Chapter 4 Resource Masters
Chapter 4Resource MastersNew York, New YorkColumbus, OhioWoodland Hills, CaliforniaPeoria, Illinois
StudentWorksTM This CD-ROM includes the entire Student Edition along with theStudy Guide, Practice, and Enrichment masters.TeacherWorksTM All of the materials found in this booklet are included for viewingand printing in the Advanced Mathematical Concepts TeacherWorksCD-ROM.Copyright The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproducethe material contained herein on the condition that such material be reproduced onlyfor classroom use; be provided to students, teachers, and families without charge;and be used solely in conjunction with Glencoe Advanced Mathematical Concepts.Any other reproduction, for use or sale, is prohibited without prior writtenpermission of the publisher.Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027ISBN: 0-07-869131-11 2 3 4 5 6 7 8 9 10Advanced Mathematical ConceptsChapter 4 Resource MastersXXX11 10 09 08 07 06 05 04
ContentsVocabulary Builder . . . . . . . . . . . . . . . . vii-ixLesson 4-7Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 149Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 151Lesson 4-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 131Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 133Lesson 4-8Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 152Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 154Lesson 4-2Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 134Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 136Chapter 4 AssessmentChapter 4 Test, Form 1A . . . . . . . . . . . . 155-156Chapter 4 Test, Form 1B . . . . . . . . . . . . 157-158Chapter 4 Test, Form 1C . . . . . . . . . . . . 159-160Chapter 4 Test, Form 2A . . . . . . . . . . . . 161-162Chapter 4 Test, Form 2B . . . . . . . . . . . . 163-164Chapter 4 Test, Form 2C . . . . . . . . . . . . 165-166Chapter 4 Extended ResponseAssessment . . . . . . . . . . . . . . . . . . . . . . . 167Chapter 4 Mid-Chapter Test . . . . . . . . . . . . . 168Chapter 4 Quizzes A & B . . . . . . . . . . . . . . . 169Chapter 4 Quizzes C & D. . . . . . . . . . . . . . . 170Chapter 4 SAT and ACT Practice . . . . . 171-172Chapter 4 Cumulative Review . . . . . . . . . . . 173Unit 1 Review . . . . . . . . . . . . . . . . . . . . 175-176Unit 1 Test . . . . . . . . . . . . . . . . . . . . . . . 177-180Lesson 4-3Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 137Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 139Lesson 4-4Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 140Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 142Lesson 4-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 143Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 145Lesson 4-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 146Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Glencoe/McGraw-HillSAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A20iiiAdvanced Mathematical Concepts
A Teacher’s Guide to Using theChapter 4 Resource MastersThe Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 4 Resource Masters include the corematerials needed for Chapter 4. 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 aPractice There is one master for each lesson.student study tool that presents the keyvocabulary 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.These problems more closely follow thestructure of the Practice section of the StudentEdition exercises. These exercises are ofaverage difficulty.When to Use These provide additionalpractice options or may be used as homeworkfor second day teaching of the lesson.When to Use Give these pages to studentsbefore beginning Lesson 4-1. Remind them toadd definitions and examples as they completeeach 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 arelearning. These are not written exclusivelyfor honors students, but are accessible for usewith all levels of students.Study Guide There is one Study Guidemaster for each lesson.When to Use Use these masters asreteaching activities for students who needadditional reinforcement. These pages can alsobe used in conjunction with the Student Editionas an instructional tool for those students whohave been absent. Glencoe/McGraw-HillWhen to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.ivAdvanced Mathematical Concepts
Assessment OptionsIntermediate AssessmentThe assessment section of the Chapter 4Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.Chapter Tests Forms 1A, 1B, and 1C Form 1 tests containmultiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with averagelevel 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.The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoringrubric is included for evaluation guidelines.Sample answers are provided forassessment. Glencoe/McGraw-Hill Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter. The SAT and ACT Practice offerscontinuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein 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.AnswersAll of the above tests include a challengingBonus question. A Mid-Chapter Test provides an option toassess the first half of the chapter. It iscomposed of free-response questions.Continuing AssessmentChapter Assessments v Page A1 is an answer sheet for the SAT andACT Practice questions that appear in theStudent Edition on page 273. 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-lessonmasters are provided as reduced pages withanswers appearing in red. Full-size answer keys are provided for theassessment options in this booklet.Advanced Mathematical Concepts
Chapter 4 Leveled WorksheetsGlencoe’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 whoare moving at a regular pace. Enrichment masters offer students the opportunity to extend theirlearning.Five Different Options to Meet the Needs ofEvery Student in a Variety of Waysprimarily skillsprimarily conceptsprimarily applicationsBASICAVERAGE1Study Guide2Vocabulary Builder3Parent and Student Study Guide (online) dvanced Mathematical Concepts
NAME DATE PERIODChapter4Reading to Learn MathematicsVocabulary BuilderThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.Vocabulary TermFoundon PageDefinition/Description/Examplecompleting the squarecomplex numberconjugatedegreedepressed polynomialDescartes’ Rule of Signsdiscriminantextraneous solutionFactor TheoremFundamental Theorem of Algebra(continued on the next page) Glencoe/McGraw-HillviiAdvanced Mathematical Concepts
NAME DATE PERIODChapter4Reading to Learn MathematicsVocabulary Builder (continued)Vocabulary TermFoundon PageDefinition/Description/Exampleimaginary numberIntegral Root Theoremleading coefficientLocation Principlelower boundlower Bound Theorempartial fractionspolynomial equationpolynomial functionpolynomial in one variablepure imaginary number(continued on the next page) Glencoe/McGraw-HillviiiAdvanced Mathematical Concepts
NAME DATE PERIODChapter34Reading to Learn MathematicsVocabulary Builder (continued)Vocabulary TermFoundon PageDefinition/Description/ExampleQuadratic Formularadical equationradical inequalityrational equationrational inequalityRational Root theoremRemainder Theoremsynthetic divisionupper boundUpper Bound Theoremzero Glencoe/McGraw-HillixAdvanced Mathematical Concepts
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NAME DATE PERIOD4-1Study GuidePolynomial FunctionsThe degree of a polynomial in one variable is the greatest exponent ofits variable. The coefficient of the variable with the greatest exponentis called the leading coefficient. If a function ƒ(x) is defined by apolynomial in one variable, then it is a polynomial function. The valuesof x for which ƒ(x) 0 are called the zeros of the function. Zeros of thefunction are roots of the polynomial equation when ƒ(x) 0. Apolynomial equation of degree n has exactly n roots in the set ofcomplex numbers.Example 1State the degree and leading coefficient of the polynomial 2 is afunction ƒ(x) 6x5 8x3 8x. Then determine whether 3zero of ƒ(x).6x5 8x3 8x has a degree of 5 and a leading coefficient of 6.2 . That is, find ƒ 2 .Evaluate the function for x 3 32 6 2 8 2 8 2 ƒ 3 3 3 3 532 x 32 1 6 2 8 2 294 333 3 02 0, 2 is a zero of ƒ(x) 6x5 8x3 8x.Since ƒ 33 Example 2 i,Write a polynomial equation of least degree with roots 0, 2and 2 i.The linear factors for the polynomial are x 0, x 2 i, and x 2 i.Find the products of these factors. i)(x 2 i) (x 0)(x 22 2i2) x(xx(x2 2) x3 2x Example 3000 2i2 2( 1) or 20State the number of complex roots of theequation 3x2 11x 4 0. Then find the roots.The polynomial has a degree of 2, so there are twocomplex roots. Factor the equation to find the roots.3x2 11x 4 0(3x 1)(x 4) 0To find each root, set each factor equal to zero.3x 1 0x 4 03x 1x 41x 3 The roots are 4 and 13 . Glencoe/McGraw-Hill131Advanced Mathematical Concepts
NAME DATE PERIOD4-1PracticePolynomial FunctionsState the degree and leading coefficient of each polynomial.1. 6a4 a3 2a2. 3p2 7p5 2p3 5Write a polynomial equation of least degree for each set of roots.3. 3, 0.5, 14. 3, 3, 1, 1, 25. 2i, 3, 36. 1, 3 i, 2 3iState the number of complex roots of each equation. Then findthe roots and graph the related function.7. 3x 5 08. x2 4 09. c 2 2c 1 010. x3 2x2 15x 011. Real Estate A developer wants to build homes on a rectangular plot of land 4 kilometers long and 3 kilometers wide. In thispart of the city, regulations require a greenbelt of uniform widthalong two adjacent sides. The greenbelt must be 10 times thearea of the development. Find the width of the greenbelt. Glencoe/McGraw-Hill132Advanced Mathematical Concepts
NAME DATE PERIOD4-1EnrichmentGraphic AdditionOne way to sketch the graphs of some polynomialfunctions is to use addition of ordinates. This method isuseful when a polynomial function f (x) can be writtenas the sum of two other functions, g(x) and h(x), thatare easier to graph. Then, each f (x) can be found bymentally adding the corresponding g(x) and h(x). Thegraph at the right shows how to construct the graph of1 31 31 2x – 8 from the graphs of g(x) – xf(x) – x 222and h(x) 1 x22– 8.In each problem, the graphs of g(x) and h(x) are shown. Use addition of ordinates tograph a new polynomial function f(x), such that f(x) g(x) h(x). Then write theequation for f(x).1.2.3.4.5.6. Glencoe/McGraw-Hill133Advanced Mathematical Concepts
NAME DATE PERIOD4-2Study GuideQuadratic EquationsA quadratic equation is a polynomial equation with a degree of 2.Solving quadratic equations by graphing usually does not yieldexact answers. Also, some quadratic expressions are not factorable.However, solutions can always be obtained by completing thesquare.Example 1Solve x2 12x 7 0 by completing the square.x2 12x 7 0Subtract 7 from each side.x2 12x 72Completethe square by adding 12 ( 12) ,2x 12x 36 7 36or 36, to each side.Factor the perfect square trinomial.(x 6)2 29 9 Take the square root of each side.x 6 2 9 Add 6 to each side.x 6 2The roots of the equation are 6 2 9 .Completing the square can be used to develop a general formula forsolving any quadratic equation of the form ax2 bx c 0. Thisformula is called the Quadratic Formula and can be used to findthe roots of any quadratic equation.Quadratic Formula b 2 2ab 4ac .If ax2 bx c 0 with a 0, x In the Quadratic Formula, the radicand b2 4ac is called thediscriminant of the equation. The discriminant tells thenature of the roots of a quadratic equation or the zeros of therelated quadratic function.Example 2Find the discriminant of 2x2 3x 7 anddescribe the nature of the roots of the equation.Then solve the equation by using the QuadraticFormula.Rewrite the equation using the standard form ax2 bx c 0.2x2 3x 7 0 a 2, b 3, and c 7The value of the discriminant b2 4ac is( 3)2 4(2)( 7), or 65.Since the value of the discriminant is greater thanzero, there are two distinct real roots.Now substitute the coefficients into the quadraticformula and solve. 3 ) (24 )( 7 ) ( 3) ( x 2(2)2 b b 2 a4 c x 2a3 6 5 x 43 6 5 and 3 6 5 . The roots are 44 Glencoe/McGraw-Hill134Advanced Mathematical Concepts
NAME DATE PERIOD4-2PracticeQuadratic EquationsSolve each equation by completing the square.1. x2 5x 141 02. 4x2 11x 7Find the discriminant of each equation and describe the nature ofthe roots of the equation. Then solve the equation by using theQuadratic Formula.4. 4x2 4x 15 03. x2 x 6 05. 9x2 12x 4 06. 3x2 2x 5 0Solve each equation.7. 2x2 5x 12 08. 5x2 14x 11 09. Architecture The ancient Greek mathematicians thought thatthe most pleasing geometric forms, such as the ratio of the heightto the width of a doorway, were created using the golden section.However, they were surprised to learn that the golden section is nota rational number. One way of expressing the golden section isA B AC . Ifby using a line segment. In the line segment shown, ACCBAB .AC 1 unit, find the ratio AC Glencoe/McGraw-Hill135Advanced Mathematical Concepts
NAME DATE PERIOD4-2EnrichmentConjugates and Absolute ValueWhen studying complex numbers, it is often convenient to representa complex number by a single variable. For example, we might letz x yi. We denote the conjugate of z by z. Thus, z x yi.We can define the absolute value of a complex number as follows.2 y 2 z x yi x There are many important relationships involving conjugates andabsolute values of complex numbers.Show that z2 zz for any complex number z.Let z x yi. Then,Examplezz (x yi)(x yi) x 2 y22 x y 2 2 z 2z Show thatis the multiplicative inverse for any2Example z nonzero complex number z.We know that z 2 zz . If z 0, then we have z z z z 1. Thus,is the multiplicative22 z inverse of z.For each of the following complex numbers, find the absolutevalue and multiplicative inverse.1. 2i2. –4 3i3. 12 5i4. 5 12i5. 1 i i6. 37. 3 3 3 i3 Glencoe/McGraw-Hill8. 2 2 2 i21369.1 2 3 i2Advanced Mathematical Concepts
NAME DATE PERIOD4-3Study GuideThe Remainder and Factor TheoremsThe RemainderTheoremExample 1If a polynomial P(x) is divided by x r, the remainder is a constant P(r ),and P(x) (x r ) Q(x) P(r )where Q(x) is a polynomial with degree one less than the degree of P(x).Divide x4 5x2 17x 12 by x 3.x3 3x2 4x 29 4 x0 3 x5 2 71 x 21 x 3 x43x 3x 3x3 5x2 3x3 9x24x2 17x4x2 12x 29x 12 29x 8775 remainderFind the value of rin this division.x r x 3 r 3r 3According to theRemainder Theorem,P(r) or P( 3) shouldequal 75.Use the Remainder Theorem to check the remainder found bylong division.P(x) x4 5x2 17x 12P( 3) ( 3)4 5( 3)2 17( 3) 12 81 45 51 12 or 75The Factor Theorem is a special case of the RemainderTheorem and can be used to quickly test for factors ofa polynomial.The FactorTheoremExample 2The binomial x r is a factor of the polynomial P(x) if and only if P(r) 0.Use the Remainder Theorem to find theremainder when 2x3 5x2 14x 8 is divided byx 2. State whether the binomial is a factor ofthe polynomial. Explain.Find ƒ(2) to see if x 2 is a factor.ƒ(x) 2x3 5x2 14x 8ƒ(2) 2(2)3 5(2)2 14(2) 8 16 20 28 8 0Since ƒ(2) 0, the remainder is 0. So thebinomial x 2 is a factor of the polynomialby the Factor Theorem. Glencoe/McGraw-Hill137Advanced Mathematical Concepts
NAME DATE PERIOD4-3PracticeThe Remainder and Factor TheoremsDivide using synthetic division.1. (3x2 4x 12) (x 5)2. (x2 5x 12) (x 3)3. (x 4 3x2 12) (x 1)4. (2x3 3x2 8x 3) (x 3)Use the Remainder Theorem to find the remainder for each division.State whether the binomial is a factor of
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term.
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .
Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the Chapter 1 Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 1 Resource Masters includes the core materials needed for Chapter 1. These materials include worksheets, extensions, and assessment options.
Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 6 Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options.
Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the Chapter 7 Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 7 Resource Masters includes the core materials needed for Chapter 7. These materials include worksheets, extensions, and assessment options.
Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 3 Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 3 Resource Masters includes the core materials needed for Chapter 3. These materials include worksheets, extensions, and assessment options.
Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 9 Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter 9 Resource Masters includes the core materials needed for Chapter 9. These materials include worksheets, extensions, and assessment options.
