ

ence

s ca

n f

ollo

w a

pat

tern

.

Fin

d t

he

nex

t th

ree

term

s in

th

e se

qu

ence

3,9

,27,

81,…

.

Stu

dy t

he

patt

ern

in

th

e se

quen

ce.

Su

cces

sive

ter

ms

are

fou

nd

bym

ult

iply

ing

the

last

giv

en t

erm

by

3.

Th

e n

ext

thre

e te

rms

are

243,

729,

2187

.

81

� 3

� 3

� 3

243

729

2187

3

� 3

� 3

� 3

927

81

Fin

d t

he

nex

t th

ree

term

sin

th

e se

qu

ence

10,

6,11

,7,1

2,8,

….

Stu

dy t

he

patt

ern

in

th

e se

quen

ce.

Ass

um

e th

at t

he

patt

ern

con

tin

ues

.

Th

e n

ext

thre

e te

rms

are

13,9

,14.

8

� 5

� 4

� 5

139

14

� 4

� 5

106

11

� 4

� 5

712

� 4

8

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

1.G

ive

the

nex

t tw

o it

ems

for

the

patt

ern

bel

ow.

Giv

e th

e n

ext

thre

e n

um

ber

s in

eac

h s

equ

ence

.

2.2,

12,7

2,43

2,…

3.

7,�

14,2

8,�

56,…

2592

,15,

552,

93,3

1211

2,�

224,

448

4.0,

10,5

,15,

10,…

5.

0,1,

3,6,

10,…

20,1

5,25

15,2

1,28

6.x

�1,

x�

2,x

�3,

…

7.x,

,,

,…

x�

4,x

�5,

x�

6,

,x � 7

x � 6x � 5

x � 4x � 3

x � 2

©G

lenc

oe/M

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w-H

ill25

6G

lenc

oe A

lgeb

ra 1

Wri

te E

qu

atio

ns

Som

etim

es a

pat

tern

can

lea

d to

a g

ener

al r

ule

th

at c

an b

e w

ritt

en a

san

equ

atio

n.

Su

pp

ose

you

pu

rch

ased

a n

um

ber

of

pac

kag

es o

f b

lan

k c

omp

act

dis

ks.

If e

ach

pac

kag

e co

nta

ins

3 co

mp

act

dis

ks,

you

cou

ld m

ake

a ch

art

to s

how

the

rela

tion

ship

bet

wee

n t

he

nu

mb

er o

f p

ack

ages

of

com

pac

t d

isk

s an

d t

he

nu

mb

er o

f d

isk

s p

urc

has

ed.U

se x

for

the

nu

mb

er o

f p

ack

ages

an

d y

for

the

nu

mb

er o

f co

mp

act

dis

ks.

Mak

e a

tabl

e of

ord

ered

pai

rs f

or s

ever

al p

oin

ts o

f th

e gr

aph

.

Th

e di

ffer

ence

in

th

e x

valu

es i

s 1,

and

the

diff

eren

ce i

n t

he

yva

lues

is

3.T

his

pat

tern

show

s th

at y

is a

lway

s th

ree

tim

es x

.Th

is s

ugg

ests

th

e re

lati

on y

�3x

.Sin

ce t

he

rela

tion

is

also

a f

un

ctio

n,w

e ca

n w

rite

th

e eq

uat

ion

in

fu

nct

ion

al n

otat

ion

as

f(x)

�3x

.

1.W

rite

an

equ

atio

n f

or t

he

fun

ctio

n i

n2.

Wri

te a

n e

quat

ion

for

th

e fu

nct

ion

in

fun

ctio

nal

not

atio

n.T

hen

com

plet

e fu

nct

ion

al n

otat

ion

.Th

en c

ompl

ete

the

tabl

e.th

e ta

ble.

f(x

) �

4x�

2f(

x)

��

3x�

4

3.W

rite

an

equ

atio

n i

n f

un

ctio

nal

not

atio

n.

4.W

rite

an

equ

atio

n i

n f

un

ctio

nal

not

atio

n.

f(x

) �

�x

�2

f(x

) �

2x�

2

x

y Ox

y

O

x�

2�

10

12

3

y10

74

1�

2�

5x

�1

01

23

4

y�

22

610

1418

Nu

mb

er o

f P

acka

ges

12

34

5

Nu

mb

er o

f C

Ds

36

912

15

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Wri

tin

g E

qu

atio

ns

fro

m P

atte

rns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Wri

tin

g E

qu

atio

ns

fro

m P

atte

rns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-H

ill25

7G

lenc

oe A

lgeb

ra 1

Lesson 4-8

Fin

d t

he

nex

t tw

o it

ems

for

each

pat

tern

.Th

en f

ind

th

e 19

th f

igu

re i

n t

he

pat

tern

.

1. 2. Fin

d t

he

nex

t th

ree

term

s in

eac

h s

equ

ence

.

3.1,

4,10

,19,

31,…

46,6

4,85

4.15

,14,

16,1

5,17

,16,

…18

,17,

19

5.29

,28,

26,2

3,19

,…14

,8,1

6.2,

3,2,

4,2,

5,…

2,6,

2

7.x,

x�

1,x

�2,

…x

�3,

x�

4,x

�5

8.y,

4y,9

y,16

y,…

25y,

36y,

49y

Wri

te a

n e

qu

atio

n i

n f

un

ctio

n n

otat

ion

for

eac

h r

elat

ion

.

9.10

.11

.

f(x

) �

�2x

f(x

) �

x�

2f(

x)

�1

�x

12.

13.

14.

f(x

) �

x�

6f(

x)

�5

�x

f(x

) �

2x�

1

x

y

O

x

y

Ox

y

O

x

y

Ox

y

Ox

y

O

;;

©G

lenc

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cGra

w-H

ill25

8G

lenc

oe A

lgeb

ra 1

1.G

ive

the

nex

t tw

o it

ems

for

the

patt

ern

.Th

en f

ind

the

21st

fig

ure

in

th

e pa

tter

n.

Fin

d t

he

nex

t th

ree

term

s in

eac

h s

equ

ence

2.�

5,�

2,�

3,0,

�1,

2,1,

4,…

3,6,

53.

0,1,

3,6,

10,1

5,…

21,2

8,36

4.0,

1,8,

27,…

64,1

25,2

165.

3,2,

4,3,

5,4,

…6,

5,7

6.a

�1,

a�

4,a

�9,

…7.

3d�

1,4d

�2,

5d�

3,…

a�

16,a

�25

,a�

366d

�4,

7d�

5,8d

�6

Wri

te a

n e

qu

atio

n i

n f

un

ctio

n n

otat

ion

for

eac

h r

elat

ion

.

8.9.

10.

f(x

) �

�x

f(x

) �

3x�

6f(

x)

�2x

�4

BIO

LOG

YF

or E

xerc

ises

11

and

12,

use

th

e fo

llow

ing

info

rmat

ion

.

Mal

e fi

refl

ies

flas

h in

var

ious

pat

tern

s to

sig

nal l

ocat

ion

and

perh

aps

to w

ard

off

pred

ator

s.D

iffe

rent

spe

cies

of

fire

flie

s ha

ve d

iffe

rent

fla

sh c

hara

cter

isti

cs,s

uch

as t

he in

tens

ity

of t

hefl

ash,

its

rate

,and

its

shap

e.T

he t

able

bel

ow s

how

s th

e ra

te a

t w

hich

a m

ale

fire

fly

is f

lash

ing.

11.W

rite

an

equ

atio

n i

n f

un

ctio

n n

otat

ion

for

th

e re

lati

on.

f(t)

�2t

,wh

ere

tis

th

eti

me

in s

eco

nd

s an

d f

(t)

is t

he

nu

mb

er o

f fl

ash

es

12.H

ow m

any

tim

es w

ill

the

fire

fly

flas

h i

n 2

0 se

con

ds?

40

13.G

EOM

ETRY

Th

e ta

ble

show

s th

e n

um

ber

of d

iago

nal

s th

at c

an b

e dr

awn

fro

m o

ne

vert

ex i

n a

pol

ygon

.Wri

te a

neq

uat

ion

in

fu

nct

ion

not

atio

n f

or t

he

rela

tion

an

d fi

nd

the

nu

mbe

r of

dia

gon

als

that

can

be

draw

n f

rom

on

e ve

rtex

in

a

12-s

ided

pol

ygon

.f(

s) �

s�

3,w

her

e s

is t

he

nu

mb

er

of

sid

es a

nd

f(s

) is

th

e n

um

ber

of

dia

go

nal

s;9

Sid

es3

45

6

Dia

go

nal

s0

12

3

Tim

e (s

eco

nd

s)1

23

45

Nu

mb

er o

f F

lash

es2

46

810

1 � 2

x

y O

x

y

O

x

y

O

;

Pra

ctic

e (

Ave

rag

e)

Wri

tin

g E

qu

atio

ns

fro

m P

atte

rns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csW

riti

ng

Eq

uat

ion

s F

rom

Pat

tern

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-8

4-8

©G

lenc

oe/M

cGra

w-H

ill25

9G

lenc

oe A

lgeb

ra 1

Lesson 4-8

Pre-

Act

ivit

yW

hy

is w

riti

ng

equ

atio

ns

from

pat

tern

s im

por

tan

t in

sci

ence

?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-8

at

the

top

of p

age

240

in y

our

text

book

.

•W

hat

is

mea

nt

by t

he

term

lin

ear

patt

ern

?T

he

po

ints

lin

e u

p a

lon

ga

stra

igh

t lin

e.

•D

escr

ibe

any

arit

hm

etic

seq

uen

ces

in t

he

data

.B

oth

th

e vo

lum

e o

fw

ater

an

d t

he

volu

me

of

ice

are

arit

hm

etic

seq

uen

ces.

Th

evo

lum

e o

f w

ater

has

a c

om

mo

n d

iffe

ren

ce o

f 11

an

d t

he

volu

me

of

ice

has

a c

om

mo

n d

iffe

ren

ce o

f 12

.

Rea

din

g t

he

Less

on

1.W

hat

is

mea

nt

by t

he

term

in

du

ctiv

e re

ason

ing?

mak

ing

a c

on

clu

sio

n b

ased

on

ap

atte

rn o

f ex

amp

les

2.F

or t

he

figu

res

belo

w,e

xpla

in w

hy

Fig

ure

5 d

oes

not

fol

low

th

e pa

tter

n.

If y

ou

fo

llow

th

e p

atte

rn o

f sh

adin

g t

he

top

tri

ang

le,t

hen

th

e b

ott

om

,th

en t

he

top

,an

d s

o o

n,F

igu

re 5

sh

ou

ld h

ave

the

top

tri

ang

le s

had

ed.

3.D

escr

ibe

the

step

s yo

u w

ould

use

to

fin

d th

e pa

tter

n i

n t

he

sequ

ence

1,5

,25,

125,

… .

Su

btr

act

succ

essi

ve t

erm

s to

fin

d o

ut

if it

is a

n a

rith

met

ic s

equ

ence

.It

isn

ot.

So

stu

dy

the

succ

essi

ve t

erm

s to

see

if m

ult

iplic

atio

n o

r d

ivis

ion

isu

sed

.Th

e p

atte

rn is

to

mu

ltip

ly e

ach

ter

m b

y 5

to g

et t

he

nex

t te

rm.

Hel

pin

g Y

ou

Rem

emb

er

4.W

hat

are

som

e ba

sic

thin

gs t

o re

mem

ber

wh

en y

ou a

re t

ryin

g to

dis

cove

r w

het

her

th

ere

is a

pat

tern

in

a s

equ

ence

of

nu

mbe

rs?

Sam

ple

an

swer

:L

oo

k to

see

wh

eth

ersu

cces

sive

ter

ms

incr

ease

or

dec

reas

e by

th

e sa

me

amo

un

t.If

no

t,lo

ok

to s

ee w

het

her

su

cces

sive

ter

ms

hav

e b

een

mu

ltip

lied

or

div

ided

by

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4.

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AB

C

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G

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F

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____

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4-8

4-8



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. C

A

D

C

C

B

C

A

C

C

D

B

{(0, �3), (1, �2),(2, �1), (3, 0)}

B

C

A

B

C

C

C

C

A

B

A

B

D

C

B

A

B

D

A

C

Chapter 4 Assessment Answer Key Form 1 Form 2APage 261 Page 262 Page 263

(continued on the next page)See Students’ Work


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: �4m2 � 10m � 6

B

D

A

A

C

C

A

B

B

D

C

A

D

C

A

D

B

B

C

C

{(6, 2), (4, 0), (12, �4)}

B

A

A

D

B

B

C

D

Chapter 4 Assessment Answer KeyForm 2A (continued) Form 2BPage 264 Page 265 Page 266

An

swer

s


1.

2.

3.4–6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

(3, �1), (�1, �1), (3, 3)

y

xO

y

xO

(3, 4.5)

(3, 0)

(2, 0)

(2, 3)(0, 3)

(0, 4.5)

3 in. by 4.5 in.

Sample answer:f(d ) � 0.04d

an

nO 1

51015

2 4 5

�10�15

an � �7n � 19;

36, 43, 50

yes; 3

25v2 � 15v � 2

8

not a function

function

y

xO

yes; 2x � 3y � �4

no

y

xO

{(�1, �5), (3, �3)}

Domain:{�5, �2, 1, 3, 7};Range: {0, 2, 3, 4}

X

1�2�5

7

43023

Y

translation

reflection

y

xO

B

AC

(2, �1); IV(�2, �3); III

(3, 1); I

Chapter 4 Assessment Answer KeyForm 2CPage 267 Page 268

{(�3, 2), (�3, �1),(0, 1), (4, 3)}; {(2, �3),(�1, �3), (1, 0), (3, 4)}

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

1.

2.

3.

4–6.

7.

8.

9.

Domain:{�2, �1, 1, 4, 8};Range: {�5, 0, 3, 7}

10.

11.

12.

13.

14.15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:y

xO

(�2, 3) (4, 3)

(�2, �2) (4, �2)

(�2, �2)

y

xO

(3, 5)

(3, 0)

(1.2, 0)

(1.2, 2)(0, 2)

(0, 5)

1.2 in. by 2 in.

f(d) � 0.05d

an

nO 1

5101520253035

2 3 4 5 6 7

an � 9n � 6;

�13, �10, �7

yes; 4

t2 � 4t � 7

39

function

not a function

y

xO

yes; 4x � 2y � 0

no

y

xO

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

{(�3, �16), (5, 8)}

{(�3, 3), (�2, 0),(0, �1), (2, 1)}; {(3, �3),(0, �2), (�1, 0), (1, 2)}

X

8�2

14

70

�53�1

Y

dilation

rotation

y

xOA

C

B

(�2, �4); III(�1, 2); II

(2, 3); I

An

swer

s


Chapter 4 Assessment Answer KeyForm 2DPage 269 Page 270

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

about 12 yrs

about 10�12

� yrs

V(t) � 2000 � 190t

an

n1

2

2 3 4 5 6 7�2�4�6�8

�10�12�14

an � 4n � 19

y � �1

yes; �12

�

�24p2 � 16p

function

not a function

y

xO

yes; 3x � 7y � �6

y

xO 2

5

{(�4, 18), (3, �6.5),(5, �13.5), (6, �17)}

{(�3, �5.4), (15, 9)}

Domain:{�2, �1, 3, 4, 6};Range: {�2, �1, 3}

X

4�2�1

6

�1

3

�23

Y

y

xO

N'

M'

P'

M

N

P

M�(1, 0 ), N�(�1, 2),P�(�2, �3);

dilation

y

xO

(–3, 1.5)

, –432( )

(0, �8)


Chapter 4 Assessment Answer KeyForm 3Page 271 Page 272

{(�3, 4), (�2, 1), (�1, �4),(1, 1), (3, 4), (3, �3)};{(4, �3), (1, �2), (�4, �1),(1, 1), (4, 3), (�3, 3)}

�(�4, �36), ��1, �31�12

��,(2, �27), �3, �25�

12

��,(6, �21)�; �(�36, �4),

��31�12

�, �1�, (�27, 2),

��25�12

�, 3�, (�21, 6)�

Chapter 4 Assessment Answer KeyPage 273, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts of pointson the coordinate plane, transformations, relations,functions, linear equations, arithmetic sequences, andpatterns.

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

• Shows an understanding of the concepts of points on thecoordinate plane, transformations, relations, functions,linear equations, arithmetic sequences, and patterns.

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

• Shows an understanding of most of the concepts of pointson the coordinate plane, transformations, relations,functions, linear equations, arithmetic sequences, andpatterns.

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

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

the final computation.• Graphs may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofpoints on the coordinate plane, transformations, relations,functions, linear equations, arithmetic sequences, andpatterns.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs are inaccurate or inappropriate.• 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

An

swer

s


Chapter 4 Assessment Answer KeyPage 273, Open-Ended Assessment

Sample Answers

1a. Sample answer: (�1, �1); quadrant III1b. Sample answer: (1, �1);

1c. The student should explain thatknowing in which quadrant a point is located determines whether the pointis to the right or left of the y-axis and above or below the x-axis,and thus, positive or negative.

2a. The four transformations are reflection,translation, dilation, and rotation.Depending on which two were chosen,the students may discuss whether theimage and preimage are the same size,whether a line or point was used toassist the movement of the figure,whether the movement preserved theorientation or reversed the orientationof the figure.

2b. The student should discuss thecharacteristics of the twotransformations that are the same. Thestudent should discuss whether thetransformations preserve size, direction,and orientation.

3a. Sample answer:{(�1, �3), (0, �1), (1, 4), (2, 5)};{(�3, �1), (�1, 0), (4, 1), (5, 2)}

3b. Sample answer:

3c. The student should identify in theirrelation where they used the samedomain element with two or moredifferent range elements.

4. Sample answer: x � y � 1; To graph theequation find two solutions for theequation, plot the two points associatedwith the two solutions, and draw astraight line through the two points.

5. The value in the range that correspondsto the element 2 in the domain isrepresented by f(2). The value f(2) isfound by substituting 2 for x in theequation.

6a.Sample answer: 2, 5, 8, 11, … ;The common difference is 3; a6 � 17

6b.Sample answer: 5, 3, 8, 6, 11, 9, 14, … ;The pattern is to subtract 2 from thefirst term to find the second term, thenadd 5 to the second term to find thethird term. Repeat the process ofsubtracting 2 then adding 5.

6c. The sequence 1, 1, 1, 1, … is a set ofnumbers whose difference betweensuccessive terms is the constant number0. Thus, this sequence is an arithmeticsequence by the definition.

X

�3�1

45

�1012

Y

y

xO

y

xO

(�1, �1) (1, �1)

In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.

x y

�1 �3

0 �3

0 �1

1 4

2 5


1. axes

2. origin

3. quadrants

4. Transformations

5. translation

6. inverse

7. linear equation

8. function

9. vertical line test

10. the planecontaining the x- and y-axes

11. a diagram that usesarrows to show howthe elements of thedomain of a relationare paired withelements of therange

1.

2.

3.

4.

5.

Quiz (Lessons 4–3 and 4–4)

Page 275

1.

Domain: {�4, 1, 2, 3} Range: {3, 4, 5, 6, 8}

2.

3.

4.

5.

1.

2.

3.

4.

5.

Quiz (Lessons 4–7 and 4–8)

Page 276

1.

2.

3.

4.

5. f(x) � 2x � 2

5, 1, �15

�

25

21, 25, 29

no

B

30

function

y

xO

yes; 2x � y � 1

{(�3, �2), (�2, 0),(�1, 2), (0, 4), (1, 6)}

{(�2, 0), (�1, 1.5),(0, 3), (2, 6), (3, 7.5)}

{(�2, �11), (2, 5)}

{(�5, 2), (�3, �2),(1, 1), (4, �3};{(2, �5), (�2, �3),(1, 1), (�3, 4)}

X

56843

3�4

21

Y

y

xO

A'

B'

C'

A

B

C

A�(�1, 2), B�(�3, 3),C� (�3, �1)

dilation

rotation

(2, �4); IV

(�3, 0); none

Chapter 4 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 4–1 and 4–2) Quiz (Lessons 4–5 and 4–6)

Page 274 Page 275 Page 276

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14. y

xO

{(�3, �17), (�1, �9),(1, �1), (2, 3), (5, 15)}

y

xAA�

B

B�

C

C�

A�(�1, 0), B�(2, �4),C�(3, �1)

not a function

{(�4, 3), (�4, �1),(�2, 2), (0, �1), (2, 3),(3, �3), (4, 1)}; domain:{�4, �2, 0, 2, 3, 4};range: {�3, �1, 1, 2, 3}

(�4, 3); II

no

�2

$12.75

x divided by 4 minus yequals negative 2 times

the quotient of x and y.

6.16

8m � 9n

�45

�

{(�3, 1), (0, 2),(3, 3), (6, 4)}:

{(�2, �7), (�1, �5),(0, �3), (2, 1), (3, 3)}

D�(1, 1), E�(4, 1), F�(2, 4);

X

�2

�1

0

�3357

Y

B

D

A

B

C

Chapter 4 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 277 Page 278

Stem Leaf0 5 81 0 2 2 2 2 2 52 0

1�0 � 10

See students’ graphs.


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17. DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

7 7

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 4

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2 8

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

3 7

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 4 Assessment Answer KeyStandardized Test Practice

Page 279 Page 280

An

swer

s

Chapter 4 Resource Masters - Morgan Park High School

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Transcript of Chapter 4 Resource Masters - Morgan Park High School