Grade 12 Pre-Calculus Mathematics (40S)
Grade 12 Precalculus Cover New:EN Cover Black Logo1/15/20088:44 AMPage 1Grade 12 Pre-CalculusMathematics (40S)A Course forIndependent Study
GRADE 12 PRE-CALCULUSMATHEMATICS (40S)A Course forIndependent Study2007Manitoba Education, Citizenship and Youth
Manitoba Education, Citizenship and Youth Cataloguing in Publication Data510Grade 12 pre-calculus mathematics (40S) : a course forindependent studyPreviously published as : Senior 4 pre-calculusmathematics (40S) : a course for distance learning.ISBN-13: 978-0-7711-3797-61. Mathematics—Programmed instruction.2. Calculus—Programmed instruction. 3. Mathematics—Study and teaching (Secondary). 4. Calculus—Studyand teaching (Secondary)—Manitoba. 5. Mathematics—Studyand teaching (Secondary)—Manitoba. 6. Calculus—Study andteaching (Secondary)—Manitoba. I. Manitoba. ManitobaEducation, Citizenship and Youth. II. Title: Senior 4pre-calculus mathematics (40S) : a course for distancelearning.Copyright 2007, the Crown in Right of Manitoba as represented by the Minister ofEducation, Citizenship and Youth.Manitoba Education, Citizenship and YouthSchool Programs Division1970 Ness AvenueWinnipeg, Manitoba R3J 0Y9.Every effort has been made to acknowledge original sources and to comply with copyrightlaw. If cases are identified where this has not been done, please notify Manitoba Education,Citizenship and Youth. Errors or omissions will be corrected in a future edition. Sincerethanks to the authors and publishers who allowed their original material to be adapted orreproduced.
Grade 12 Pre-Calculus toba Education, Citizenship and Youth gratefully acknowledges the contributions of thefollowing individuals in the development of Grade 12 Pre-Calculus Mathematics: A Course forIndependent Study (40S).WriterWilliam KorytowskiConsultantWinnipeg, ManitobaHarry DmytryshynConsultant(Field Validation Version)Winnipeg S.D. No. 1Carolyn Wilkinson(Final Version)Winnipeg, ManitobaContent EditorConsultantMembers of the Development TeamJohn BarsbyDave KlassenViviane LeonardJayesh ManierDon NicholHilliard SawchukDon TrimAlan WellsAbdou DaoudiKatharine TetlockWayne WattSt. John’s-Ravenscourt SchoolR.D. Parker CollegiateCollège BeliveauOak Park High SchoolWhitemouth SchoolConsultantUniversity of ManitobaConsultantBureau de l’éducationfrançaiseSchool Programs DivisionSchool Programs DivisionIndependent SchoolsMystery Lake S.D. No. 2355St. Boniface S.D. No. 4Assiniboine South S.D. No. 3Agassiz S.D. No. 13Winnipeg, ManitobaUniversity of ManitobaWinnipeg, ManitobaManitoba Education, Training and YouthManitoba Education, Training and YouthManitoba Education, Training and YouthManitoba Education, Training and Youth StaffSchool Programs DivisionCarole BilykProject Leader(Final Version)Curriculum UnitProgram Development BranchLee-Ila BotheCoordinatorProduction Support UnitProgram Development BranchPaul CuthbertProject Manager(Field Validation Version)Distance Learning andInformation Technologies UnitProgram Development BranchLynn HarrisonDesktop PublisherProduction Support UnitProgram Development BranchGilles LandryProject Manager(Final Version)Distance Learning andInformation Technologies UnitProgram Development BranchKatharine TetlockProject Leader(Field Validation Version)Distance Learning andInformation Technologies UnitProgram Development Branch
Grade 12 Pre-Calculus MathematicsContentsvContentsAcknowledgements iiiIntroductionIntroduction 3Formula Sheet 9Module Test Cover SheetsModule 1: TransformationsIntroduction 3Lesson 1: Translations 5Lesson 2: Reflections and Symmetry 13Lesson 3: Absolute Value and Piece-Wise Functions 29Lesson 4: Stretches and Compressions 39Lesson 5: Reciprocals 51Lesson 6: Combinations of Transformations 61Review 73Module 1 Answer KeyModule 2: Circular Functions IIntroduction 3Lesson 1: Trigonometric Values of Special Angles 5Lesson 2: The Unit Circle 13Lesson 3: The Circular Functions 21Lesson 4: Radian Measures 35Lesson 5: Sine and Cosine Graphs 47Lesson 6: Graphs of the Remaining Four Circular Functions 61Lesson 7: The Inverse Circular Functions 65Review 81Module 2 Answer KeyModule 3: Circular Functions IIIntroduction 3Lesson 1: Elementary Identities 5Lesson 2: Using Elementary Identities 15Lesson 3: Sum and Difference Identities 21Lesson 4: Double Angle Identities 29Lesson 5: Modelling with Trigonometric Functions 35Review 43Module 3 Answer Key
viContentsGrade 12 Pre-Calculus MathematicsModule 4: Exponential and Logarithmic Functions IIntroduction 3Lesson 1: Graphing Exponential Functions 5Lesson 2: A Special Exponential Function 13Lesson 3: The Logarithmic Function 19Lesson 4: An Algebraic Approach to Logarithms 25Lesson 5: The Logarithmic Theorems 31Lesson 6: Finding Logarithms Using a Calculator 39Lesson 7: Solving Exponential and Logarithmic Equations 45Review 51Module 4 Answer KeyModule 5: Exponential and Logarithmic Functions IIIntroduction 3Lesson 1: Applications of Exponential Functions 5Lesson 2: More Applications of Exponential Functions 15Lesson 3: Geometric Sequences 25Lesson 4: The Sum of a Geometric Series 33Lesson 5: The Sum of an Infinite Geometric Series 41Review 49Module 5 Answer KeyModule 6: Permutations and CombinationsIntroduction 3Lesson 1: Fundamental Counting Principle 5Lesson 2: The Formula nPr and Factorial Notation 15Lesson 3: Grouped Permutations and CircularPermutations 25Lesson 4: Like Objects 35Lesson 5: Combinations 41Lesson 6: Binomial Theorem 51Review 61Module 6 Answer Key
Grade 12 Pre-Calculus MathematicsContentsModule 7: ProbabilityIntroduction 3Lesson 1: Review of Probability Ideas 5Lesson 2: The Probability Laws 13Lesson 3: Two Special Cases 21Lesson 4: Conditional Probability 25Lesson 5: Practice Problems of Compound Events 31Lesson 6: Probabilities Using Permutations andCombinations 37Review 41Module 7 Answer KeyModule 8: Conic SectionsIntroduction 3Lesson 1: Review of Circles 5Lesson 2: The Ellipse and Hyperbola 11Lesson 3: Parabolas and Classifying the Conic Sections 23Lesson 4: Converting from General to Standard Form 29Review 37Module 8 Answer KeyModule TestsModule 1Module 2Module 3Module 4Module 5Module 6Module 7Module 8Hand-inHand-inHand-inHand-inModule Tests Answer KeyModule 2Module 4Module 6Module 8vii
viiiContentsNotesGrade 12 Pre-Calculus Mathematics
GRADE 12 PRE-CALCULUSMATHEMATICS (40S)Introduction
Grade 12 Pre-Calculus MathematicsIntroduction3Grade 12 Pre-Calculus MathematicsIntroductionWelcome to the Grade 12 Pre-Calculus Mathematics course fordistance learning offered through the School Programs Division,Manitoba Education, Citizenship and Youth. It is expectedthat you have successfully completed Grade 11 PreCalculus Mathematics.As a student in a course for distance learning, you have takenon a dual role—that of a student and a teacher. As a student,you are responsible for mastering the lessons and completingthe exercises assigned at the end of each lesson. As a teacher,you are responsible for checking your work carefully and notingthe nature of your errors. Finally, you must work diligently toovercome your difficulties.You should seek out a study partner for this course. Moststudents find that a study partner helps them get through thecourse with greater success. This study partner can help youcorrect your assignments and module self-tests, as well as helpyou prepare for the examinations.Grade 12 Pre-Calculus Mathematics is one of four possibleGrade 12 mathematics courses in Manitoba (the other threecourses are Grade 12 Applied Mathematics, Grade 12 ConsumerMathematics, and Grade 12 Accounting Systems). This coursecontains many new and interesting topics in mathematics. Youwill need to use many of the skills and procedures that you havealready learned to solve some of the problems you will find inthe exercises. On completion of this course, successful studentswill be well prepared to study post-secondary mathematics.This course is divided into eight modules. Each module containslessons that are followed by assignments. It is recommendedthat you complete all of the assigned exercises. Answerkeys are provided to the exercises and are found at the end ofeach module.
4IntroductionGrade 12 Pre-Calculus MathematicsThe eight modules are as follows:Module 1: TransformationsModule 2: Circular Functions IModule 3: Circular Functions IIModule 4: Exponential and Logarithmic Functions IModule 5: Exponential and Logarithmic Functions IIModule 6: Permutations and CombinationsModule 7: ProbabilityModule 8: Conic SectionsThe table of contents outlines the topics found in this course.Every student enrolled in Grade 12 Pre-Calculus Mathematicsis required to complete all eight modules. Each module endswith a test.These tests should be written without the aid of any books. Yourperformance on these tests will give you an indication of howwell you understand the material. Your study partner can helpyou by marking some of these tests. Answer keys for ModuleTests 2, 4, 6, and 8 are provided in the Module Tests AnswerKeys section at the end of the course. You should then correctall errors and use these tests to help you prepare for themidterm and final examinations. Note that Module Tests 1, 3, 5,and 7 are to be sent to the Tutor/Marker as soon as each one iscompleted. Therefore, no answer keys are provided for thesetests.Tutor/MarkerThe person who marks your tests and exams is your tutor/marker. This person is also available to help you with yourlearning. When you register for the course, you will receive aletter that gives you the name and contact information for thetutor/marker for Grade 12 Pre-Calculus Mathematics. Pleasetake advantage of all the resources provided by the IndependentStudy Office.Note:Module Tests 1, 3, 5,and 7 are to be sent tothe Tutor/Marker assoon as each one iscompleted.Address:ISO Tutor/Marker555 Main StreetWINKLER MBR6W 1C4Your Tutor/Markerwill review the resultswith you.
Grade 12 Pre-Calculus MathematicsIntroduction5Calculator UseYou will need a scientific calculator for this course. A graphingcalculator may be helpful but it is not necessary. Referencesmade to the graphing calculator in the course are in optionalsections. Many of the exercises ask for exact answers wherecalculator use is not required.You are permitted to use a scientific calculator for all tests andexams. You are not asked questions requiring a graphingcalculator.Formula SheetA formula sheet is provided for the exams. You may also use itfor your tests. A copy of the formula sheet is included at the endof this introductory section.EvaluationYour final mark in this course will be based on the results offour hand-in tests, and two examinations: a midterm and afinal.The value of these tests and examinations are shown below.First Term:Hand-in Tests 1 and 3, 10% eachMidterm Examination after Module 4(Based on Modules 1 through 4)20%20%Second Term:Hand-in Tests 5 and 7, 10% eachFinal Examination after Module 8(Based on Modules 1 through 8)20%40%Total100%The module tests after Modules 1, 3, 5, and 7 must becompleted, and then sent in to your Tutor/Marker. You shouldalso note that the final examination is cumulative, meaningthat it is based on the entire course.You are required to send a coloured cover sheet with eachhand-in test. Cover sheets can be found after page 6 of theIntroduction.
6IntroductionGrade 12 Pre-Calculus MathematicsGuide GraphicsGraphics have been placed inside the margins of the course toidentify a specific task. Each graphic has a specific purpose tohelp guide you.The significance of each guide graphic is described below.Assignment: You are required to do theassignment questions that accompany thisgraphic.Note: This graphic will appear when there is adirection or explanation that you should notecarefully.Test Time: This graphic alerts students that itis time to write a test or prepare for a test.Send In: This graphic indicates that you mustsend in the assignment or self-test forcorrecting.Study/Review: This graphic is to remind youthat you should review your material for a testor examination.Check: Check your answers against AnswerKey provided for this lesson.Exam Time: When this graphic appears, it istime to write an examinationCautionary NoteSome of the activities in this course involve chance andprobability. In some families and communities, the connectionbetween probability and gambling may be problematic; forexample, parents/guardians may not approve of playing cards,dice, or prize money. As an alternative, students can usenumbered index cards, number cubes, or points or credits.
Grade 12 Pre-Calculus MathematicsIntroduction7Applying for Exams If you are attending school, ask your Independent StudyOption (ISO) facilitator to add your name to the ISOexamination eligibility list. Do this at least three weeks priorto the next scheduled examination week. If you are not attending school, check the ExaminationRequest Form for options available to you. Fill in this formand mail or fax three weeks before you are ready to write theGrade 12 Pre-Calculus Mathematics Midterm or FinalExamination. The address is:ISO Tutor/Marker555 Main StreetWINKLER MB R6W 1C4Fax: 204-325-1719Contact InformationUse the following mailing address for any materials that youare forwarding to your tutor/marker:Distance Learning and InformationTechnologies Unit555 Main StreetWINKLER MB R6W 1C4Telephone: (204) 325-1700Toll-free: (800) 465-9915Fax: (204) 325-1719Website: www.edu.gov.mb.ca/k12/dl/iso/index.html This website contains information concerning policies andprocedures for the Independent Study Option as well as otheruseful information. It also includes a forms section where youcan download the application form for writing exams.
8IntroductionNotesGrade 12 Pre-Calculus Mathematics
Grade 12 Pre-Calculus MathematicsIntroduction9Formula SheetSenior 4 Pre-Calculus Mathematics (40S)s θrsin 2 θ cos 2 θ 1tan 2 θ 1 sec 2 θ r A P 1 n A Pe rtnte 2.718 281 cot θ csc θ22sin (α β ) sin α cos β cos α sin βcos (α β ) cos α cos β sin α sin βtan (α β ) tan α tan β1 tan α tan βlog a ( MN ) log a M log a N M log a log a M log a N N log a ( M n ) n log a Mlog b Mlog b alog a M sin (α β ) sin α cos β cos α sin βcos (α β ) cos α cos β sin α sin βtan (α β ) tan α tan β1 tan α tan βsin 2α 2 sin α cos αcos 2α cos 2 α sin 2 α( x h) ( y k ) r222( x h ) ( y k ) 1, a b2a2b2( x h)cos 2α 2 cos 2 α 1a2n!( n r )!C ( n, r ) or nCr n!r !( n r )!tk 1 nCk a n k b kP ( A or B ) P ( A ) P ( B ) P ( A and B )P ( A and B ) P ( A ) P ( B A )(y k) 2(y k) 2( x h)2 1, a ba2( x h)P ( n, r ) or n Pr 2b2cos 2α 1 2 sin 2 α2 tan αtan 2α 1 tan 2 α22 1b2(y k)a22 1b2( y k ) a ( x h)2( x h) a ( y k )2tn t1r n 1Sn t1 (1 r n )1 r Sn t1 tn r1 rS t1, r 11 rt1 ( r n 1)r 1
Grade 12 Pre-Calculus MathematicsModule 1, Lesson 15Lesson 1TranslationsOutcomesUpon completing this lesson, you will be able to describe how a translation affects the graph and properties ofa function sketch the translation of a function state the translation that produced a new sketch from thegiven sketchOverviewYou have encountered translations when you sketched thefollowing three functions in Grade 11 Pre-CalculusMathematics.yyxx1-1y x2- 1y x2y1x1y (x - 1)2
6Module 1, Lesson 1Grade 12 Pre-Calculus MathematicsDefining Translations2In y x – 1, each y-value is 1 less than the corresponding22y-value in y x . Therefore the graph of y x – 1 is 1 unit lower22than the graph of y x . We say that the graph of y x has2been translated 1 unit down. The function y x – 1 is a2vertical translation of the function, y x .22Similarly, y (x – 1) is a horizontal translation of y x .2The graph of y x has been shifted one unit to the right.Definition: A translation is a transformation of a geometricfigure in which every point is moved the same distance in thesame direction.Example 12Use the sketch of y x to sketch2a) y x 2b) y (x 2)2Note: A graphing calculator is not required for any part of thiscourse.Solutiona)b)yy43432211x-3 -2 -12As evident from the graphs, the sketch of y x 2 is a vertical2translation of two units up, and the graph of y (x 2) is ahorizontal translation of two units to the left of the graph of2y x.Notice that with the vertical translation the “2” is outside thebrackets, and affects the y-values by an amount of 2. With thehorizontal translation, the “2” is inside the brackets and affectsthe x-values, not by an amount of 2 but rather by –2.x
Grade 12 Pre-Calculus MathematicsModule 1, Lesson 17Vertical translations are often easier and more natural tounderstand. Is there some way to remember in which directionto shift a horizontal translation? One suggestion is to follow thevertex or the intercepts, whichever is more convenient. In theabove examples, the smallest value of a squared quantity iszero, thus producing the vertex. Let’s follow the vertex as shownin the chart below.QuadraticEquationy xThe Value of xWhich Makes y 0Effects onGraph0basic graph2–22 units left222 units right2y (x 2)y (x – 2)Example 23Use the given sketch of f(x) x to sketch the followingfunctions. In this example, there is no vertex. Try following thex-intercept instead. Remember that the x-intercepts are alsocalled the zeros of the function.yf(x) x32113a) g(x) x – 2b) h(x) (x – 1)33c) k(x) (x 3)2x
8Module 1, Lesson 1Grade 12 Pre-Calculus MathematicsSolutionya)b)g(x) x3 - 21h(x) (x - 1)3yx2-1-21-1yc)21-3-2-112x-1-2k(x) (x 3)3Example 3Use the given sketch of f(x) to sketch the following functions.Again, try translating the x- and y-intercepts as a tool formaking the graphing easier.yf(x)11-1-1a) g(x) f(x 3)b) m(x) f(x) 2c) n(x) f(x – 5)xx
Grade 12 Pre-Calculus MathematicsModule 1, Lesson 19Solutionya)g(x)1-4 Given a function, f(x), the effect of a translation on f(x) issummarized in the chart below.TranslationEffect on GraphEffect on (x, y)f(x) kVertical translation of k units: up if k 0 down if k 0(x, y k)f(x – h)Horizontal translation of h units: to the left if h 0 to the right if h 0(x h, y)
10Module 1, Lesson 1Grade 12 Pre-Calculus MathematicsAssignment1. Given the sketch of f(x) drawn below, sketch each of thefollowing functions.a) f(x – 5)b) f(x) – 5c) f(x) 5d) f(x 5)e) f(x – 5) – 5f) f(x 5) 5y(1, 1)1f(x)lx122. Let f(x) x 2. Sketch each of the following functions.a) f(x)b) f(x) – 6c) f(x 1)d) f(x – 2) – 33. For each of the functions in Question 2 state the propertiesof the function: the domain, the range, and the values of theintercepts. (Recall: To find the y-intercept, let x 0 andsolve for y. To find the x-intercept, let y 0 and solve for x.For more practice, see Lesson 3.)4. Each of graphs (a), (b), and (c) represents a translation of thegiven function, g(x), drawn below. Write an expression foreach new function in terms of g(x).y1(1, 1)l12xg(x)
Grade 12 Pre-Calculus MathematicsModule 1, Lesson 1a)11y1-1b)1xc)yy(1, 2)2l11-131x-11x25. If f(x) x 3x – x 6, write an unsimplified equation fora) g(x) which has the same graph as f(x) moved two units tothe left.b) h(x) which has the same graph as f(x) moved three unitsdown.c) m(x) which has the same graph as f(x) moved two units tothe right and one unit up.6. How are the graphs of g ( x ) xx 3and n ( x ) related ?xx 3(For more practice, see Lesson 3, Assignment, questions 3, 4,and 5.)
12Module 1, Lesson 13Grade 12 Pre-Calculus Mathematics7. Below is the graph of f(x) x – x. Sketch the graph of3g(x) (x 2) – (x 2).yf(x)1-1-11x8. Is the translation of a function still a function?Check your answers in the Module 1 Answer Key.
Grade 12 Precalculus Cover New:EN Cover Black Logo 1/15/2008 8:44 AM Page 1. GRADE 12 PRE-CALCULUS MATHEMATICS (40S) A Course for Independent Study 2007 Manitoba Education, Citizenship and Youth. Manitoba Educati
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