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Chapter 3Resource Masters
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StudentWorksTM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice, and Enrichment masters.
TeacherWorksTM All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorksCD-ROM.
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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 3-2Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 90Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Lesson 3-3Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 93Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Lesson 3-4Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 96Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Lesson 3-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 99Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Lesson 3-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 102Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Lesson 3-7Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 105Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Lesson 3-8Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 108Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 110
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
Vocabulary Term Foundon Page Definition/Description/Example
end behavior
even function
everywhere discontinuous
extremum
horizontal asymptote
horizontal line test
image point
increasing function
infinite discontinuity
inverse function
inverse process
(continued on the next page)
Reading to Learn MathematicsVocabulary Builder (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
Reading to Learn MathematicsVocabulary Builder (continued)
NAME _____________________________ DATE _______________ PERIOD ________
3
Vocabulary Term Foundon Page Definition/Description/Example
inversely proportional
inverse relations
inverse variation
jump discontinuity
line symmetry
maximum
minimum
monotonicity
odd function
parent graph
point discontinuity
(continued on the next page)
© Glencoe/McGraw-Hill ix Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill x Advanced Mathematical Concepts
Vocabulary Term Foundon Page Definition/Description/Example
point of inflection
point symmetry
rational function
relative extremum
relative maximum
relative minimum
slant asymptote
symmetry with respect to the origin
vertical asymptote
Reading to Learn MathematicsVocabulary Builder (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
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.
© Glencoe/McGraw-Hill 88 Advanced Mathematical Concepts
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)
© Glencoe/McGraw-Hill 89 Advanced Mathematical Concepts
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.
© Glencoe/McGraw-Hill 90 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
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.
© Glencoe/McGraw-Hill 91 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
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.
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© Glencoe/McGraw-Hill 92 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
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.
© Glencoe/McGraw-Hill 93 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
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
© Glencoe/McGraw-Hill 94 Advanced Mathematical Concepts
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?
PracticeNAME _____________________________ DATE _______________ PERIOD ________
3-3
© Glencoe/McGraw-Hill 95 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
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
© Glencoe/McGraw-Hill 96 Advanced Mathematical Concepts
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.
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
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
© Glencoe/McGraw-Hill 97 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
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?
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© Glencoe/McGraw-Hill 98 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
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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.
© Glencoe/McGraw-Hill 99 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
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).
© Glencoe/McGraw-Hill 100 Advanced Mathematical Concepts
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?
PracticeNAME _____________________________ DATE _______________ PERIOD ________
3-5
© Glencoe/McGraw-Hill 101 Advanced Mathematical Concepts
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.
© Glencoe/McGraw-Hill 102 Advanced Mathematical Concepts
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.
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
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
© Glencoe/McGraw-Hill 103 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 104 Advanced Mathematical Concepts
"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
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3-6
© Glencoe/McGraw-Hill 105 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 106 Advanced Mathematical Concepts
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 ________
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© Glencoe/McGraw-Hill 107 Advanced Mathematical Concepts
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
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© Glencoe/McGraw-Hill 108 Advanced Mathematical Concepts
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
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3-8
© Glencoe/McGraw-Hill 109 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 110 Advanced Mathematical Concepts
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
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3-8
© Glencoe/McGraw-Hill 111 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 112 Advanced Mathematical Concepts
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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 113 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 114 Advanced Mathematical Concepts
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)
16. Which is true for the graph of y � �xx2
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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 115 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 116 Advanced Mathematical Concepts
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)
16. Which is true for the graph of y � �xx2
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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 117 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 118 Advanced Mathematical Concepts
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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 119 Advanced Mathematical Concepts
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. __________________
Find the inverse of each function and state whether the inverse is a function.
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
© Glencoe/McGraw-Hill 120 Advanced Mathematical Concepts
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 � 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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 121 Advanced Mathematical Concepts
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. __________________
Find the inverse of each function and state whether the inverse is a function.
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
© Glencoe/McGraw-Hill 122 Advanced Mathematical Concepts
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 � 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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 123 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 124 Advanced Mathematical Concepts
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. __________________
Find the inverse of each function and state whether the inverse is a function.
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)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
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 ________
© Glencoe/McGraw-Hill 125 Advanced Mathematical Concepts
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 ________
© Glencoe/McGraw-Hill 126 Advanced Mathematical Concepts
Chapter
3
Chapter
3
© Glencoe/McGraw-Hill 127 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill 128 Advanced Mathematical Concepts
Chapter 3 SAT and ACT Practice (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
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�
© Glencoe/McGraw-Hill 129 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill A2 Advanced Mathematical Concepts
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
87654321
87654321
87654321
© G
lenc
oe/M
cGra
w-H
ill88
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Sy
mm
etr
y a
nd
Co
ord
ina
te G
rap
hs
Det
erm
ine
wh
eth
er t
he
gra
ph
of
each
fu
nct
ion
is s
ymm
etri
c w
ith
res
pec
t to
th
e or
igin
.
1.ƒ
(x)�
��x12 �
yes
2.ƒ
(x)�
x5�
2no
3.ƒ
(x)�
x3�
4xye
s4.
ƒ(x
)�� 3
x �2
x�
no
Det
erm
ine
wh
eth
er t
he
gra
ph
of
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
y�
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)
© Glencoe/McGraw-Hill A3 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill89
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
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
.
Answers (Lesson 3-2)
© Glencoe/McGraw-Hill A4 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill91
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
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
h o
f ƒ
(x)
g(x
) is
the
gra
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
mer
Cos
tsD
uri
ng
her
fre
e ti
me,
Jil
l bab
y-si
ts
the
nei
ghbo
rhoo
d ch
ildr
en. S
he
char
ges
$4.5
0 fo
r ea
ch
wh
ole
hou
r or
an
y fr
acti
on o
f an
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
© G
lenc
oe/M
cGra
w-H
ill92
Ad
vanc
ed M
athe
mat
ical
Con
cep
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.
Answers (Lesson 3-3)
© Glencoe/McGraw-Hill A5 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill94
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Gra
ph
s o
f N
on
line
ar
Ine
qu
alit
ies
Det
erm
ine
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
lenc
oe/M
cGra
w-H
ill95
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
So
me
Pa
ram
etr
ic G
rap
hs
For
som
e cu
rves
, th
e co
ordi
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
ndi
tion
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 .
Ch
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
0.5
2428
148
482
9664
314
448
419
20
Gra
ph
eac
h c
urv
e.
1.x
�3t
, y
�2.
x�
t2�
1,
y�
t3�
112 �t
3-3
Answers (Lesson 3-4)
© Glencoe/McGraw-Hill A6 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill97
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Inve
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)�
3�x�
�2
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
© G
lenc
oe/M
cGra
w-H
ill98
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
3-4
VE
____
1728
HA
LF
____
____
229
627
IS
E__
____
3133
14O
NE
____
__20
1134
FY
____
367
RA
TI
O__
____
____
2416
1910
4E __ 18 L
EF
T__
____
__35
321
8
WH
EN
____
____
139
225
ST
RA
IG
HT
____
____
____
____
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
S
13
W15
H27
F32
T31
I
10
I22
E34
E
11
N23
A35
L
12
G24
R36
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.
Answers (Lesson 3-5)
© Glencoe/McGraw-Hill A7 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill10
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
oe/M
cGra
w-H
ill10
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
Answers (Lesson 3-6)
© Glencoe/McGraw-Hill A8 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
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
oe/M
cGra
w-H
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
3-6
Answers (Lesson 3-7)
© Glencoe/McGraw-Hill A9 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill10
7A
dva
nced
Mat
hem
atic
al C
once
pts
En
ric
hm
en
t
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
3-7
© G
lenc
oe/M
cGra
w-H
ill10
6A
dva
nced
Mat
hem
atic
al C
once
pts
Gra
ph
s o
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
____
____
____
___
PE
RIO
D__
____
__
3-7
Answers (Lesson 3-8)
© Glencoe/McGraw-Hill A10 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill11
0A
dva
nced
Mat
hem
atic
al C
once
pts
Re
ad
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
© G
lenc
oe/M
cGra
w-H
ill10
9A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
e
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
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
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© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts
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;
© Glencoe/McGraw-Hill A13 Advanced Mathematical Concepts
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→∞
© Glencoe/McGraw-Hill A14 Advanced Mathematical Concepts
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.
© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts
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
© Glencoe/McGraw-Hill A17 Advanced Mathematical Concepts
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