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Chapter 3Resource Masters

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ISBN: 0-07-869130-3 Advanced Mathematical ConceptsChapter 3 Resource Masters

1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04

© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

Vocabulary Builder . . . . . . . . . . . . . . . . . vii-x

Lesson 3-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 87Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 89




Lesson 3-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 99Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Lesson 3-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 102Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 104



Chapter 3 AssessmentChapter 3 Test, Form 1A . . . . . . . . . . . . 111-112Chapter 3 Test, Form 1B . . . . . . . . . . . . 113-114Chapter 3 Test, Form 1C . . . . . . . . . . . . 115-116Chapter 3 Test, Form 2A . . . . . . . . . . . . 117-118Chapter 3 Test, Form 2B . . . . . . . . . . . . 119-120Chapter 3 Test, Form 2C . . . . . . . . . . . . 121-122Chapter 3 Extended Response

Assessment . . . . . . . . . . . . . . . . . . . . . . . 123Chapter 3 Mid-Chapter Test . . . . . . . . . . . . . 124Chapter 3 Quizzes A & B . . . . . . . . . . . . . . . 125Chapter 3 Quizzes C & D. . . . . . . . . . . . . . . 126Chapter 3 SAT and ACT Practice . . . . . 127-128Chapter 3 Cumulative Review . . . . . . . . . . . 129

SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1

SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A17

Contents

© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

A Teacher’s Guide to Using theChapter 3 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 3 Resource Masters include the corematerials needed for Chapter 3. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.

All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.

Vocabulary Builder Pages vii-x include a student study tool that presents the key vocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.

When to Use Give these pages to studentsbefore beginning Lesson 3-1. Remind them toadd definitions and examples as they completeeach lesson.

Study Guide There is one Study Guide master for each lesson.

When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can alsobe used in conjunction with the Student Editionas an instructional tool for those students whohave been absent.

Practice There is one master for each lesson.These problems more closely follow the structure of the Practice section of the StudentEdition exercises. These exercises are ofaverage difficulty.

When to Use These provide additional practice options or may be used as homeworkfor second day teaching of the lesson.

Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for usewith all levels of students.

When to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.

© Glencoe/McGraw-Hill v Advanced Mathematical Concepts

Assessment Options

The assessment section of the Chapter 3Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessments

Chapter Tests• Forms 1A, 1B, and 1C Form 1 tests contain

multiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with average-level students, and Form 1C is intended foruse with basic-level students. These testsare similar in format to offer comparabletesting situations.

• Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.

All of the above tests include a challengingBonus question.

• The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoringrubric is included for evaluation guidelines.Sample answers are provided for assessment.

Intermediate Assessment• A Mid-Chapter Test provides an option to

assess the first half of the chapter. It iscomposed of free-response questions.

• Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter.

Continuing Assessment• The SAT and ACT Practice offers

continuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitative-comparison, and grid-in questions. Bubble-in and grid-in answer sections are providedon the master.

• The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.

Answers• Page A1 is an answer sheet for the SAT and

ACT Practice questions that appear in theStudent Edition on page 203. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they mayencounter in test taking.

• The answers for the lesson-by-lesson masters are provided as reduced pages withanswers appearing in red.

• Full-size answer keys are provided for theassessment options in this booklet.

primarily skillsprimarily conceptsprimarily applications

BASIC AVERAGE ADVANCED

Study Guide

Vocabulary Builder

Parent and Student Study Guide (online)

Practice

Enrichment

4

5

3

2

Five Different Options to Meet the Needs of Every Student in a Variety of Ways

1

© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts

Chapter 3 Leveled Worksheets

Glencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow.

• Study Guide masters provide worked-out examples as well as practiceproblems.

• Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords.

• Practice masters provide average-level problems for students who are moving at a regular pace.

• Enrichment masters offer students the opportunity to extend theirlearning.

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.

© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

Vocabulary Term Foundon Page Definition/Description/Example

absolute maximum

absolute minimum

asymptotes

constant function

constant of variation

continuous

critical point

decreasing function

direct variation

discontinuous

(continued on the next page)

Reading to Learn MathematicsVocabulary Builder

NAME _____________________________ DATE _______________ PERIOD ________Chapter

3

© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts


end behavior

even function

everywhere discontinuous

extremum

horizontal asymptote

horizontal line test

image point

increasing function

infinite discontinuity

inverse function

inverse process


Reading to Learn MathematicsVocabulary Builder (continued)


3


NAME _____________________________ DATE _______________ PERIOD ________

3


inversely proportional

inverse relations

inverse variation

jump discontinuity

line symmetry

maximum

minimum

monotonicity

odd function

parent graph

point discontinuity


© Glencoe/McGraw-Hill ix Advanced Mathematical Concepts


3

© Glencoe/McGraw-Hill x Advanced Mathematical Concepts


point of inflection

point symmetry

rational function

relative extremum

relative maximum

relative minimum

slant asymptote

symmetry with respect to the origin

vertical asymptote



3

© Glencoe/McGraw-Hill 87 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

3-1

Symmetry and Coordinate GraphsOne type of symmetry a graph may have is point symmetry. Acommon point of symmetry is the origin. Another type ofsymmetry is line symmetry. Some common lines of symmetryare the x-axis, the y-axis, and the lines y � x and y � �x.

point x-axis y-axis y � x y � �x

Example 1 Determine whether ƒ(x) � x3 is symmetric withrespect to the origin.

If ƒ(�x) � �ƒ(x), the graph has point symmetry.

Find ƒ(�x). Find �ƒ(x).ƒ(�x) � (�x)3 �ƒ(x) � �x3

ƒ(�x) � �x3

The graph of ƒ(x) � x3 is symmetric with respectto the origin because ƒ(�x) � �ƒ(x).

Example 2 Determine whether the graph of x2 � 2 � y2 issymmetric with respect to the x-axis, the y-axis,the line y � x, the line y � �x, or none of these.

Substituting (a, b) into the equation yields a2 � 2 � b2. Check to see if each test produces an equation equivalent to a2 � 2 � b2.

x-axis a2 � 2 � (�b)2 Substitute (a, �b) into the equation.a2 � 2 � b2 Equivalent to a2 � 2 � b2

y-axis (�a)2 � 2 � b2 Substitute (�a, b) into the equation.a2 � 2 � b2 Equivalent to a2 � 2 � b2

y � x (b)2 � 2 � (a)2 Substitute (b, a) into the equation.a2 � 2 � b2 Not equivalent to a2 � 2 � b2

y � �x (�b)2 � 2 � (�a)2 Substitute (�b, �a) into the equation.b2 � 2 � a2 Simplify.a2 � 2 � b2 Not equivalent to a2 � 2 � b2

Therefore, the graph of x2 � 2 � y2 is symmetric with respect to the x-axis and the y-axis.


Symmetry and Coordinate Graphs

Determine whether the graph of each function is symmetric with respect to the origin.

1. ƒ(x) � ��x12� 2. ƒ(x) � x5 � 2

3. ƒ(x) � x3 � 4x 4. ƒ(x) � �3x�

2

x�

Determine whether the graph of each equation is symmetric with respect to the x-axis, the y-axis, the line y � x, the line y � �x, or none of these.

5. x � y � 6 6. x2 � y � 2

7. xy � 3 8. x3 � y2 � 4

9. y � 4x 10. y � x2 � 1

11. Is ƒ(x) � �x� an even function, an odd function, or neither?

Refer to the graph at the right for Exercises 12 and 13.

12. Complete the graph so that it is the graph of an odd function.

13. Complete the graph so that it is the graph of an even function.

14. Geometry Cameron told her friend Juanita that the graph of � y� � 6 � �3x� has the shape of a geometric f igure. Determine whether the graph of � y� � 6 � �3x� is symmetric with respect to the x-axis, the y-axis, both, or neither. Then make a sketch of the graph.Is Cameron correct?

PracticeNAME _____________________________ DATE _______________ PERIOD ________

3-1

x

f x( )

O

(3, 3)

(4, 2)(1, 1)


EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

3-1

Symmetry in Three-Dimensional FiguresA solid figure that can be superimposed, point forpoint, on its mirror image has a plane of symmetry.A symmetrical solid object may have a finite orinfinite number of planes of symmetry. The chair inthe illustration at the right has just one plane ofsymmetry; the doughnut has infinitely manyplanes of symmetry, three of which are shown.

Determine the number of planes of symmetry for each object anddescribe the planes.

1. a brick

2. a tennis ball

3. a soup can

4. a square pyramid

5. a cube

Solid figures can also have rotational symmetry. For example, theaxis drawn through the cube in the illustration is a fourfold axis ofsymmetry because the cube can be rotated about this axis into fourdifferent positions that are exactly alike.6. How many four-fold axes of symmetry does a cube have?

Use a die to help you locate them.

7. A cube has 6 two-fold axes of symmetry. In the space at the right,draw one of these axes.



3-2

Families of GraphsA parent graph is a basic graph that is transformed to create othermembers in a family of graphs. The transformed graph may appear ina different location, but it will resemble the parent graph.

A reflection flips a graph over a line called the axis of symmetry.A translation moves a graph vertically or horizontally.A dilation expands or compresses a graph vertically orhorizontally.

Example 1 Describe how the graphs of ƒ(x) � �x� andg(x) � ��x� � 1 are related.

The graph of g(x) is a reflection of the graph ofƒ(x) over the y-axis and then translated down 1 unit.

Example 2 Use the graph of the given parent functionto sketch the graph of each related function.

a. ƒ(x) � x3; y � x3 � 2

When 2 is added to the parent function, the graph of the parent function moves up 2 units.

b. ƒ(x) � �x�; y � 3�x�

The parent function is expanded vertically by afactor of 3, so the vertical distance between thesteps is 3 units.

c. ƒ(x) � x; y � 0.5x

When x is multiplied by a constant greaterthan 0 but less than 1, the graph compressesvertically, in this case, by a factor of 0.5.

d. ƒ(x) � x2; y � x2 � 4

The parent function is translated down 4 unitsand then any portion of the graph below the x-axis is reflected so that it is above the x-axis.



Families of Graphs

Describe how the graphs of ƒ(x) and g(x) are related.1. ƒ(x) � x2 and g(x) � (x � 3)2 � 1 2. ƒ(x) � x and g(x) � �2x

Use the graph of the given parent function to describe the graph of each related function.

3. ƒ(x) � x3 4. ƒ(x) � �x�a. y � 2x3 a. y � �x� �� 3� � 1

b. y � �0.5(x � 2)3 b. y � ��x� � 2

c. y � (x � 1)3 c. y � �0�.2�5�x� � 4

Sketch the graph of each function.

5. ƒ(x) � �(x � 1)2 � 1 6. ƒ(x) � 2x � 2 � 3

7. Consumer Costs During her free time, Jill baby-sits the neighborhood children. She charges $4.50 for each whole hour or any fraction of an hour. Write and graph a function that shows the cost of x hours of baby-sitting.

3-2



3-2

A graph G′ is isomorphic to a graph G if the following conditions hold.

1. G and G′ have the same number of vertices and edges.2. The degree of each vertex in G is the same as the degree of each corresponding vertex in G′.3. If two vertices in G are joined by k (k � 0) edges, then the two corresponding vertices in G′ are

also joined by k edges.

Isomorphic GraphsA graph G is a collection of points in which a pair of points, called vertices, are connected by a set of segments or arcs, called edges. The degree of vertex C, denoteddeg (C), is the number of edges connected to that vertex. We say two graphs are isomorphic if they have the same structure. The definition below will help youdetermine whether two graphs are isomorphic.

Example In the graphs below HIJKLMN ↔ TUVWXYZ.Determine whether the graphs are isomorphic.

Number of vertices in G: 7 Number of vertices in G′: 7Number of edges in G: 10 Number of edges in G′: 10deg (H): 3 deg (I): 3 deg (T): 1 deg (U): 3deg (J): 3 deg (K): 3 deg (V): 3 deg (W): 3deg (L): 4 deg (M): 3 deg (X): 4 deg (Y): 3deg (N): 1 deg (Z): 3

Since there are the same number of vertices and the same number of edgesand there are five vertices of degree 3, one vertex of degree 4, and onevertex of degree 1 in both graphs, we can assume they are isomorphic.

Each graph in Row A is isomorphic to one graph in Row B. Match the graphs that are isomorphic.Row A 1. 2. 3.

Row B a. b. c.



Graphs of Nonlinear InequalitiesGraphing an inequality in two variables identifies all orderedpairs that satisfy the inequality. The first step in graphingnonlinear inequalities is graphing the boundary.

Example 1 Graph y � �x� �� 3� � 2.

The boundary of the inequality is the graph of y ��x� �� 3� � 2.To graph the boundary curve, start with the parent graph y ��x�. Analyze the boundary equation to determine how theboundary relates to the parent graph.

y � �x� �� 3� � 2 ↑ ↑

move 3 units right move 2 units up

Since the boundary is not included in the inequality, thegraph is drawn as a dashed curve.

The inequality states that the y-values of the solution areless than the y-values on the graph of y ��x� �� 3� � 2.Therefore, for a particular value of x, all of the points in theplane that lie below the curve have y-values less than �x� �� 3� � 2. This portion of the graph should be shaded.

To verify numerically, test a point not on the boundary.

y � �x� �� 3� � 20 ?

� �4� �� 3� � 2 Replace (x, y) with (4, 0).0 � 3 ✓ True

Since (4, 0) satisfies the inequality, the correct region isshaded.

Example 2 Solve x � 3 � 2 � 7.

Two cases must be solved. In one case, x � 3 is negative, andin the other, x � 3 is positive.

Case 1 If a � 0, then a � �a. Case 2 If a � 0, then a � a.�(x � 3) � 2 � 7 x � 3 � 2 � 7

�x � 3 � 2 � 7 x � 5 � 7�x � 6 x � 12

x � �6

The solution set is {xx � �6 or x � 12}.

3-3


Graphs of Nonlinear Inequalities

Determine whether the ordered pair is a solution for the given inequality. Write yes or no.

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

Graph each inequality.

4. y � 2 � x � 1 � 5. y � 2(x � 1)2

6. y � �x� �� 2� � 1 7. y � (x � 3)3

Solve each inequality.

8. � 4x � 10 � � 6 9. � x � 5 � � 2 � 6 10. � 2x � 2 � � 1 � 7

11. Measurement Instructions for building a birdhouse warn that the platform, which ideally measures 14.75 cm2, should not vary in size by more than 0.30 cm2. If it does, the preconstructed roof for the birdhouse will not f it properly.a. Write an absolute value inequality that represents the range of

possible sizes for the platform. Then solve for x to f ind the range.

b. Dena cut a board 14.42 cm2. Does the platform that Dena cut f it within the acceptable range?


3-3



Some Parametric GraphsFor some curves, the coordinates x and y can be written as functions of a third variable. The conditions determining the curve are given by two equations, rather than by a single equation in x and y. The thirdvariable is called a parameter, and the two equations are called para-metric equations of the curve.

For the curves you will graph on this page, the parameter is t and the parametric equations of each curve are in the form x � f (t) and y � g(t).

Example Graph the curve associated with the parametricequations x � 48t and y � 64t � 16t2.Choose values for t and make a table showing the values of all three variables. Then graph thex- and y-values.

t x y

–1 –48 –800 0 00.5 24 281 48 482 96 643 144 484 192 0

Graph each curve.

1. x � 3t, y � 2. x � t2 � 1, y � t3 � 112�t

3-3


Inverse Functions and RelationsTwo relations are inverse relations if and only if one relationcontains the element (b, a) whenever the other relationcontains the element (a, b). If the inverse of the function ƒ(x)is also a function, then the inverse is denoted by ƒ�1(x).

Example 1 Graph ƒ(x) � �14�x3 � 3 and its inverse.

To graph the function, let y � ƒ(x). To graphƒ�1(x), interchange the x- and y-coordinates ofthe ordered pairs of the function.

You can use the horizontal line test to determine if theinverse of a relation will be a function. If every horizontal lineintersects the graph of the relation in at most one point, thenthe inverse of the relation is a function.

You can find the inverse of a relation algebraically. First, lety � ƒ(x). Then interchange x and y. Finally, solve the resulting equation for y.

Example 2 Determine if the inverse of ƒ(x) � (x � 1)2 � 2 is a function. Then find the inverse.

Since the line y � 4 intersects the graph of ƒ(x) at more than one point, the function fails thehorizontal line test. Thus, the inverse of ƒ(x) is not a function.

y � (x � 1)2 � 2 Let y � ƒ(x).x � ( y � 1)2 � 2 Interchange x and y.

x � 2 � ( y � 1)2 Isolate the expression containing y.�x� �� 2� � y � 1 Take the square root of each side.

y � 1 �x� �� 2� Solve for y.


3-4

ƒ (x) � �14

�x3 � 3

x y�3 �9.75�2 �5�1 �3.25

0 �31 �2.752 �13 3.75

ƒ�1(x)x y

�9.75 �3�5 �2�3.25 �1�3 0�2.75 1�1 2

3.75 3



Inverse Functions and Relations

Graph each function and its inverse.

1. ƒ(x) � (x � 1)3 � 1 2. ƒ(x) � 3�x � � 2

Find the inverse of ƒ(x). Then state whether the inverse is also a function.

3. ƒ(x) � �4x2 � 1 4. ƒ(x) � �3x� �� 1� 5. ƒ(x) � �

(x �

43)2

�

Graph each equation using the graph of the given parent function.

6. y � ��x� �� 3� � 1, p(x) � x2 7. y � 2 � �5

x� �� 2�, p(x) � x5

8. Fire Fighting Airplanes are often used to drop water on forest f ires in an effort tostop the spread of the fire. The time t it takes the water to travel from height h tothe ground can be derived from the equation h � �12�gt2 where g is the accelerationdue to gravity (32 feet/second2).

a. Write an equation that will give time as a function of height.

b. Suppose a plane drops water from a height of 1024 feet. How many seconds will ittake for the water to hit the ground?

3-4



3-4

__ __17 28

__ __ __ __2 29 6 27

__ __ __31 33 14__ __ __20 11 34__ __36 7__ __ __ __ __24 16 1910 4__18

__ __ __ __35 3 21 8

__ __ __ __13 9 22 5__ __ __ __ __ __ __ __26 32 30 23 25 12 15 1

1

14

25

2 3

16

4

17

28

5

18

29

6 7

19

26 30

20

8 9

21

33

13 15

27 3231

10

22

34

11

23

35

12

24

36

An Inverse AcrosticThe puzzle on this page is called an acrostic. To solve the puzzle,work back and forth between the clues and the puzzle box. You mayneed a math dictionary to help with some of the clues.

1. If a relation contains the element (e, v), then theinverse of the relation must contain the element( __, __ ).

2. The inverse of the function 2x is found by computing __ of x.

3. The first letter and the last two letters of themeaning of the symbol f –1 are __ .

4. This is the product of a number and its multiplicative inverse.

5. If the second coordinate of the inverse of (x, f (x )) isy, then the first coordinate is read “__ of __”.

6. The inverse ratio of two numbers is the __ of thereciprocals of the numbers.

7. If is a binary operation on set S andx e � e x � x for all x in S, then an identity element for the operation is __.

8. To solve a matrix equation, multiply each side ofthe matrix equation on the __ by the inversematrix.

9. Two variables are inversely proportional __ theirproduct is constant.

10. The graph of the inverse of a linear function is a __ line.

From President Franklin D. Roosevelt’s inaugural address during the Great Depression; delivered March 4, 1933.



3-5

Continuity and End Behavior

Functions can be continuous or discontinuous. Graphs thatare discontinuous can exhibit infinite discontinuity, jumpdiscontinuity, or point discontinuity.

Example 1 Determine whether each function is continuousat the given x-value. Justify your answer usingthe continuity test.

a. ƒ(x) � 2x� 3; x � 2(1) The function is defined at x � 2;

ƒ(2) � 7.(2) The tables below show that y

approaches 7 as x approaches 2 from the left and that y approaches 7 as x approaches 2 from the right.

(3) Since the y-values approach 7 as x approaches 2 from both sides and ƒ(2) � 7, the function is continuous at x � 2.

The end behavior of a function describes what the y�values doasxbecomes greater and greater. In general, the end behavior ofany polynomial function can be modeled by the function made upsolely of the term with the highest power of x and its coefficient.

Example 2 Describe the end behavior of p(x) � �x5 � 2x3 � 4.

Determine ƒ(x) � anxn where xn is the term in p(x) withthe highest power of x and an is its coefficient.ƒ(x) � �x5 xn � x5 an � �1

Thus, by using the table on page 163 of your text, youcan see that when an is negative and n is odd, the endbehavior can be stated as p(x) → �∞ as x → ∞ and p(x) → ∞ as x → �∞.

b. ƒ(x) � �x22�x

1�; x � 1

Start with the first condition in the continuity test. The function is notdefined at x � 1 because substituting 1for x results in a denominator of zero. Sothe function is discontinuous at x �1.

c. ƒ(x) � �This function fails the second part of thecontinuity test because the values ofƒ(x) approach 1 as x approaches 2 fromthe left, but the values of ƒ(x) approach5 as x approaches 2 from the right.

x � 2x � 2

ifif

2x � 1x � 1

x y � ƒ(x)

2.1 7.2

2.01 7.02

2.001 7.002

x y � ƒ(x)

1.9 6.8

1.99 6.98

1.999 6.998

A function is continuous at x � c if it satisfies the following three conditions.

(1) the function is defined at c; in other words, ƒ(c) exists;

(2) the function approaches the same y-value to the left and right of x � c; and

(3) the y-value that the function approaches from each side is ƒ(c).


Continuity and End Behavior

Determine whether each function is continuous at the given x-value. Justify your answer using the continuity test.

1. y � �32x2�; x � �1 2. y � �x

2 �2x � 4� ; x � 1

3. y � x3 � 2x � 2; x � 1 4. y � �xx��

24� ; x � �4

Describe the end behavior of each function.

5. y � 2x5 � 4x 6. y � �2x6 � 4x4 � 2x � 1

7. y � x4 � 2x3 � x 8. y � �4x3 � 5

Given the graph of the function, determine the interval(s) for which the function is increasing and the interval(s) for which the function is decreasing.

9.

10. Electronics Ohm’s Law gives the relationship between resistance R, voltage

E, and current I in a circuit as R � �EI� . If the voltage remains constant but the current keeps increasing in the circuit, what happens to the resistance?


3-5


NAME _____________________________ DATE _______________ PERIOD ________

Reading MathematicsThe following selection gives a definition of a continuous function as it might be defined in a college-level mathematics textbook. Noticethat the writer begins by explaining the notation to be used for various types of intervals. It is a common practice for college authors to explain their notation, since, although a great deal of the notation is standard, each author usually chooses the notation he or she wishes to use.

Use the selection above to answer these questions.

1. What happens to the four inequalities in the first paragraph when a � b?

2. What happens to the four intervals in the first paragraph when a � b?

3. What mathematical term makes sense in this sentence?If f(x) is not ____?__ at x0, it is said to be discontinuous at x0.

4. What notation is used in the selection to express the fact that a number x is contained in the interval I?

5. In the space at the right, sketch the graph of the function f(x) defined as follows:

f(x) �� if x � �0, �

1, if x � � , 16. Is the function given in Exercise 5 continuous on the

interval [0, 1]? If not, where is the function discontinuous?

1�2

1�2

1�2

Enrichment3-5

Throughout this book, the set S, called the domain of definition of a function, will usually be an interval. An interval is a set of numbers satisfying one of the four inequalities a � x � b,a � x � b, a � x � b, or a � x � b. In these inequalities, a � b. The usual notations for theintervals corresponding to the four inequalities are, respectively, (a, b), [a, b), (a, b], and [a, b].

An interval of the form (a, b) is called open, an interval of the form [a, b) or (a, b] is called half-open or half-closed, and an interval of the form[a, b] is called closed.

Suppose I is an interval that is either open, closed, or half-open. Suppose ƒ(x) is a functiondefined on I and x0 is a point in I. We say that the function f (x) is continuous at the point x0 ifthe quantity |ƒ(x) � ƒ(x0)| becomes small as x � I approaches x0.


Critical Points and ExtremaCritical points are points on a graph at which a line drawntangent to the curve is horizontal or vertical. A critical point may bea maximum, a minimum, or a point of inflection. A point ofinflection is a point where the graph changes its curvature. Graphscan have an absolute maximum, an absolute minimum, arelative maximum, or a relative minimum. The general term formaximum or minimum is extremum (plural, extrema).Example 1 Locate the extrema for the graph of y � ƒ(x).

Name and classify the extrema of the function.

a.

The function has an absolute maximum at (0, 2). The absolute maximum is the greatest value that a function assumes over its domain.

b.

The function has an absolute minimum at (�1, 0). The absolute minimum is the least value that a function assumes over its domain.

By testing points on both sides of a critical point, you candetermine whether the critical point is a relative maximum, arelative minimum, or a point of inflection.

Example 2 The function ƒ(x) � 2x6 � 2x4 � 9x2 has a criticalpoint at x � 0. Determine whether the criticalpoint is the location of a maximum, a minimum,or a point of inflection.

Because 0 is greater than both ƒ(x � 0.1) and ƒ(x � 0.1), x � 0 is the location of a relative maximum.


3-6

c.

The relative maximum and minimummay not be the greatest and the least y-value for the domain, respectively, butthey are the greatest and least y-value onsome interval of the domain. The functionhas a relative maximum at (�2, �3) and arelative minimum at (0, �5). Because the graph indicates that the functionincreases or decreases without bound as x increases or decreases, there is neitheran absolute maximum nor an absolute minimum.

x x � 0.1 x � 0.1 ƒ( x � 0.1) ƒ( x) ƒ( x � 0.1) Type ofCritical Point

0 �0.1 0.1 �0.0899 0 �0.0899 maximum


Practice

NAME _____________________________ DATE _______________ PERIOD ________

Critical Points and Extrema

Locate the extrema for the graph of y � ƒ(x). Name and classify the extrema of the function.

1. 2.

3. 4.

Determine whether the given critical point is the location of a maximum, a minimum, or a point of inflection.

5. y � x2 � 6x � 1, x � 3 6. y � x2 � 2x � 6, x � 1 7. y � x4 � 3x2 � 5, x � 0

8. y � x5 � 2x3 � 2x2, x � 0 9. y � x3 � x2 � x, x � �1 10. y � 2x3 � 4, x � 0

11. Physics Suppose that during an experiment you launch a toy rocket straight upward from a heightof 6 inches with an initial velocity of 32 feet per second. The height at any time t can be modeled by the function s(t) � �16t2 � 32t � 0.5 where s(t) is measured in feet and t is measured in seconds.Graph the function to f ind the maximum height obtained by the rocket before it begins to fall.

3-6


"Unreal" EquationsThere are some equations that cannot be graphed on the real-number coordinate system. One example is the equationx2 – 2x � 2y2 � 8y � 14 � 0. Completing the squares in x and ygives the equation (x – 1)2 � 2(y � 2)2 � –5.

For any real numbers, x and y, the values of (x – 1)2 and 2(y � 2)2

are nonnegative. So, their sum cannot be –5. Thus, no real values of x and y satisfy the equation; only imaginary values can be solutions.

Determine whether each equation can be graphed on the real-number plane. Write yes or no.

1. (x � 3)2 � (y � 2)2 � �4 2. x2 � 3x � y2 � 4y � �7

3. (x � 2)2 � y2 � 6y � 8 � 0 4. x2 � 16 � 0

5. x4 � 4y2 � 4 � 0 6. x2 � 4y2 � 4xy � 16 � 0

In Exercises 7 and 8, for what values of k :

a. will the solutions of the equation be imaginary?

b. will the graph be a point?

c. will the graph be a curve?

d. Choose a value of k for which the graph is a curve and sketch the curve on the axes provided.

7. x2 � 4x � y2 � 8y � k � 0 8. x2 � 4x � y2 � 6y � k � 0

Enrichment

NAME _____________________________ DATE _______________ PERIOD ________

3-6


Study Guide

NAME _____________________________ DATE _______________ PERIOD ________

Graphs of Rational Functions

Example 1 Determine the asymptotes for the graph of ƒ(x) � �2x

x��

31�.

Since ƒ(�3) is undefined, there may be a vertical asymptote at x � �3. To verify that x � �3 is a vertical asymptote, check to see that ƒ(x) → ∞ or ƒ(x) → �∞ as x → �3 from either the left or the right.

The values in the table confirm that ƒ(x) → �∞ as x → �3 from the right, so there is a vertical asymptote at x � �3.

Example 2 Determine the slant asymptote for ƒ(x) � �3x2

x�

�2x

1� 2�.

First use division to rewrite the function.

As x → ∞, �x �3

1� → 0. Therefore, the graph of ƒ(x)

will approach that of y � 3x � 1. This means thatthe line y � 3x � 1 is a slant asymptote for thegraph of ƒ(x).

3-7

One way to find the horizontalasymptote is to let ƒ(x) � y and solve for x in terms of y. Then find where the function isundefined for values of y.

y � �2xx��

31�

y(x � 3) � 2x � 1xy � 3y � 2x � 1xy � 2x � �3y � 1x(y � 2) � �3y � 1

x � ��

y3y

�

�

21

�

The rational expression ��

y3y

�

�

21

� is undefined for y � 2. Thus, thehorizontal asymptote is the line y � 2.

x ƒ(x)�2.9 �68�2.99 �698�2.999 �6998�2.9999 �69998

A rational function is a quotient of two polynomial functions.

The line x � a is a vertical asymptote for a function ƒ(x) if ƒ(x) → ∞ or ƒ(x) → �∞ as x → afrom either the left or the right.

The line y � b is a horizontal asymptote for a function ƒ(x) if ƒ(x) → b as x → ∞ or as x → �∞.

A slant asymptote occurs when the degree of the numerator of a rational function is exactlyone greater than that of the denominator.

3x � 1x � 13�x�2�� 2�x� �� 2� → ƒ(x) � 3x � 1 � �x �

31�

3x2 � 3xx � 2x � 1

3


Graphs of Rational FunctionsDetermine the equations of the vertical and horizontal asymptotes, if any, of each function.

1. ƒ(x) � �x2

4� 1� 2. ƒ(x) � �2

xx�

�

11� 3. g(x) � �

(x �

x1�

)(x3� 2)

�

Use the parent graph ƒ(x) � �1x

� to graph each equation. Describe the transformation(s) that have taken place. Identify the new locations of the asymptotes.

4. y � �x �3

1� � 2 5. y � ��x �4

3� � 3

Determine the slant asymptotes of each equation.

6. y � �5x2 �x �

102x � 1� 7. y � �xx

2

��

1x�

8. Graph the function y � �x2

x�

�x

1� 6�.

9. Physics The illumination I from a light source is given by the formula I � �

dk

2�, where k is a constant and d is distance. As the

distance from the light source doubles, how does the illuminationchange?

Practice

NAME _____________________________ DATE _______________ PERIOD ________

3-7


Enrichment

NAME _____________________________ DATE _______________ PERIOD ________

Slant AsymptotesThe graph of y � ax � b, where a � 0, is called a slant asymptote of y � f (x) if the graph of f (x) comes closer and closer to the line asx → ∞ or x → – ∞.

For f (x) � 3x � 4 � , y � 3x � 4 is a slant asymptote because

f (x) � (3x � 4) � , and → 0 as x → ∞ or x → – ∞.

Example Find the slant asymptote of f (x) � .

–2 1 8 15 Use synthetic division.–2 –12

1 6 3

y � � x � 6 �

Since → 0 as x → ∞ or x → –∞,

y � x � 6 is a slant asymptote.

Use synthetic division to find the slant asymptote for each of the following.

1. y �

2. y �

3. y �

4. y �

5. y �ax2 � bx � c��

x � d

ax2 � bx � c��

x � d

x2 � 2x � 18��

x � 3

x2 � 3x � 15��

x � 2

8x2 � 4x � 11��

x � 5

3�x � 2

3�x � 2

x2 � 8x � 15��

x � 2

x2 � 8x � 15��

x � 2

2�x

2�x

2�x

3-7


Direct, Inverse, and Joint VariationA direct variation can be described by the equation y � kxn.The k in this equation is called the constant of variation.To express a direct variation, we say that y varies directly asxn. An inverse variation can be described by the equation y � �x

kn� or xny � k. When quantities are inversely

proportional, we say they vary inversely with each other.Joint variation occurs when one quantity varies directly asthe product of two or more other quantities and can bedescribed by the equation y � kxnzn.

Example 1 Suppose y varies directly as x and y � 14 when x � 8.a. Find the constant of variation and write an

equation of the form y � kxn.b. Use the equation to find the value of y when

x � 4.

a. The power of x is 1, so the direct variationequation is y � kx.

y � kx14 � k(8) y � 14, x � 8

1.75 � k Divide each side by 8.The constant of variation is 1.75. Theequation relating x and y is y � 1.75x.

b. y � 1.75xy � 1.75(4) x � 4y � 7When x � 4, the value of y is 7.

Example 2 If y varies inversely as x and y � 102 when x � 7,find x when y � 12.

Use a proportion that relates the values.

�xy1

2

n

� � �xy2

1

n

�

�172� � �10

x2� Substitute the known values.

12x � 714 Cross multiply.x � �71

124� or 59.5 Divide each side by 12.

When y � 12, the value of x is 59.5.

Study Guide

NAME _____________________________ DATE _______________ PERIOD ________

3-8


Practice

NAME _____________________________ DATE _______________ PERIOD ________

Direct, Inverse, and Joint VariationWrite a statement of variation relating the variables of each equation. Then name the constant of variation.

1. ��xy2� � 3 2. E � IR

3. y � 2x 4. d � 6t2

Find the constant of variation for each relation and use it to write anequation for each statement. Then solve the equation.

5. Suppose y varies directly as x and y � 35 when x � 5. Find y when x � 7.

6. If y varies directly as the cube of x and y � 3 when x � 2, find x when y � 24.

7. If y varies inversely as x and y � 3 when x � 25, find x when y � 10.

8. Suppose y varies jointly as x and z, and y � 64 when x � 4 and z � 8.Find y when x � 7 and z � 11.

9. Suppose V varies jointly as h and the square of r, and V � 45� when r � 3 and h � 5. Find r when V � 175� and h � 7.

10. If y varies directly as x and inversely as the square of z, and y � �5 when x � 10 andz � 2, find y when x � 5 and z � 5.

11. Finances Enrique deposited $200.00 into a savings account. The simple interest I on his account varies jointly as the time t in years and the principal P.After one quarter (three months), the interest on Enrique’s account is $2.75.Write an equation relating interest, principal, and time. Find the constant of variation. Then find the interest after three quarters.

3-8


Reading Mathematics: Interpreting Conditional StatementsThe conditional statement below is written in “if-then” form. It has the form p → q where p is the hypothesis and q is the consequent.

If a matrix A has a determinant of 0, then A�1 does not exist.It is important to recognize that a conditional statement need notappear in “if-then” form. For example, the statement

Any point that lies in Quadrant I has a positive x-coordinate.can be rewritten as

If the point P(x, y) lies in Quadrant I, then x is positive.Notice that P lying in Quadrant I is a sufficient condition for itsx-coordinate to be positive. Another way to express this is to say that P lying in Quadrant I guarantees that its x-coordinate is positive. On the other hand, we can also say that x being positive is a necessary condition for P to lie in Quadrant I. In other words, Pdoes not lie in Quadrant I if x is not positive.

To change an English statement into “if-then” form requires that you understand the meaning and syntax of the English statement.Study each of the following equivalent ways of expressing p → q.• If p then q • p implies q• p only if q • only if q, p• p is a sufficient condition for q • not p unless q• q is a necessary condition for p.

Rewrite each of the following statements in “if-then” form.

1. A consistent system of equations has at least one solution.

2. When the region formed by the inequalities in a linear programming application is unbounded, an optimal solution for the problem may not exist.

3. Functions whose graphs are symmetric with respect to the y-axis are called even functions.

4. In order for a decimal number d to be odd, it is sufficient that d end in the digit 7.

Enrichment

NAME _____________________________ DATE _______________ PERIOD ________

3-8


Chapter 3 Test, Form 1A

NAME _____________________________ DATE _______________ PERIOD ________

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

1. The graph of the equation y � x3 � x is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these

2. If the graph of a function is symmetric about the origin, which of the 2. ________following must be true?A. ƒ(x) � ƒ(�x) B. ƒ(�x) � �ƒ(x) C. ƒ(x) � �ƒ(x)� D. ƒ(x) � �ƒ(

1x)�

3. The graph of an odd function is symmetric with respect to which 3. ________of the following?A. the x-axis B. the y-axis C. the line y � x D. none of these

4. Given the parent function p(x) � �x�, what transformations 4. ________occur in the graph of p(x) � 2�x � 3� � 4?A. vertical expansion by factor B. vertical compression by factor of

of 2, left 3 units, up 4 units 0.5, down 3 units, left 4 unitsC. vertical expansion by factor D. vertical compression by factor of

of 2, right 3 units, up 4 units 0.5, down 3 units, left 4 units

5. Which of the following results in the graph of ƒ(x) � x2 being expanded 5. ________vertically by a factor of 3 and reflected over the x-axis?A. ƒ(x) � �13�x2 B. ƒ(x) � �3x2 C. ƒ(x) � ��x

12� � 3 D. ƒ(x) � ��13�x2

6. Which of the following represents the graph of ƒ(x) � �x�3 � 1? 6. ________A. B. C. D.

7. Solve �2x � 5� � 7. 7. ________A. x � �1 or x � 6 B. �1 � x � 6C. x � 6 D. x � �1

8. Choose the inequality shown by the graph. 8. ________A. y � 2�x� � 1B. y � �2�x� � 1C. y � 2�x� � 1D. y � �2�x� � 1

9. Find the inverse of ƒ(x) � 2�x� � 3. 9. ________A. ƒ�1(x) � ��x �

23��2 B. ƒ�1(x) � ��x �

23��2

C. ƒ�1(x) � �12��x� � 3 D. ƒ�1(x) � �12��x� � 310. Which graph represents a function whose inverse is also a function? 10. ________

A. B. C. D.

Chapter

3


11. Describe the end behavior of ƒ(x) � 2x4 � 5x � 1. 11. ________A. x → �, ƒ(x)→ � B. x→ �, ƒ(x) → �

x→ ��, ƒ(x) → �� x → ��, ƒ(x) → �C. x → �, ƒ(x) → �� D. x → �, ƒ(x) → ��

x → ��, ƒ(x) → � x → ��, ƒ(x) → ��

12. Which type of discontinuity, if any, is shown 12. ________in the graph at the right?A. jump B. infiniteC. point D. The graph is continuous.

13. Choose the statement that is true for the graph of ƒ(x) � x3 � 12x. 13. ________A. ƒ(x) increases for x � �2. B. ƒ(x) decreases for x � �2.C. ƒ(x) increases for x � 2. D. ƒ(x) decreases for x � 2.

14. Which type of critical point, if any, is present in the graph of 14. ________ƒ(x) � (�x � 4)5 � 1?A. maximum B. minimumC. point of inflection D. none of these

15. Which is true for the graph of ƒ(x) � �x3 � 3x � 2? 15. ________A. relative maximum at (1, 0) B. relative minimum at (1, 0)C. relative maximum at (�1, �4) D. relative minimum at (0, �2)

16. Which is true for the graph of y � �xx2

2��

49�? 16. ________

A. vertical asymptotes at x � 3 B. horizontal asymptotes at y � 2C. vertical asymptotes at x � 2 D. horizontal asymptote at y � 1

17. Find the equation of the slant asymptote for the graph of 17. ________

y � .

A. y � x � 4 B. y � x � 1 C. y � x D. y � 1

18. Which of the following could be the function 18. ________represented by the graph at the right?A. ƒ(x) � �xx

��

12� B. ƒ(x) � �

((xx

�

�

13))((xx

�

� 23))

�

C. ƒ(x) � �xx��

12� D. ƒ(x) � �

((xx

�

�

31))((xx

�

�

23))

�

19. Chemistry The volume V of a gas varies inversely as pressure P is 19. ________exerted. If V � 3.5 liters when P � 5 atmospheres, find V when P � 8 atmospheres.A. 2.188 liters B. 5.600 liters C. 11.429 liters D. 17.5 liters

20. If y varies jointly as x and the cube of z, and y � 378 when x � 4 and 20. ________z � 3, find y when x � 9 and z � 2.A. y � 283.5 B. y � 84 C. y � 567 D. y � 252

Bonus An even function ƒ has a vertical asymptote at x � 3 and a Bonus: ________maximum at x � 0. Which of the following could be ƒ ?

A. ƒ(x) � �x2 �x

9� B. ƒ(x) � �x �x

3� C. ƒ(x) � �x2x�

2

9� D. ƒ(x) � �x4 �x2

81�

x3 � 4x2 � 2x � 5��x2 � 2

Chapter 3 Test, Form 1A (continued)


3


Chapter 3 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. The graph of the equation y � x4 � 3x2 is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these

2. If the graph of a function is symmetric with respect to the y-axis, 2. ________which of the following must be true?A. ƒ(x) � ƒ(�x) B. ƒ(x) � �ƒ(x) C. ƒ(x) ��ƒ(x)� D. ƒ(x) � �ƒ(

1x)�

3. The graph of an even function is symmetric with respect to which of 3. ________the following?A. the x-axis B. the y-axis C. the line y � xD. none of these

4. Given the parent function p(x) � x2, what translations occur in 4. ________the graph of p(x) � (x � 7)2 � 3?A. right 7 units, up 3 units B. down 7 units, left 3 unitsC. left 7 units, up 3 units D. right 7 units, down 3 units

5. Which of the following results in the graph of ƒ(x) � x3 being expanded 5. ________vertically by a factor of 3?A. ƒ(x) � x3 � 3 B. ƒ(x) � �13�x3 C. ƒ(x) � 3x3 D. ƒ(x) � ��13�x3

6. Which of the following represents the graph of ƒ(x) � �x2 � 1�? 6. ________A. B. C. D.

7. Solve �2x � 4� � 6. 7. ________A. x � �1 or x � 5 B. �1 � x � 5C. x � 5 D. x � �1

8. Choose the inequality shown by the graph. 8. ________A. y � �x � 2� � 1 B. y � �x � 2� � 1C. y � �x � 2� � 1 D. y � �x � 2� � 1

9. Find the inverse of ƒ(x) � �x �1

2�. 9. ________

A. ƒ�1(x) � �x �1

2� B. ƒ�1(x) � �1x� � 2

C. ƒ�1(x) � x � 2 D. ƒ�1(x) � �1x� � 2

10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.

Chapter

3


11. Which type of discontinuity, if any, is shown 11. ________in the graph at the right?A. jump B. infiniteC. point D. The graph is continuous.

12. Describe the end behavior of ƒ(x) � 2x3 � 5x � 1 12. ________A. x → �, ƒ(x) → � B. x → �, ƒ(x) → �

x → ��, ƒ(x) → �� x → ��, ƒ(x) → �C. x → �, ƒ(x) → �� D. x → �, ƒ(x) → ��

x → ��, ƒ(x) → � x → ��, ƒ(x) → ��

13. Choose the statement that is true for the graph of ƒ(x) � �(x � 2)2. 13. ________A. ƒ(x) increases for x � �2. B. ƒ(x) decreases for x � �2.C. ƒ(x) increases for x � 2. D. ƒ(x) decreases for x � 2.

14. Which type of critical point, if any, is present in the graph of 14. ________ƒ(x) � (�x � 4)3?A. maximum B. minimumC. point of inflection D. none of these

15. Which is true for the graph of ƒ(x) � x3 � 3x � 2? 15. ________A. relative maximum at (1, 0) B. relative minimum at (�1, 4)C. relative maximum at (�1, 4) D. relative minimum at (0, 2)


2��

49�? 16. ________

A. vertical asymptotes, x � 3 B. horizontal asymptotes, y � 2C. vertical asymptotes, x � 2 D. horizontal asymptote, y � 0

17. Find the equation of the slant asymptote for the graph of 17. ________y � �3x2

x+

�2x

1� 3�.

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

18. Which of the following could be the function 18. ________represented by the graph at the right?A. ƒ(x) � �x �

x2� B. ƒ(x) � �(x �

x(x3)

�

(x3�

)2)�

C. ƒ(x) � �x �x

2� D. ƒ(x) � �(x �

x(x3)

�

(x3�

)2)�

19. Chemistry The volume V of a gas varies inversely as pressure P is 19. ________exerted. If V � 4 liters when P � 3.5 atmospheres, find V when P � 2.5 atmospheres.A. 5.6 liters B. 2.188 liters C. 2.857 liters D. 10.0 liters

20. If y varies jointly as x and the cube root of z, and y � 120 when x � 3 20. ________and z � 8, find y when x � 4 and z � 27.A. y � 540 B. y � 240 C. y � 60 D. y � 26�23�

Bonus If ƒ( g(x)) � x and ƒ(x) � 3x � 4, find g(x). Bonus: ________A. g(x) � �x �

34� B. g(x) � �x +

34� C. g(x) � �3

x� � 4 D. g(x) � �3x� � 4

Chapter 3 Test, Form 1B (continued)


3


Chapter 3 Test, Form 1C

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. The graph of the equation y � x2 � 3 is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these

2. If the graph of a relation is symmetric about the line y � x and the 2. ________point (a, b) is on the graph, which of the following must also be on the graph?A. (�a, b) B. (a, �b) C. (b, a) D. (�a, �b)

3. The graph of an odd function is symmetric with respect to which 3. ________of the following?A. the x-axis B. the y-axis C. the origin D. none of these

4. Given the parent function p(x) � �x�, what transformation occurs in 4. ________the graph of p(x) � �x� �� 2� � 5?A. right 2 units, up 5 units B. up 2 units, right 5 units C. left 2 units, down 5 units D. down 2 units, left 5 units

5. Which of the following results in the graph of ƒ(x) � x2 being 5. ________expanded vertically by a factor of 4?A. ƒ(x) � x2 � 4 B. ƒ(x) � x2 � 4 C. ƒ(x) � 4x2 D. ƒ(x) � �14�x2

6. Which of the following represents the graph of ƒ(x) � �x3�? 6. ________A. B. C. D.

7. Solve �x � 4� � 6. 7. ________A. x � �2 B. x � 10C. �2 � x � 10 D. x � �2 or x � 10

8. Choose the inequality shown by the graph. 8. ________A. y � x2 � 1 B. y � (x � 1)2

C. y � x2 � 1 D. y � (x � 1)2

9. Find the inverse of ƒ(x) � 2x � 4. 9. ________A. ƒ�1(x) � �2x

1� 4� B. ƒ�1(x) � �x �

24�

C. ƒ�1(x) � �2x� � 4 D. ƒ�1(x) � �2

x� � 4

10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.

Chapter

3


11. Which type of discontinuity, if any, is shown 11. ________in the graph at the right?A. jumpB. infiniteC. pointD. The graph is continuous.

12. Describe the end behavior of ƒ(x) � �x2. 12. ________A. x → ∞, ƒ(x) → ∞ B. x → ∞, ƒ(x) → ∞

x → �∞, ƒ(x) → �∞ x → �∞, ƒ(x) → ∞C. x → ∞, ƒ(x) → �∞ D. x → ∞, ƒ(x) → �∞

x → �∞, ƒ(x) → ∞ x → �∞, ƒ(x) → �∞13. Choose the statement that is true for the graph of ƒ(x) � (x � 1)2. 13. ________

A. ƒ(x) increases for x � �1. B. ƒ(x) decreases for x � �1.C. ƒ(x) increases for x � 1. D. ƒ(x) decreases for x � 1.

14. Which type of critical point, if any, is present in the graph 14. ________of ƒ(x) � x3 � 1?A. maximum B. minimumC. point of inflection D. none of these

15. Which is true for the graph of ƒ(x) � x3 � 3x? 15. ________A. relative maximum at (1, �2) B. relative minimum at (�1, 2)C. relative maximum at (�1, 2) D. relative minimum at (0, 0)


2��

94�? 16. ________

A. vertical asymptotes at x � 3 B. horizontal asymptotes at y � 2C. vertical asymptotes at x � 2 D. horizontal asymptote at y � 0

17. Find the equation of the slant asymptote for the graph of y � �xx2

��

35�. 17. ________

A. x � 3 B. y � x � 3 C. y � x D. y � x � 3

18. Which of the following could be the function 18. ________represented by the graph?A. ƒ(x) � �x �

14� B. ƒ(x) � �(x �

x2�)(x

2� 4)�

C. ƒ(x) � �xx��

24� D. ƒ(x) � �(x �

x2�)(x

2� 4)�

19. Chemistry The volume V of a gas varies inversely as pressure 19. ________P is exerted. If V � 4 liters when P � 3 atmospheres, find V when P � 7 atmospheres.A. 1.714 liters B. 5.25 liters C. 9.333 liters D. 1.5 liters

20. If y varies inversely as the cube root of x, and y � 12 when x � 8, 20. ________find y when x � 1.A. y � 6144 B. y � 24 C. y � 6 D. y � �12

38�

Bonus The graph of ƒ(x) � �x21� c� has a vertical asymptote at x � 3. Bonus: ________

Find c.A. 9 B. 3 C. �3 D. �9

Chapter 3 Test, Form 1C (continued)


3


Chapter 3 Test, Form 2A

NAME _____________________________ DATE _______________ PERIOD ________

Determine whether the graph of each equation is symmetric withrespect to the origin, the x-axis, the y-axis, the line y � x, the liney � �x, or none of these.

1. xy � �4 1. __________________

2. x � 5y2 � 2 2. __________________

3. Determine whether the function ƒ(x) � �x2 �x

4� is odd, even, 3. __________________or neither.

4. Describe the transformations relating the graph of 4. __________________y � �2x3 � 4 to its parent function, y � x3.

5. Use transformations of the parent graph p(x) � �1x� 5.to sketch the graph of p(x) � ��

1x�� 1.

6. Graph the inequality y � 2x2 � 1. 6.

7. Solve �5 � 2x� � 11. 7. __________________

Find the inverse of each function and state whether the inverse is a function.

8. ƒ(x) � �x �x

2� 8. __________________

9. ƒ(x) � x2 � 4 9. __________________

10. Graph ƒ(x) � x3 � 2 and its inverse. State whether the 10. __________________inverse is a function.

Chapter

3


Determine whether each function is continuous at the given x-value. If discontinuous, state the type of discontinuity (point,jump, or infinite).

11. ƒ(x) � � ; x � 1 11. __________________

12. ƒ(x) � �xx2

��

39�; x � �3 12. __________________

13. Describe the end behavior of y � �3x4 � 2x. 13. __________________

14. Locate and classify the extrema for the graph of 14. __________________y � x4 � 3x2 � 2.

15. The function ƒ(x) � x3 � 3x2 � 3x has a critical point 15. __________________when x � 1. Identify the point as a maximum, a minimum, or a point of inf lection, and state its coordinates.

16. Determine the vertical and horizontal asymptotes for 16. __________________the graph of y � �x3 �

x2

5�x2

4� 6x�.

17. Find the slant asymptote for y � �3x2

x�

�5x

2� 1�. 17. __________________

18. Sketch the graph of y � �x3 �x2

x2�

�1

12x�. 18.

19. If y varies directly as x and inversely as the square root 19. __________________of z, and y � 8 when x � 4 and z � 16, find y when x � 10 and z � 25.

20. Physics The kinetic energy Ek of a moving object, 20. __________________measured in joules, varies jointly as the mass mof the object and the square of the speed v. Find the constant of variation k if Ek is 36 joules, m is 4.5 kilograms,and v is 4 meters per second.

Bonus Given the graph of p(x), Bonus:sketch the graph of y � �2p��12�(x � 2) � 2.

x2 � 1 if x � 1�x3 � 2 if x � 1

Chapter 3 Test, Form 2A (continued)


3


Chapter 3 Test, Form 2B

NAME _____________________________ DATE _______________ PERIOD ________

Determine whether the graph of each equation is symmetric withrespect to the origin, the x-axis, the y-axis, the line y � x, the liney � �x, or none of these.

1. xy � 2 1. __________________

2. y � 5x3 � 2x 2. __________________

3. Determine whether the function ƒ(x) � �x� is odd, even, 3. __________________or neither.

4. Describe the transformations relating the graph of 4. __________________y � �12�(x � 3)2 to its parent function, y � x2.

5. Use transformations of the parent graph of p(x) � �x� 5.to sketch the graph of p(x) � ��x� � 3.

6. Graph the inequality y � (x � 2)3. 6.

7. Solve �2x � 4� � 10. 7. __________________


8. ƒ(x) � x3 � 4 8. __________________

9. ƒ(x) � (x � 1)2 9. __________________

10. Graph ƒ(x) � x2 � 2 and its inverse. State whether the 10. __________________inverse is a function.

Chapter

3



11. ƒ(x) � � ; x � 1 11. __________________

12. ƒ(x) � �xx2

��

39�; x � �3 12. __________________

13. Describe the end behavior of y � 3x3 � 2x. 13. __________________

14. Locate and classify the extrema for the graph 14. __________________of y � x4 � 3x2.

15. The function ƒ(x) � �x3 � 6x2 � 12x � 7 has a critical 15. __________________point when x � �2. Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.

16. Determine the vertical and horizontal asymptotes for the 16. __________________graph of y � �x

x3

2

��

x42�.

17. Find the slant asymptote for y � �2x2

x�

�5x

3� 2�. 17. __________________

18. Sketch the graph of y � �xx2 �

�31x�. 18.

19. If y varies directly as x and inversely as the square of z, 19. __________________and y � 8 when x � 4 and z � 3, find y when x � 6 and z � �2.

20. Geometry The volume V of a sphere varies directly as 20. __________________the cube of the radius r. Find the constant of variation kif V is 288� cubic centimeters and r is 6 centimeters.

Bonus Determine the value of k such that Bonus: __________________

ƒ(x) � � is continuous when x � 2.2x2 if x � 2x � k if x � 2

x � 1 x � 1

ifif

x2 � 1 �x � 3

Chapter 3 Test, Form 2B (continued)


3


Chapter 3 Test, Form 2C

NAME _____________________________ DATE _______________ PERIOD ________

Determine whether the graph of each equation is symmetric with respect to the origin, the x-axis, the y-axis, the line y � x, the line y � �x, or none of these.

1. y � 2�x� 1. __________________

2. y � x3 � x 2. __________________

3. Determine whether the function ƒ(x) � x2 � 2 is odd, even, 3. __________________or neither.

4. Describe the transformation relating the graph of 4. __________________y � (x � 1)2 to its parent function, y � x2.

5. Use transformations of the parent graph p(x) � x3 5.to sketch the graph of p(x) � (x � 2)3 � 1.

6. Graph the inequality y � x2 � 1. 6.

7. Solve �x � 4� � 8. 7. __________________


8. ƒ(x) � x2 8. __________________

9. ƒ(x) � x3 � 1 9. __________________

10. Graph ƒ(x) � �2x � 4 and its inverse. State whether the 10. __________________inverse is a function.

Chapter

3



11. ƒ(x) � � ; x � 0 11. __________________

12. ƒ(x) � �xx2�

�

39

� ; x � �3 12. __________________

13. Describe the end behavior of y � x4 � x2. 13. __________________

14. Locate and classify the extrema for the graph of 14. __________________y � �x4 � 2x2.

15. The function ƒ(x) � x3 � 3x has a critical point when x � 0. 15. __________________Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.

16. Determine the vertical and horizontal asymptotes for the 16. __________________graph of y � �

xx2 �

�

245

�.

17. Find the slant asymptote for y � �x2

x�

�x

1� 2�. 17. __________________

18. Sketch the graph of y � �x2x� 4�. 18.

19. If y varies directly as the square of x, and y � 200 19. __________________when x � 5, find y when x � 2.

20. If y varies inversely as the cube root of x, and y � 10 20. __________________when x � 27, find y when x � 8.

Bonus Determine the value of k such that Bonus: __________________ƒ(x) � 3x2 � kx � 4 is an even function.

x2 � 1 if x � 0� x if x � 0

Chapter 3 Test, Form 2C (continued)


3


Chapter 3 Open-Ended Assessment

NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. a. Draw the parent graph and each of the three transformations that would result in the graph shown below. Write the equation for each graph and describe the transformation.

b. Draw the parent graph of a polynomial function. Then draw a transformation of the graph. Describe the type of transformation you performed. Use equations as needed to clarify your answer.

c. Sketch the graph of y � �x(x �x3�)(x

4� 4)�. Describe how you

determined the shape of the graph and its vertical asymptotes.

d. Write the equation of a graph that has a vertical asymptote at x � 2 and point discontinuity at x � �1. Describe your reasoning.

2. The critical points of ƒ(x) � �15�x5 � �14�x4 � �23�x3 � 2 are at x � 0,2, and �1. Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection.Explain why x � 3 is not a critical point of the function.

3. Mrs. Custer has 100 bushels of soybeans to sell. The current price for soybeans is $6.00 a bushel. She expects the market price of a bushel to rise in the coming weeks at a rate of $0.10 per week. For each week she waits to sell, she loses 1 bushel due to spoilage.

a. Find a function to express Mrs. Custer’s total income from selling the soybeans. Graph the function and determine when Mrs. Custer should sell the soybeans in order to maximizeher income. Justify your answer by showing that selling ashort time earlier or a short time later would result in lessincome.

b. Suppose Mrs. Custer loses 5 bushels per week due to spoilage. How would this affect her selling decision?

Chapter

3


Determine whether the graph of each equation issymmetric with respect to the origin, the x-axis, they-axis, the line y � x, the line y � �x, or none of these.

1. x � y2 � 1 1. __________________

2. xy � 8 2. __________________

3. Determine whether the function ƒ(x) � 2(x � 1)2 � 4x is 3. __________________odd, even, or neither.

4. Describe the transformations that relate the graph 4. __________________of p(x) � (0.5x)3 � 1 to its parent function p(x) � x3.

5. Use transformations of the parent graph p(x) � �x� 5.to sketch the graph of p(x) � 2�x � 3�.

6. Graph the inequality y � �x� �� 1�. 6.

7. Solve �x � 4� � 6. 7. __________________


8. ƒ(x) � x3 � 1 8. __________________

9. ƒ(x) � ��1x� 9. __________________

10. Graph ƒ(x) � (x � 3)2 and its inverse. State whether 10. __________________the inverse is a function.

Chapter 3 Mid-Chapter Test (Lessons 3-1 through 3-4)


3

Determine whether the graph of each equation is symmetric with respect to the origin, the x-axis, the y-axis, the line y � x, the line y � �x, or none of these.

1. y � (x � 2)3 2. y � �x14� 1. __________________

2. __________________

3. Determine whether the function ƒ(x) � x3 � 4x is 3. __________________odd, even, or neither.

4. Describe how the graphs of ƒ(x) � �x� and g(x) � �4�x� 4. __________________are related.

5. Use transformations of the parent graph of p(x) � x2 to 5.sketch the graph of p(x) � (x � 1)2 � 2.

1. Graph y � �x2 � 1. 1.

2. Solve �x � 3� � 4. 2. __________________

3. Find the inverse of ƒ(x) � 3x � 6. Is the inverse a function? 3. __________________

4. Graph ƒ(x) � (x � 1)2 � 2 and its inverse. 4.

5. What is the equation of the line that acts as a line of 5. __________________symmetry between the graphs of ƒ(x) and ƒ�1(x)?

Chapter 3 Quiz B (Lessons 3-3 and 3-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 3 Quiz A (Lessons 3-1 and 3-2)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

3

Chapter

3

Determine whether the function is continuous at the given x-value. If discontinuous, state the type of discontinuity ( jump, infinite, or point).

1. ƒ(x) � � ; x � 0 1. __________________

2. ƒ(x) � �x2

1� 1� ; x � 1 2. __________________

3. Describe the end behavior of y � x4 � 3x � 1. 3. __________________

4. Locate and classify the extrema for the graph of 4. __________________y � 2x3 � 6x.

5. The function ƒ(x) � �x55� � �43�x3 has a critical point at x � 2. 5. __________________

Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.

1. Determine the equations of the vertical and horizontal 1. __________________asymptotes, if any, for the graph of y � �x2 �

x5

2

x � 6� .

2. Find the slant asymptote for ƒ(x) � �x2 �

x �4x

2� 3�. 2. __________________

3. Use the parent graph ƒ(x) � �1x� to graph the function 3. __________________g(x) � �x �

23�. Describe the transformation(s) that have

taken place. Identify the new locations of the asymptotes.

Find the constant of variation for each relation and use it to write an equation for each statement. Then solve the equation.

4. If y varies directly as x, and y � 12 when x � 8, find y when 4. __________________x � 14.

5. If y varies jointly as x and the cube of z, and y � 48 when 5. __________________x � 3 and z � 2, find y when x � �2 and z � 5.

x2 + 1 if x � 0x if x � 0

Chapter 3 Quiz D (Lessons 3-7 and 3-8)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 3 Quiz C (Lessons 3-5 and 3-6)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

3

Chapter

3


Chapter 3 SAT and ACT Practice

NAME _____________________________ DATE _______________ PERIOD ________

After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

Multiple Choice1. If x � y � 4, what is the value of

�y � x� � �x � y�?A 0B 4C 8D 16E It cannot be determined from the

information given.

2. If 2x � 5 � 4y and 12y � 6x � 15, thenwhich of the following is true?A 2y � 5B 2x � 5C y � 2x � 5D x � 2y � 5E 2y � x � 5

3. If x � �45�, then 3(x � 1) � 4x is how many fifths?A 13B 15C 17D 23E 25

4. If x2 � 3y � 10 and �xy� � �1, then

which of the following could be a value of x?A �5B �2C �1D 2E 3

5. 168:252 as 6: ?A 4B 5C 9D 18E 36

6. (3.1 � 12.4) � (3.1 � 12.4) �A �24.8B �6.2C 0D 6.2E 24.8

7. If x � 5z and y � �15z1+ 2�, then which of

the following is equivalent to y?A �x +

52�

B �31x�

C �x �1

2�

D �3x1� 2�

E None of these

8. If �x �

xy

� � 3 and �z �

zy

� � 5, then which

of the following is the value of �xz�?

A �12�

B �53�

C �53�

D 2E 15

9. �1 � �12 � � �14 � � �18 � � �1 16� � �3 1

2� �A �3�

86��

B ��86�2��

C ��42��

D �14�

E �3�81�4��

Chapter

3

The larger valueof x in theequation

x2 � 7x � 12 � 0

The smallervalue of x in the

equationx2 � 7x � 12 � 0


Chapter 3 SAT and ACT Practice (continued)


310. If 12 and 18 each divide R without a

remainder, which of the following could be the value of R?A 24B 36C 48D 54E 120

11. If x � y � 5 and x � y � 3, then x2 � y2 �A 2B 4C 8D 15E 16

12. If n � 0, then which of the followingstatements are true?I. �

3��n� � 0

II. �1� �� n� � 1III. �

5n�3� � 0

A I onlyB III onlyC I and III onlyD II and III onlyE None of the statements are true.

13. If x* is defined to be x � 2, which of thefollowing is the value of (3* � 4*)*?A 0B 1C 2D 3E 4

14. Which of the following is the total costof 3�12� pounds of apples at $0.56 perpound and 4�12� pounds of bananas at$0.44 per pound?A $2.00B $3.00C $3.94D $4.00E $4.06

15. If a and b are real numbers where a � b and a � b � 0, which of the following MUST be true?A a � 0B b � 0C �a� � �b�D ab � 0E It cannot be determined from the

information given.

16. For all x, (3x3 � 13x2 � 6x � 8)(x � 4) �A 3x4 � 25x3 � 46x2 � 16x � 32B 3x4 � 25x3 � 46x2 � 32x � 32C 3x4 � x3 � 46x2 � 16x � 32D 3x4 � 25x3 � 58x2 � 32x � 32E 3x4 � 25x3 � 58x2 � 16x � 32

17–18. Quantitative ComparisonA if the quantity in Column A is

greaterB if the quantity in Column B is

greaterC if the two quantities are equalD if the relationship cannot be

determined from the informationgiven

Column A Column B

17.

18. (2x � 3)2 4x(x � 3)

19. Grid-In For nonzero real numbers a, b, c, and d, d � 2a, a � 2b, and b � 2c. What is the value of �dc�?

20. Grid-In If � �34�, what is the

value of x?

3��4 � �x �

x1�


Chapter 3, Cumulative Review (Chapters 1-3)

NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of {(�2, 2), (0, 2), (2, 2)}. Then 1. __________________state whether the relation is a function. Write yes or no.

2. If ƒ(x) � �x �3

1� and g(x) � x � 1, find ƒ( g(x)). 2. __________________

3. Write the standard form of the equation of the line that is 3. __________________parallel to the line with equation y � 2x � 3 and passes through the point at (�1, 5).

4. Graph ƒ(x) � �x � 1�. 4.

Solve each system of equations algebraically.

5. 3x � 5y � 21 5. __________________x � y � 5

6. x � 2y � z � 7 6. __________________3x � y � z � 22x � 3y � 2z � 7

7. How many solutions does a consistent and independent 7. __________________system of linear equations have?

8. Find the value of � �. 8. __________________

9. Determine whether the graph of y � x3 � 2x is symmetric 9. __________________with respect to the origin, the x-axis, the y-axis, or none of these.

10. Describe the transformations relating the graph of 10. __________________y � 2(x � 1)2 to the graph of y � x2.

11. Solve �3x � 6� � 9. 11. __________________

12. Find the inverse of the function ƒ(x) � �x �3

2�. 12. __________________

13. State the type of discontinuity ( jump, infinite, or point) 13. __________________that is present in the graph of ƒ(x) � �x�.

14. Determine the horizontal asymptote for the graph of 14. __________________ƒ(x) � �xx

��

35�.

15. If y varies inversely as the square root of x, and y � 20 15. __________________when x � 9, find y when x � 16.

�7�2

35

Chapter

3

© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts

SAT and ACT Practice Answer Sheet(10 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

0 0 0

.. ./ /

.

99 9 9

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1


SAT and ACT Practice Answer Sheet(20 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

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87654321

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mm

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rap

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Det

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2.ƒ

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x �2

x�

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wh

eth

er t

he

gra

ph

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each

eq

uat

ion

is s

ymm

etri

c w

ith

res

pec

t to

th

e x-

axis

, th

e y-

axis

, th

e lin

e y

�x,

th

e lin

e y

��

x, o

r n

one

of t

hes

e.

5.x

�y

�6

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x6.

x2�

y�

2y-

axis

7.xy

�3

y�

x an

dy

��

x8.

x3�

y2�

4x-

axis

9.y

�4x

none

of

thes

e10

.y

�x2

�1

y-ax

is

11.

Is ƒ

(x)�

�x�a

n e

ven

fu

nct

ion

, an

odd

fu

nct

ion

, or

nei

ther

?ev

en

Ref

er t

o th

e g

rap

h a

t th

e ri

gh

t fo

r E

xerc

ises

12

and

13.

12.C

ompl

ete

the

grap

h s

o th

at it

is t

he

grap

h

of a

n o

dd f

un

ctio

n.

13.C

ompl

ete

the

grap

h s

o th

at it

is t

he

grap

h

of a

n e

ven

fu

nct

ion

.

14.G

eom

etry

Cam

eron

tol

d h

er f

rien

d Ju

anit

a th

at t

he

grap

h o

f �y

� �6

��3

x� h

as t

he

shap

e of

a g

eom

etri

c fi

gure

. Det

erm

ine

wh

eth

er t

he

grap

h o

f �y

��6

��3

x�is

sym

met

ric

wit

h

resp

ect

to t

he

x-ax

is, t

he

y-ax

is, b

oth

, or

nei

ther

. Th

en m

ake

a sk

etch

of

the

grap

h.

Is C

amer

on c

orre

ct?

x-ax

is a

nd y

-axi

s;

yes,

the

gra

ph

is t

hat

of

a d

iam

ond

or

par

alle

log

ram

.

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

3-1

13 12

x

fx(

) O

(3,3

)

(4,2

)(1

,1)

Answers (Lesson 3-1)


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____

____

____

____

____

____

____

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____

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3-1

Sy

mm

etr

y in

Th

ree

-Dim

en

sio

na

l F

igu

res

Aso

lid

figu

re t

hat

can

be

supe

rim

pose

d, p

oin

t fo

rpo

int,

on

its

mir

ror

imag

e h

as a

pla

ne

of s

ymm

etry

.A

sym

met

rica

l sol

id o

bjec

t m

ay h

ave

a fi

nit

e or

infi

nit

e n

um

ber

of p

lan

es o

f sy

mm

etry

. Th

e ch

air

in t

he

illu

stra

tion

at

the

righ

t h

as ju

st o

ne

plan

e of

sym

met

ry; t

he

dou

ghn

ut

has

infi

nit

ely

man

ypl

anes

of

sym

met

ry, t

hre

e of

wh

ich

are

sh

own

.

Det

erm

ine

the

nu

mb

er o

f p

lan

es o

f sy

mm

etry

for

eac

h o

bje

ct a

nd

des

crib

e th

e p

lan

es.

1.a

bric

k3

pla

nes

of

sym

met

ry; e

ach

pla

ne is

par

alle

l to

a p

air

of

op

po

site

fac

es.

2.a

ten

nis

bal

lA

n in

finit

e nu

mb

er o

f p

lane

s; e

ach

pla

ne p

asse

sth

roug

h th

e ce

nter

.3.

a so

up

can

an in

finit

e nu

mb

er o

f p

lane

s p

assi

ng t

hro

ugh

the

cent

ral a

xis,

plu

s o

ne p

lane

cut

ting

the

cen

-te

r o

f th

e ax

is a

t ri

ght

ang

les

4.a

squ

are

pyra

mid

4 p

lane

s al

l pas

sing

thr

oug

h th

e to

p v

erte

x:

2 p

lane

s ar

e p

aral

lel t

o a

pai

r o

f o

pp

osi

te e

dg

eso

f th

e b

ase

and

the

oth

er 2

cut

alo

ng t

he d

iag

o-

nals

of

the

squa

re b

ase.

5.a

cube

9 p

lane

s: 3

pla

nes

are

par

alle

l to

pai

rs o

f o

pp

o-

site

fac

es a

nd t

he o

ther

6 p

ass

thro

ugh

pai

rs o

fo

pp

osi

te e

dg

es.

Sol

id f

igu

res

can

als

o h

ave

rota

tion

al s

ymm

etry

. For

exa

mpl

e, t

he

axis

dra

wn

th

rou

gh t

he

cube

in t

he

illu

stra

tion

is a

fou

rfol

d ax

is o

fsy

mm

etry

bec

ause

th

e cu

be c

an b

e ro

tate

d ab

out

this

axi

s in

to f

our

diff

eren

t po

siti

ons

that

are

exa

ctly

ali

ke.

6.H

ow m

any

fou

r-fo

ld a

xes

of s

ymm

etry

doe

s a

cube

hav

e?U

se a

die

to

hel

p yo

u lo

cate

th

em.

3; e

ach

axis

pas

ses

thro

ugh

the

cent

ers

of

ap

air

of

op

po

site

fac

es.

7.A

cube

has

6 t

wo-

fold

axe

s of

sym

met

ry. I

n t

he

spac

e at

th

e ri

ght,

draw

on

e of

th

ese

axes

.



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Fa

mili

es

of

Gra

ph

s

Des

crib

e h

ow t

he

gra

ph

s of

ƒ(x

) an

d g

(x)

are

rela

ted

.1.

ƒ(x

)�x2

and

g(x)

�(x

�3)

2�

12.

ƒ(x

)�x

an

d g(

x)�

�2

xg

(x) i

s th

e g

rap

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(x)

g(x

) is

the

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ph

of

ƒ(x)

ref

lect

edtr

ansl

ated

left

3 u

nits

and

o

ver

the

x-ax

is a

nd c

om

pre

ssed

do

wn

1 un

it.

hori

zont

ally

by

a fa

cto

r o

f 0.

5.

Use

th

e g

rap

h o

f th

e g

iven

par

ent

fun

ctio

n t

o d

escr

ibe

the

gra

ph

of

each

re

late

d f

un

ctio

n.

3.ƒ

(x)�

x34.

ƒ(x

)��

x�a.

y�

2x3

a.y

� �

x��

3��

1ex

pan

ded

ver

tica

lly b

y tr

ansl

ated

left

3 u

nits

and

up

a

fact

or

of

21

unit

b.

y�

�0.

5(x

�2)

3b

.y

��

��x�

�2

refle

cted

ove

r th

e x-

axis

, re

flect

ed o

ver

the

y-ax

is,

tran

slat

ed r

ight

2 u

nits

, tr

ansl

ated

do

wn

2 un

its

com

pre

ssed

ver

tica

lly b

y a

fact

or

of

0.5

c.y

�(

x�

1)3

c.y

��

0�.2�

5�x��

4tr

ansl

ated

left

1 u

nit,

ex

pan

ded

ho

rizo

ntal

lyp

ort

ion

bel

ow

the

x-a

xis

by

a fa

cto

r o

f 4,

re

flect

ed s

o t

hat

it is

tran

slat

ed d

ow

n 4

unit

sab

ove

the

x-a

xis

Ske

tch

th

e g

rap

h o

f ea

ch f

un

ctio

n.

5.ƒ

(x)�

�(x

�1)

2�

16.

ƒ(x

)�2

x�

2�

3

7.C

onsu

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Cos

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uri

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her

fre

e ti

me,

Jil

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he

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$4.5

0 fo

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ch

wh

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hou

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an

y fr

acti

on o

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hou

r. W

rite

an

d gr

aph

a

fun

ctio

n t

hat

sh

ows

the

cost

of

xh

ours

of

baby

-sit

tin

g.

f(x)

��4.

5if

�x��

x4.

5�x

�1�

if �x

��x

3-2

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Con

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ts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

3-2

Agr

aph

G′i

s is

omor

phic

to a

gra

ph G

if th

e fo

llow

ing

cond

ition

s ho

ld.

1.G

and

G′h

ave

the

sam

e nu

mbe

r of

ver

tices

and

edg

es.

2.T

he d

egre

e of

eac

h ve

rtex

in G

is th

e sa

me

as th

e de

gree

of e

ach

corr

espo

ndin

g ve

rtex

in G

′.3.

If tw

o ve

rtic

es in

Gar

e jo

ined

by

k(k

�0)

edg

es, t

hen

the

two

corr

espo

ndin

g ve

rtic

es in

G′a

real

so jo

ined

by

ked

ges.

ba

c

Iso

mo

rph

ic G

rap

hs

Agr

aph

Gis

a c

olle

ctio

n o

f po

ints

in w

hic

h a

pai

r of

poi

nts

, cal

led

vert

ices

, are

co

nn

ecte

d by

a s

et o

f se

gmen

ts o

r ar

cs, c

alle

d ed

ges.

Th

e d

egre

eof

ver

tex

C, d

enot

edde

g (C

), is

th

e n

um

ber

of e

dges

con

nec

ted

to t

hat

ver

tex.

We

say

two

grap

hs

are

isom

orp

hic

if t

hey

hav

e th

e sa

me

stru

ctu

re.

Th

e de

fin

itio

n b

elow

wil

l hel

p yo

ude

term

ine

wh

eth

er t

wo

grap

hs

are

isom

orph

ic.

Exa

mp

leIn

th

e gr

aph

s b

elow

HIJ

KL

MN

↔T

UV

WX

YZ

.D

eter

min

e w

het

her

th

e gr

aph

s ar

e is

omor

ph

ic.

Nu

mbe

r of

ver

tice

s in

G: 7

Nu

mbe

r of

ver

tice

s in

G′:

7N

um

ber

of e

dges

in G

: 10

Nu

mbe

r of

edg

es in

G′:

10de

g (H

): 3

deg

(I):

3de

g (T

): 1

deg

(U):

3de

g(J

): 3

deg

(K):

3de

g (V

): 3

deg

(W):

3de

g (L

): 4

deg

(M):

3de

g (X

): 4

deg

(Y):

3de

g (N

): 1

deg

(Z):

3

Sin

ce t

here

are

the

sam

e nu

mbe

r of

ver

tice

s an

dth

e sa

me

num

ber

of e

dges

and

ther

e ar

e fi

ve v

erti

ces

of d

egre

e 3,

on

e ve

rtex

of

degr

ee 4

, an

d on

eve

rtex

of

degr

ee 1

in b

oth

gra

phs,

we

can

ass

um

e th

ey a

re is

omor

phic

.

Eac

h g

rap

h in

Row

A is

isom

orp

hic

to

one

gra

ph

in R

ow B

. M

atch

th

e g

rap

hs

that

are

isom

orp

hic

.R

ow A

1.2.

3.

Row

B

a.b

.c.



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Ad

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Con

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Gra

ph

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on

line

ar

Ine

qu

alit

ies

Det

erm

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wh

eth

er t

he

ord

ered

pai

r is

a s

olu

tion

for

th

e g

iven

ineq

ual

ity.

W

rite

yes

or

no.

1.y

�(x

�2)

2�

3, (

�2,

6)

2.y

�(x

�3)

3�

2, (

4, 5

)3.

y�

� 2x

�4

��1,

(�

4, 1

)ye

sno

yes

Gra

ph

eac

h in

equ

alit

y.

4.y

�2

� x�

1 �

5.y

�2(

x�

1)2

6.y

��

x��

2��

17.

y�

(x�

3)3

Sol

ve e

ach

ineq

ual

ity.

8.� 4

x�

10 �

�6

9.� x

�5

� �2

�6

10.

� 2x

�2

��1

�7

{x�1

�x

�4}

{x�x

��

9 o

r x

��

1}{x

��3

�x

�5}

11.

Mea

sure

men

tIn

stru

ctio

ns

for

buil

din

g a

bird

hou

se w

arn

th

at t

he

plat

form

, wh

ich

idea

lly

mea

sure

s 14

.75

cm2 ,

sh

ould

not

var

y in

siz

e by

mor

e th

an 0

.30

cm2 .

If

it d

oes,

th

e pr

econ

stru

cted

roo

f fo

r th

e bi

rdh

ouse

wil

l not

fit

pro

perl

y.a.

Wri

te a

n a

bsol

ute

val

ue

ineq

ual

ity

that

rep

rese

nts

th

e ra

nge

of

poss

ible

siz

es f

or t

he

plat

form

. Th

en s

olve

for

xto

fin

d th

e ra

nge

. �x

�14

.75

��0.

30; {

x�1

4.45

�x

�15

.05}

b.

Den

a cu

t a

boar

d 14

.42

cm2 .

Doe

s th

e pl

atfo

rm t

hat

Den

a cu

t fi

t w

ith

in t

he

acce

ptab

le r

ange

?no

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

3-3

© G

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Ad

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Con

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Enr

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NA

ME

____

____

____

____

____

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____

_ D

ATE

____

____

____

___

PE

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D__

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__

So

me

Pa

ram

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ic G

rap

hs

For

som

e cu

rves

, th

e co

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nat

es x

and

yca

n b

e w

ritt

en a

s fu

nct

ion

s of

a t

hir

d va

riab

le.

Th

e co

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s de

term

inin

g th

e cu

rve

are

give

n

by t

wo

equ

atio

ns,

rat

her

th

an b

y a

sin

gle

equ

atio

n in

xan

d y.

Th

eth

ird

vari

able

is c

alle

d a

para

met

er, a

nd

the

two

equ

atio

ns

are

call

edpa

ram

etri

c eq

uat

ion

sof

th

e cu

rve.

For

th

e cu

rves

you

wil

l gra

ph o

n t

his

pag

e, t

he

para

met

er is

t a

nd

the

para

met

ric

equ

atio

ns

of e

ach

cu

rve

are

in t

he

form

x�

f(t)

an

d

y�

g(t

).

Exa

mp

le

Gra

ph

th

e cu

rve

asso

ciat

ed w

ith

th

e p

aram

etri

ceq

uat

ion

sx

�48

tan

dy

�64

t�

16t2 .

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oose

val

ues

for

tan

d m

ake

a ta

ble

show

ing

the

valu

es o

f al

l th

ree

vari

able

s. T

hen

gra

ph t

he

x- a

nd

y-v

alu

es.

tx

y

–1–4

8–8

00

00

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rse

Fu

nc

tio

ns

an

d R

ela

tio

ns

Gra

ph

eac

h f

un

ctio

n a

nd

its

inve

rse.

1.ƒ

(x)�

(x�

1)3

�1

2.ƒ

(x)�

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Fin

d t

he

inve

rse

of ƒ

(x).

Th

en s

tate

wh

eth

er t

he

inve

rse

is a

lso

a fu

nct

ion

.

3.ƒ

(x)�

�4x

2�

14.

ƒ(x)

��3

x��

1�5.

ƒ(x

)� � (x

�4 3)2

�

inve

rse:

ƒ�

1 (x)

�x3

�1

inve

rse:

y�

3�

�2� xx� �

y�

��1 2� �

1��

x�ye

sno

no

Gra

ph

eac

h e

qu

atio

n u

sin

g t

he

gra

ph

of

the

giv

en p

aren

t fu

nct

ion

.

6.y

��

�x�

��3�

�1,

p(x)

�x2

7.y

�2

��5

x��

2�,p

(x)�

x5

8.F

ire

Fig

hti

ng

Air

plan

es a

re o

ften

use

d to

dro

p w

ater

on

for

est

fire

s in

an

eff

ort

tost

op t

he

spre

ad o

f th

e fi

re.T

he

tim

e t

it t

akes

th

e w

ater

to

trav

el f

rom

hei

ght

hto

the

grou

nd

can

be

deri

ved

from

th

e eq

uat

ion

h�

�1 2� gt2

wh

ere

gis

th

e ac

cele

rati

ondu

e to

gra

vity

(32

fee

t/se

con

d2).

a.W

rite

an

equ

atio

n t

hat

wil

l giv

e ti

me

as a

fu

nct

ion

of

hei

ght.

t�

�� 1h 6� �o

r

b.

Su

ppos

e a

plan

e dr

ops

wat

er f

rom

a h

eigh

t of

102

4 fe

et.H

ow m

any

seco

nds

wil

l it

take

for

th

e w

ater

to

hit

th

e gr

oun

d?8

seco

nds

�h �

�4

3-4

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Ad

vanc

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athe

mat

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Con

cep

ts

Enr

ichm

ent

NA

ME

____

____

____

____

____

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ATE

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____

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____

____

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2632

3023

2512

151

1

T14

E25

I

2

H3

E16

A

4

O17

V28

E

5

N18

E29

A

6

L7

Y19

T26

S30

R

20

O

8

T9

H21

F33

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13

W15

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F32

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10

I22

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11

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12

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F

An

In

vers

e A

cro

stic

Th

e pu

zzle

on

th

is p

age

is c

alle

d an

acr

osti

c. T

o so

lve

the

puzz

le,

wor

k ba

ck a

nd

fort

h b

etw

een

th

e cl

ues

an

d th

e pu

zzle

box

. Yo

u m

ayn

eed

a m

ath

dic

tion

ary

to h

elp

wit

h s

ome

of t

he

clu

es.

1.If

a r

elat

ion

con

tain

s th

e el

emen

t (e

, v),

th

en t

he

inve

rse

of t

he

rela

tion

mu

st c

onta

in t

he

elem

ent

( __

, __

).2.

Th

e in

vers

e of

th

e fu

nct

ion

2x

is f

oun

d by

co

mpu

tin

g __

of

x.3.

Th

e fi

rst

lett

er a

nd

the

last

tw

o le

tter

s of

th

em

ean

ing

of t

he

sym

bol f

– 1ar

e __

.4.

Th

is is

th

e pr

odu

ct o

f a

nu

mbe

r an

d it

s m

ult

ipli

cati

ve in

vers

e.5.

If t

he

seco

nd

coor

din

ate

of t

he

inve

rse

of (

x, f

(x))

isy,

th

en t

he

firs

t co

ordi

nat

e is

rea

d “_

_ of

__”

.6.

Th

e in

vers

e ra

tio

of t

wo

nu

mbe

rs is

th

e __

of

the

reci

proc

als

of t

he

nu

mbe

rs.

7.If

is

a b

inar

y op

erat

ion

on

set

San

dx

e

�e

x

�x

for

allx

in S

, th

en a

n id

enti

ty

elem

ent

for

the

oper

atio

n is

__.

8.To

sol

ve a

mat

rix

equ

atio

n, m

ult

iply

eac

h s

ide

ofth

e m

atri

x eq

uat

ion

on

th

e __

by

the

inve

rse

mat

rix.

9.Tw

o va

riab

les

are

inve

rsel

y pr

opor

tion

al _

_ th

eir

prod

uct

is c

onst

ant.

10.T

he

grap

h o

f th

e in

vers

e of

a li

nea

r fu

nct

ion

is a

__

lin

e.

Fro

m P

resi

den

t F

ran

kli

n D

. Roo

seve

lt’s

in

augu

ral

add

ress

d

uri

ng

the

Gre

at D

epre

ssio

n; d

eliv

ered

Mar

ch 4

, 193

3.



© G

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1A

dva

nced

Mat

hem

atic

al C

once

pts

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Re

ad

ing

Ma

the

ma

tic

sT

he

foll

owin

g se

lect

ion

giv

es a

def

init

ion

of

a co

nti

nu

ous

fun

ctio

n a

s it

mig

ht

be d

efin

ed in

a c

olle

ge-l

evel

mat

hem

atic

s te

xtbo

ok.

Not

ice

that

th

e w

rite

r be

gin

s by

exp

lain

ing

the

not

atio

n t

o be

use

d fo

r va

riou

s ty

pes

of in

terv

als.

It

is a

com

mon

pra

ctic

e fo

r co

lleg

e au

thor

s to

exp

lain

th

eir

not

atio

n, s

ince

, alt

hou

gh a

gre

at d

eal o

f th

e n

otat

ion

is

sta

nda

rd, e

ach

au

thor

usu

ally

ch

oose

s th

e n

otat

ion

he

or s

he

wis

hes

to

use

.

Use

th

e se

lect

ion

ab

ove

to a

nsw

er t

hes

e q

ues

tion

s.

1.W

hat

hap

pen

s to

th

e fo

ur

ineq

ual

itie

s in

th

e fi

rst

para

grap

h w

hen

a�

b?O

nly

the

last

ineq

ualit

y ca

n b

e sa

tisf

ied

. 2.

Wh

at h

appe

ns

to t

he

fou

r in

terv

als

in t

he

firs

t pa

ragr

aph

wh

en a

�b?

The

firs

t in

terv

al is

ø a

nd t

he o

ther

s re

duc

e to

the

po

int

a�

b.

3.W

hat

mat

hem

atic

al t

erm

mak

es s

ense

in t

his

sen

ten

ce?

If f

(x)

is n

ot __

__?__

at

x 0, it

is s

aid

to b

e di

scon

tin

uou

s at

x0.

cont

inuo

us4.

Wh

at n

otat

ion

is u

sed

in t

he

sele

ctio

n t

o ex

pres

s th

e fa

ct t

hat

a n

um

ber

xis

co

nta

ined

in t

he

inte

rval

I?

x�

I5.

In t

he

spac

e at

th

e ri

ght,

ske

tch

th

e gr

aph

of

th

e fu

nct

ion

f(x

) de

fin

ed a

s fo

llow

s:

f(x)

��if

x�

�0,�

1, if

x �

�, 1

6.

Is t

he

fun

ctio

n g

iven

in E

xerc

ise

5 co

nti

nu

ous

on t

he

inte

rval

[0,

1]?

If

not

, wh

ere

is t

he

fun

ctio

n d

isco

nti

nu

ous?

1 � 2

1 � 21 � 2

Enr

ichm

ent

3-5

Thr

ough

out t

his

book

, the

set

S, c

alle

d th

e do

mai

n of

def

initi

on o

f a fu

nctio

n, w

ill u

sual

ly b

e an

inte

rval

. An

inte

rval

is a

set

of n

umbe

rs s

atis

fyin

g on

e of

the

four

ineq

ualit

ies

a�

x�

b,

a�

x�

b,a

�x

�b

,or

a�

x�

b.

In th

ese

ineq

ualit

ies,

a�

b.

The

usu

al n

otat

ions

for

the

inte

rval

s co

rres

pond

ing

to th

e fo

ur in

equa

litie

s ar

e, r

espe

ctiv

ely,

(a,

b),

[a, b

), (

a, b

], an

d [a

, b].

An

inte

rval

of t

he fo

rm (

a,b

) is

cal

led

open

, an

inte

rval

of t

he fo

rm

[a, b

)or

(a, b

] is

cal

led

half-

open

orha

lf-cl

osed

, and

an

inte

rval

of t

he fo

rm[a

,b] i

s ca

lled

clos

ed.

Sup

pose

Iis

an in

terv

al th

at is

eith

er o

pen,

clo

sed,

or

half-

open

. Sup

pose

ƒ(x

)is

a fu

nctio

nde

fined

on

Iand

x 0 is

a p

oint

inI.

We

say

that

the

func

tion

f(x)

is c

ontin

uous

at t

he p

oint

x 0if

the

quan

tity

|ƒ(x

)�

ƒ(x

0)|

beco

mes

sm

all a

s x

�Ia

ppro

ache

sx 0.

No

; it

is d

isco

ntin

uous

at

x�

.1 � 2

© G

lenc

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0A

dva

nced

Mat

hem

atic

al C

once

pts

Co

ntin

uity

an

d E

nd

Be

ha

vio

r

Det

erm

ine

wh

eth

er e

ach

fu

nct

ion

is c

onti

nu

ous

at t

he

giv

en x

-val

ue.

Ju

stif

y yo

ur

answ

er u

sin

g t

he

con

tin

uit

y te

st.

1.y

� � 32 x2�

; x�

�1

2.y

��x

2�

2x�

4�

; x�

1

Yes;

the

fun

ctio

n is

def

ined

Ye

s; t

he f

unct

ion

isat

x�

�1;

yap

pro

ache

s �2 3�

as

def

ined

at

x�

1; y

xap

pro

ache

s�

1 fr

om

bo

th

app

roac

hes

3 as

xap

pro

ache

s

sid

es; ƒ

(�1)

��2 3� .

1 fr

om

bo

th s

ides

; ƒ(1

)�3.

3.y

�x3

�2x

�2;

x�

14.

y�

�x x� �

2 4�

; x�

�4

Yes;

the

fun

ctio

n is

def

ined

N

o; t

he f

unct

ion

is

at x

�1;

y a

pp

roac

hes

1 as

un

def

ined

at

x�

�4.

x

app

roac

hes

1 fr

om

bo

th s

ides

;ƒ(

1)�

1.

Des

crib

e th

e en

d b

ehav

ior

of e

ach

fu

nct

ion

.

5.y

�2x

5�

4x6.

y�

�2x

6�

4x4

�2x

�1

y→

∞as

x→

∞,

y→

�∞

as x

→∞

,y

→�

∞as

x→

�∞

y→

�∞

as x

→�

∞

7.y

�x4

�2x

3�

x8.

y�

�4x

3�

5y

→∞

as x

→∞

,y

→�

∞as

x→

∞,

y→

∞as

x→

�∞

y→

∞as

x→

�∞

Giv

en t

he

gra

ph

of

the

fun

ctio

n, d

eter

min

e th

e in

terv

al(s

) fo

r w

hic

h t

he

fun

ctio

n

is in

crea

sin

g a

nd

th

e in

terv

al(s

) fo

r w

hic

h t

he

fun

ctio

n is

dec

reas

ing

.

9.in

crea

sing

fo

r x

��

1 an

d x

� 1

; d

ecre

asin

g f

or

�1�

x�

1

10.E

lect

ron

ics

Oh

m’s

Law

giv

es t

he

rela

tion

ship

bet

wee

n r

esis

tan

ce R

,vol

tage

E,a

nd

curr

ent

Iin

a c

ircu

it a

s R

��E I�

.If

the

volt

age

rem

ain

s co

nst

ant

but

the

curr

ent

keep

s in

crea

sin

g in

th

e ci

rcu

it, w

hat

hap

pen

s to

th

e re

sist

ance

? R

esis

tanc

e d

ecre

ases

and

ap

pro

ache

s ze

ro.

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

3-5



© G

lenc

oe/M

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w-H

ill10

4A

dva

nced

Mat

hem

atic

al C

once

pts

"Un

rea

l" E

qu

atio

ns

Th

ere

are

som

e eq

uat

ion

s th

at c

ann

ot b

e gr

aph

ed o

n t

he

real

-nu

mbe

r co

ordi

nat

e sy

stem

. On

e ex

ampl

e is

th

e eq

uat

ion

x2–

2x�

2y2

�8y

�14

�0.

Com

plet

ing

the

squ

ares

in x

and

ygi

ves

the

equ

atio

n (

x –

1)2

�2

(y�

2)2

�– 5

.

For

an

y re

al n

um

bers

, xan

d y,

th

e va

lues

of

(x –

1)2

and

2(y

�2)

2

are

non

neg

ativ

e. S

o, t

hei

r su

m c

ann

ot b

e – 5

. T

hu

s, n

o re

al v

alu

es o

f x

and

ysa

tisf

y th

e eq

uat

ion

; on

ly im

agin

ary

valu

es c

an b

e so

luti

ons.

Det

erm

ine

wh

eth

er e

ach

eq

uat

ion

can

be

gra

ph

ed o

n t

he

real

-nu

mb

er p

lan

e. W

rite

yes

or

no.

1.(x

�3)

2�

(y�

2)2

��

42.

x2�

3x�

y2�

4y�

�7

nono

3.(x

�2)

2�

y2�

6y�

8�

04.

x2�

16�

0ye

sno

5.x4

�4y

2�

4�

06.

x2�

4y2

�4x

y�

16�

0no

no

In E

xerc

ises

7 a

nd

8, f

or w

hat

val

ues

of

k :

a. w

ill t

he

solu

tion

s of

th

e eq

uat

ion

be

imag

inar

y?

b.

will

th

e g

rap

h b

e a

poi

nt?

c. w

ill t

he

gra

ph

be

a cu

rve?

d.

Ch

oose

a v

alu

e of

k f

or w

hic

h t

he

gra

ph

is a

cu

rve

and

ske

tch

th

e cu

rve

on t

he

axes

pro

vid

ed.

7.x2

�4x

�y2

�8y

�k

�0

8.x2

�4x

�y2

�6y

�k

�0

a. k

> 2

0; b

. k=

20;

a. k

< –

13;

b. k

= –

13;

c. k

< 2

0;c.

k>

–13

;d

.d

.

En

ric

hm

en

t

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

3-6

© G

lenc

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ill10

3A

dva

nced

Mat

hem

atic

al C

once

pts

Pra

ctic

e

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Cri

tic

al

Po

ints

an

d E

xtre

ma

Loca

te t

he

extr

ema

for

the

gra

ph

of

y�

ƒ(x)

. Nam

e an

d c

lass

ify

the

extr

ema

of t

he

fun

ctio

n.

1.2.

rela

tive

max

imum

: (�

2, 1

)ab

solu

te m

inim

um: (

�2,

0)

rela

tive

min

imum

: (1,

�2)

abso

lute

min

imum

: (�

6,�

6)

3.4.

abso

lute

max

imum

: (�

1, 3

)re

lati

ve m

axim

um: (

0, 2

)re

lati

ve m

inim

um: (

2,�

2)

Det

erm

ine

wh

eth

er t

he

giv

en c

riti

cal p

oin

t is

th

e lo

cati

on o

f a

max

imu

m, a

m

inim

um

, or

a p

oin

t of

infl

ecti

on.

5.y

�x2

�6x

�1,

x�

36.

y�

x2�

2x�

6, x

�1

7.y

�x4

�3x

2�

5, x

�0

min

imum

min

imum

min

imum

8.y

�x5

�2x

3�

2x2 ,

x�

09.

y�

x3�

x2�

x, x

��

110

.y

�2x

3�

4, x

�0

max

imum

max

imum

po

int

of

infle

ctio

n

11.

Ph

ysic

sS

upp

ose

that

du

rin

g an

exp

erim

ent

you

la

un

ch a

toy

roc

ket

stra

igh

t u

pwar

d fr

om a

hei

ght

of 6

inch

es w

ith

an

init

ial v

eloc

ity

of 3

2 fe

et p

er

seco

nd.

Th

e h

eigh

t at

an

y ti

me

tca

n b

e m

odel

ed b

y th

e fu

nct

ion

s(t

)��

16t2

�32

t�0.

5 w

her

es(

t) is

m

easu

red

in f

eet

and

tis

mea

sure

d in

sec

onds

. G

raph

th

e fu

nct

ion

to

fin

d th

e m

axim

um

hei

ght

obta

ined

by

the

rock

et b

efor

e it

beg

ins

to f

all.

16

.5 f

t

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NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Sla

nt

Asy

mp

tote

sT

he

grap

h o

f y

�ax

�b,

wh

ere

a

0, is

cal

led

a sl

ant

asym

ptot

e of

y

�f(

x) if

th

e gr

aph

of

f(x)

com

es c

lose

r an

d cl

oser

to

the

lin

e as

x→

∞or

x→

–∞

.

For

f(x

)�3x

�4

�,

y�

3x�

4 is

a s

lan

t as

ympt

ote

beca

use

f(x)

�(3

x�

4)�

, an

d →

0 as

x→

∞or

x→

–∞

.

Exa

mp

leF

ind

th

e sl

ant

asym

pto

te o

ff

(x)

�.

– 21

815

Use

syn

thet

ic d

ivis

ion

.– 2

– 12

16

3

y�

�x

�6

�

Sin

ce

→0

as

x→

∞or

x→

–∞

,

y�

x�

6 i

s a

slan

t as

ympt

ote.

Use

syn

thet

ic d

ivis

ion

to

fin

d t

he

slan

t as

ymp

tote

for

eac

h o

f th

e fo

llow

ing

.

1.y

� y�

8x�

44

2.y

� y�

x�

5

3.y

� y�

x�

1

4.y

� y�

ax�

b�

ad

5.y

� y�

ax�

b�

ad

ax2

�bx

�c

��

x�

d

ax2

�bx

�c

��

x�

d

x2�

2x�

18�

�x

�3

x2�

3x�

15�

�x

�2

8x2

�4x

�11

��

x�

5

3� x

�2

3� x

� 2

x2�

8x�

15�

�x

�2

x2�

8x�

15�

�x

�2

2 � x2 � x

2 � x

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Mat

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once

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Gra

ph

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f R

atio

na

l F

un

ctio

ns

Det

erm

ine

the

equ

atio

ns

of t

he

vert

ical

an

d h

oriz

onta

l asy

mp

tote

s, if

an

y, o

f ea

ch f

un

ctio

n.

1.ƒ

(x)�

� x24 �

1�

2.ƒ

(x)�

�2 xx ��

11�

3.g(

x)�� (x

�

x 1� )(x3 �

2)�

y�

0x

��

1, y

�2

x�

�1,

x�

2, y

�0

Use

th

e p

aren

t g

rap

h ƒ

(x)�

�1 x�to

gra

ph

eac

h e

qu

atio

n. D

escr

ibe

the

tran

sfor

mat

ion

(s)

that

hav

e ta

ken

pla

ce. I

den

tify

th

e n

ew lo

cati

ons

of t

he

asym

pto

tes.

4.y

�� x

�31

��

25.

y�

�� x

�43

��

3

exp

and

ed v

erti

cally

by

are

flect

ed o

ver

x-ax

is;

fact

or

of

3; t

rans

late

d le

ftex

pan

ded

ver

tica

lly b

y a

1 un

it a

nd d

ow

n 2

unit

s;fa

cto

r o

f 4;

tra

nsla

ted

rig

htx

��

1, y

��

23

unit

s an

d u

p 3

uni

ts; x

�3,

y�

3

Det

erm

ine

the

slan

t as

ymp

tote

s of

eac

h e

qu

atio

n.

6.y

��5

x2� x

�102x

�1

�7.

y�

�x x2

��1x

�

y�

5x

y�

x�

2

8.G

raph

th

e fu

nct

ion

y�

�x2

x��x

1�6

�.

9.P

hys

ics

Th

e il

lum

inat

ion

Ifr

om a

ligh

t so

urc

e is

giv

en b

y th

e fo

rmu

la I

�� dk 2�

, wh

ere

kis

a c

onst

ant

and

dis

dis

tan

ce. A

s th

e di

stan

ce f

rom

th

e li

ght

sou

rce

dou

bles

, how

doe

s th

e il

lum

inat

ion

chan

ge?

It d

ecre

ases

by

one

fo

urth

.

Pra

ctic

e

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

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RIO

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____

__

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once

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Re

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ing

Ma

the

ma

tic

s: I

nte

rpre

tin

g

Co

nd

itio

na

l S

tate

me

nts

Th

e co

ndi

tion

al s

tate

men

t be

low

is w

ritt

en in

“if

-th

en”

form

.It

has

th

e fo

rm p

→q

wh

ere

pis

th

e h

ypot

hes

is a

nd

qis

th

e co

nse

quen

t.If

a m

atri

x A

has

a d

eter

min

ant

of 0

,th

en A

�1

doe

s n

ot e

xist

.It

is im

port

ant

to r

ecog

niz

e th

at a

con

diti

onal

sta

tem

ent

nee

d n

otap

pear

in “

if-t

hen

”fo

rm.F

or e

xam

ple,

the

stat

emen

tA

ny

poi

nt

that

lie

s in

Qu

adra

nt

I h

as a

pos

itiv

e x-

coor

din

ate.

can

be

rew

ritt

en a

sIf

th

e p

oin

t P

(x,y

) li

es i

n Q

uad

ran

t I,

then

xis

pos

itiv

e.N

otic

e th

at P

lyin

g in

Qu

adra

nt

I is

a s

uff

icie

nt

con

diti

on f

or it

sx-

coor

din

ate

to b

e po

siti

ve.

An

oth

er w

ay t

o ex

pres

s th

is is

to

say

that

Ply

ing

in Q

uad

ran

t I

guar

ante

esth

at it

s x-

coor

din

ate

is

posi

tive

.O

n t

he

oth

er h

and,

we

can

als

o sa

y th

at x

bein

g po

siti

ve is

a

nec

essa

ryco

ndi

tion

for

Pto

lie

in Q

uad

ran

t I.

In o

ther

wor

ds,P

does

not

lie

in Q

uad

ran

t I

if x

is n

ot p

osit

ive.

To

chan

ge a

n E

ngl

ish

sta

tem

ent

into

“if

-th

en”

form

req

uir

es t

hat

yo

u u

nde

rsta

nd

the

mea

nin

g an

d sy

nta

x of

th

e E

ngl

ish

sta

tem

ent.

Stu

dy e

ach

of

the

foll

owin

g eq

uiv

alen

t w

ays

of e

xpre

ssin

g p

→q.

• If

pth

en q

• p

impl

ies

q•

pon

ly if

q•

only

if q

,p•

pis

a s

uff

icie

nt

con

diti

on f

or q

• n

ot p

un

less

q•

qis

a n

eces

sary

con

diti

on f

or p

.

Rew

rite

eac

h o

f th

e fo

llow

ing

sta

tem

ents

in “

if-t

hen

” fo

rm.

1.A

con

sist

ent s

yste

m o

f equ

atio

ns h

as a

t lea

st o

ne s

olut

ion.

If a

sys

tem

of

equa

tio

ns is

co

nsis

tent

, the

n th

e sy

stem

has

at

leas

t o

ne s

olu

tio

n.2.

Wh

en t

he

regi

on f

orm

ed b

y th

e in

equ

alit

ies

in a

lin

ear

prog

ram

min

g ap

plic

atio

n is

un

bou

nde

d,an

opt

imal

sol

uti

on

for

the

prob

lem

may

not

exi

st.

If t

he r

egio

n fo

rmed

by

the

ineq

ualit

ies

in a

line

ar

pro

gra

mm

ing

ap

plic

atio

n is

unb

oun

ded

, the

n an

o

pti

mal

so

luti

on

for

the

pro

ble

m m

ay n

ot

exis

t.3.

Fun

ctio

ns w

hose

gra

phs

are

sym

met

ric

wit

h re

spec

t to

the

y-a

xis

are

calle

d ev

en fu

ncti

ons.

If t

he g

rap

h o

f a

func

tio

n is

sym

met

ric

wit

h re

spec

t to

the

y-a

xis,

the

n th

e fu

ncti

on

is e

ven.

4.In

ord

er f

or a

dec

imal

nu

mbe

r d

to b

e od

d,it

is s

uff

icie

nt

that

den

d in

th

e di

git

7.If

a d

ecim

al n

umb

er d

end

s in

the

dig

it 7

, the

n d

is o

dd

.

En

ric

hm

en

t

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

3-8

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once

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NA

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____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

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RIO

D__

____

__

Dir

ec

t, I

nve

rse

, a

nd

Jo

int

Vari

atio

nW

rite

a s

tate

men

t of

var

iati

on r

elat

ing

th

e va

riab

les

of e

ach

eq

uat

ion

. T

hen

nam

e th

e co

nst

ant

of v

aria

tion

.

1.�

�x y2 ��

32.

E�

IR

yva

ries

dir

ectl

y as

the

Eva

ries

join

tly

as

squa

re o

f x;

��1 3�

I and

R; 1

3.y

�2x

4.d

�6t

2

yva

ries

dir

ectl

y as

x; 2

dva

ries

dir

ectl

y as

th

e sq

uare

of

t; 6

Fin

d t

he

con

stan

t of

var

iati

on f

or e

ach

rel

atio

n a

nd

use

it t

o w

rite

an

equ

atio

n f

or e

ach

sta

tem

ent.

Th

en s

olve

th

e eq

uat

ion

.

5.S

upp

ose

yva

ries

dir

ectl

y as

xan

d y

�35

wh

en x

�5.

Fin

d y

wh

en x

�7.

7; y

�7x

;49

6.If

yva

ries

dir

ectl

y as

th

e cu

be o

f x

and

y�

3 w

hen

x�

2, f

ind

xw

hen

y�

24.

�3 8� ; y

��3 8� x

3; 4

7.If

yva

ries

inve

rsel

y as

xan

d y

�3

wh

en x

�25

, fin

d x

wh

en y

�10

.75

; y�

�7 x5 �;7

.5

8.S

upp

ose

yva

ries

join

tly

as x

and

z, a

nd

y�

64 w

hen

x�

4 an

d z

�8.

F

ind

yw

hen

x�

7 an

d z

�11

.2;

y�

2x

z;15

4

9.S

upp

ose

Vva

ries

join

tly

as h

and

the

squ

are

of r,

and

V�

45�

wh

en r

�3

and

h�

5. F

ind

rw

hen

V�

175�

and

h�

7.�

; V�

�r2

h;5

10.I

f y

vari

es d

irec

tly

as x

and

inve

rsel

y as

th

e sq

uar

e of

z, a

nd

y�

�5

wh

en x

�10

an

dz

�2,

fin

d y

wh

en x

�5

and

z�

5.�

2; y

��

2�zx 2�

; �0.

4

11.

Fin

an

ces

En

riqu

e de

posi

ted

$200

.00

into

a s

avin

gs a

ccou

nt.

Th

e si

mpl

e in

tere

st I

on h

is a

ccou

nt

vari

es jo

intl

y as

th

e ti

me

tin

yea

rs a

nd

the

prin

cipa

lP.

Aft

er o

ne

quar

ter

(th

ree

mon

ths)

, th

e in

tere

st o

n E

nri

que’

s ac

cou

nt

is $

2.75

. W

rite

an

equ

atio

n r

elat

ing

inte

rest

, pri

nci

pal,

and

tim

e. F

ind

the

con

stan

t of

va

riat

ion

. Th

en f

ind

the

inte

rest

aft

er t

hre

e qu

arte

rs.

I�kP

t; 0

.055

; $8.

25

3-8


Page 111

1. C

2. B

3. D

4. C

5. B

6. D

7. A

8. D

9. A

10. D

Page 112

11. B

12. B

13. C

14. C

15. A

16. D

17. A

18. B

19. A

20. D

Bonus: D

Page 113

1. B

2. A

3. B

4. A

5. C

6. C

7. B

8. A

9. B

10. D

Page 114

11. C

12. A

13. C

14. C

15. C

16. A

17. D

18. B

19. A

20. B

Bonus: B

Chapter 3 Answer KeyForm 1A Form 1B


Chapter 3 Answer Key

Page 115

1. B

2. C

3. C

4. C

5. C

6. C

7. C

8. A

9. B

10. D

Page 116

11. A

12. D

13. A

14. C

15. C

16. C

17. B

18. D

19. A

20. B

Bonus: A

Page 117

1. y � x, y � �x,the origin

2. x-axis

3. odd

4.

5.

6.

7. x 8

8. ƒ�1(x) � �12�x

x� ; yes

9. ƒ�1(x) � � �x� �� 4� ;no

10. yes

Page 11811. discontinuous

jump12. discontinuous

infinite13. x→ ∞, y→ �∞,

x→ �∞, y→�∞14. rel. max. at (0, 2);

abs. min. at (�1.22, �0.25)

15. point of inflectionat (1, 1)

16. horizontal at y � 0

17. y � 3x � 1

18.

19. y � 16

20. k � �21�

Bonus:

Form 1C Form 2A

reflected over the x-axis, expandedvertically by factorof 2, translated up 4 units

x � �3 or

vertical at x � 0, 3;


Page 119

1. y � x, y � �x,the origin

2. origin

3. even

4.

5.

6.

7. �3 � x � 7

8.

9.

10. no

Page 120

11. continuous

12.

13.

14.

15.

16.

17. y � 2x � 1

18.

19. y � 27

20. k � �43

��

Bonus: k � 6

Page 121

1. y-axis

2. origin

3. even

4.

5.

6.

7. �4 � x � 12

8. ƒ�1(x) � ��x�; no

9.

10. yes

Page 122

11.

12. continuous

13.

14. abs. max. at (0, 0)

15. point of inflectionat (0, 0)

16. vertical at x � �5;horizontal at y � 0

17. y � x

18.

19. y � 32

20. y � 15

Bonus: k � 0

Chapter 3 Answer KeyForm 2B Form 2C

translated right3 units,compressedvertically by afactor of 0.5

ƒ�1(x) � �3x� �� 4�;

yes

ƒ�1(x) � 1��x� ; no

discontinuous;point

x→ ∞, y→ �∞,x→ ∞, y→∞

rel. max. at (0, 0);abs. min. at(�1.22, �2.25)

point of inflectionat (�2, 1)

vertical at x � 0,1; horizontal at y � 0

translated right1 unit

ƒ�1(x) � �3

x� �� 1�;yes

discontinuous;jump

x→ ∞, y→ ∞, x→ �∞, y→∞


Chapter 3 Answer KeyCHAPTER 3 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts parentgraph, transformation asymptote, hole in graph, critical point, maximum, minimum, and point of inflection.

• Uses appropriate strategies to solve problems and transform graphs.

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

2 Satisfactory, • Shows understanding of the concepts parent graph, with Minor transformation, asymptote, hole in graph, critical point,Flaws maximum, minimum, and point of inflection.

• Uses appropriate strategies to solve problems andtransform graphs.

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

1 Nearly • Shows understanding of most of the concepts parentSatisfactory, graph, transformation, asymptote, hole in graph, criticalwith Serious point, maximum, minimum, and point of inflection.Flaws • May not use appropriate strategies to solve problems and

transform graphs.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts parentgraph, transformation, asymptote, hole in graph, critical point, maximum, minimum, and point of inflection.

• May not use appropriate strategies to solve problems and transform graphs.

• Computations are incorrect.• Written explanations are not satisfactory.• Graphs are not accurate or appropriate.• Does not satisfy requirements of problems.


Page 1231a.

1. parent graph 2. transformation

3. transformation 4. transformation

The parent graph y � x2 is reflectedover the x-axis and translated 5 unitsright and 2 units down.

1b.1. parent graph 2. transformation

3. transformation

The parent graph y � x3 is translated 2 units left and 1 unit up.

1c.

Since x � 4 is a common factor of the numerator and the denominator,the graph has point discontinuity at x � 4. Because y increases ordecreases without bound close to x � �3 and x � 0, there are verticalasymptotes at x � �3 and x � 0.

1d. ƒ( x) � �(x �

x2�)(x

1� 1)

�

ƒ( x) � �x �

12

�, except at

x � �1, where it is undefined.

ƒ( x) � �x �

12

� is undefined at x � 2. It approaches �∞ to the left of x � 2 and �∞ to the right.

2. �1, �16303�� is a maximum. (0, 2) is a

point of inflection. 2, � �1145�� is a

minimum. The value x � 3 is not acritical point because a line drawntangent to the graph at this point isneither horizontal nor vertical. Thegraph is increasing at x � 2.99 and at x � 3.01.

3a. ƒ( x) � total income, x � number of weeks ƒ( x) � (6.00 � 0.10x)(100 � x)

� �0.10x2 � 4x � 600

Examining the graph of the functionreveals a maximum at x � 20. ƒ( 20) � 640, ƒ( 19.9) � 639.999, andƒ( 20.1) � 639.999, so ƒ( 19.9) � ƒ( 20)and ƒ( 20) � ƒ( 20.1), which showsthat the function has a maximum of640 when x � 20.

3b. She should sell immediately becausethe income function would always bedecreasing.

Chapter 3 Answer KeyOpen-Ended Assessment


Mid-Chapter TestPage 124

1. x-axis2. y � x, y � �x, origin

3. neither

4.

5.

6.

7. x � �2 or x 10

8.

9. ƒ�1(x) � ��1x�; yes

10. no

Quiz APage 125

1. none of these

2. y-axis

3. odd

4.

5.

Quiz BPage 125

1.

2. {x|x � �7 or x � 1}

3. ƒ�1(x) � �x �3

6� ; yes

4.

5. y � x

Quiz CPage 126

1. discontinuous; jump

2. discontinuous; infinite

3. x →∞, y →∞, x →�∞, y →∞

4. rel. max. (�1, 4); rel. min. (1, �4)

5. min. at (2, �4.27)

Quiz DPage 126

1. vertical at x � 2 and 3, horizontal at y � 1

2. y � x � 2

3.

4. 1.5; y � 1.5x; 21

5. 2; y � 2xz3; �500

Chapter 3 Answer Key

translated up 1 unit,expanded horizontallyby a factor of 2

ƒ�1(x) � �3

x� �� 1�; yes

g(x) is the reflection of ƒ(x) overthe x-axis and is expandedvertically by a factor of 4

translated right 3 units,expanded vertically by afactor of 2; verticalasymptote at x � 3,horizontal asymptoteunchanged at y � 0


Page 127

1. C

2. B

3. A

4. B

5. C

6. A

7. D

8. A

9. E

Page 128

10. B

11. D

12. E

13. B

14. C

15. A

16. D

17. A

18. A

19. 8

20. �1

Page 129

1. D � {�2, 0, 2}; R � {2}; yes

2. �x3�

3. 2x � y � 7 � 0

4.

5. (2, 3)

6. (2, �1, 3)

7. one

8. 29

9. origin

10.

11. �1 � x � 5

12. ƒ�1(x) � �x3� � 2

13. jump

14. y � 1

15. 15

Chapter 3 Answer KeySAT/ACT Practice Cumulative Review

expanded vertically by afactor of 2, translated right1 unit

chapter 3 resource mastersrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812... · chapter 3...

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