GLOBAL EDITION Precalculus Concepts Through Functions A Unit Circle Approach to Trigonometry THIRD EDITION .JDIBFM4VMMJWBO*** www.com Precalculus Concepts Through Functions A Unit Circle Approach To Trigonometry Third Edition Global Edition Michael Sullivan Chicago State University Michael Sullivan, lll Joliet Junior College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.com QA Manager, Assessment Content: Marty Wright Editor in Chief: Anne Kelly Senior Marketing Manager: Michelle Cook Acquisitions Editor: Dawn Murrin Marketing Manager: Peggy Sue Lucas Assistant Editor: Joseph Colella Marketing Assistant: Justine Goulart Senior Managing Editor: Karen Wernholm Senior Author Support/Technology Specialist: Joe Vetere Associate Managing Editor: Tamela Ambush Procurement Manager: Vincent Scelta Senior Production Project Manager: Peggy McMahon Procurement Specialist: Debbie Rossi Digital Assets Manager: Marianne Groth Text Design: Tamara Newnam Associate Media Producer: Marielle Guiney Production Coordination, Associate Director of Design, Head, Learning Asset Acquisition, Global Edition: Laura Dent USHE EMSS/HSC/EDU: Andrea Nix Assistant Acquisitions Editor, Global Edition: Aditee Agarwal Image Manager: Rachel Youdelman Asistant Project Editor, Global Edition: Mrithyunjayan Nilayamgode Photo Research: Integra, Inc. 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The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequen- tial damages in connection with, or arising out of, the furnishing, performance, or use of these programs. ISBN 10: 1-292-05874-9 ISBN 13: 978-1-292-05874-0 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 Typeset in Times Ten by Cenveo® Publisher Services Printed and bound by Courier Kendallville in The United States of America www.com For Michael S., Kevin, and Marissa (Sullivan) Shannon, Patrick, and Ryan (Murphy) Maeve, Sean, and Nolan (Sullivan) Kaleigh, Billy, and Timmy (O’Hara) The Next Generation www.com Contents To the Student 15 Preface to the Instructor 17 Prepare for Class ‘‘Read the Book’’ 21 Practice ‘‘Work the Problems’’ 22 Review ‘‘Study for Quizzes and Tests’’ 23 Resources for Success 24 Applications Index 25 F Foundations: A Prelude to Functions 33 F.1 The Distance and Midpoint Formulas 34 6TFUIF%JTUBODF'PSNVMBr6TFUIF. 2 Graphs of Equations in Two Variables; Intercepts; Symmetry 41 (SBQI&RVBUJPOTCZ1MPUUJOH1PJOUTr'JOE*OUFSDFQUTGSPNB(SBQIr'JOE *OUFSDFQUTGSPNBO&RVBUJPOr5FTUBO&RVBUJPOGPS4ZNNFUSZr,OPX)PX to Graph Key Equations F. 3 Lines 51 $BMDVMBUFBOE*OUFSQSFUUIF4MPQFPGB-JOFr(SBQI-JOFT(JWFOB1PJOU BOEUIF4MPQFr'JOEUIF&RVBUJPOPGB7FSUJDBM-JOFr6TFUIF1PJOU4MPQF 'PSNPGB-JOF *EFOUJGZ)PSJ[POUBM-JOFTr'JOEUIF&RVBUJPOPGB-JOF (JWFO5XP1PJOUTr8SJUFUIF&RVBUJPOPGB-JOFJO4MPQF*OUFSDFQU'PSN r*EFOUJGZUIF4MPQFBOEy*OUFSDFQUPGB-JOFGSPN*UT&RVBUJPOr(SBQI -JOFT8SJUUFOJO(FOFSBM'PSN6TJOH*OUFSDFQUTr'JOE&RVBUJPOTPG 1BSBMMFM-JOFTr'JOE&RVBUJPOTPG1FSQFOEJDVMBS-JOFT F. 4 Circles 66 8SJUFUIF4UBOEBSE'PSNPGUIF&RVBUJPOPGB$JSDMFr(SBQIB$JSDMF r8PSLXJUIUIF(FOFSBM'PSNPGUIF&RVBUJPOPGB$JSDMF Chapter Project 73 1 Functions and Their Graphs 74 1.1 Functions 75 %FUFSNJOF8IFUIFSB3FMBUJPO3FQSFTFOUTB'VODUJPOr'JOEUIF7BMVFPGB 'VODUJPOr'JOEUIF%PNBJOPGB'VODUJPO%FGJOFECZBO&RVBUJPOr'PSN the Sum, Difference, Product, and Quotient of Two Functions 1.2 The Graph of a Function 88 *EFOUJGZUIF(SBQIPGB'VODUJPOr0CUBJO*OGPSNBUJPOGSPNPSBCPVUUIF Graph of a Function 1.3 Properties of Functions 98 %FUFSNJOF&WFOBOE0EE'VODUJPOTGSPNB(SBQIr*EFOUJGZ&WFOBOE 0EE'VODUJPOTGSPNUIF&RVBUJPOr6TFB(SBQIUP%FUFSNJOF8IFSFB 'VODUJPOJT*ODSFBTJOH %FDSFBTJOH PS$POTUBOUr6TFB(SBQIUP-PDBUF -PDBM.BYJNBBOE-PDBM.JOJNBr6TFB(SBQIUP-PDBUFUIF"CTPMVUF .BYJNVNBOEUIF"CTPMVUF.JOJNVNr6TFB(SBQIJOH6UJMJUZUP"QQSPYJNBUF Local Maxima and Local Minima and to Determine Where a Function is *ODSFBTJOHPS%FDSFBTJOHr'JOEUIF"WFSBHF3BUFPG$IBOHFPGB'VODUJPO 5 www.4 Library of Functions; Piecewise-defined Functions 110 (SBQIUIF'VODUJPOT-JTUFEJOUIF-JCSBSZPG'VODUJPOTr(SBQI Piecewise-defined Functions 1.5 Graphing Techniques: Transformations 121 (SBQI'VODUJPOT6TJOH7FSUJDBMBOE)PSJ[POUBM4IJGUTr(SBQI'VODUJPOT 6TJOH$PNQSFTTJPOTBOE4USFUDIFTr(SBQI'VODUJPOT6TJOH3FGMFDUJPOT about the x-Axis and the y-Axis 1.6 Mathematical Models: Building Functions 133 Build and Analyze Functions 1.7 Building Mathematical Models Using Variation 138 $POTUSVDUB.PEFM6TJOH%JSFDU7BSJBUJPOr$POTUSVDUB.PEFM6TJOH *OWFSTF7BSJBUJPOr$POTUSVDUB.PEFM6TJOH+PJOUPS$PNCJOFE7BSJBUJPO Chapter Review 143 Chapter Test 147 Chapter Projects 148 2 Linear and Quadratic Functions 150 2.1 Properties of Linear Functions and Linear Models 151 (SBQI-JOFBS'VODUJPOTr6TF"WFSBHF3BUFPG$IBOHFUP*EFOUJGZ-JOFBS 'VODUJPOTr%FUFSNJOF8IFUIFSB-JOFBS'VODUJPO*T*ODSFBTJOH %FDSFBTJOHPS$POTUBOUr'JOEUIF;FSPPGB-JOFBS'VODUJPOr#VJME-JOFBS Models from Verbal Descriptions 2.2 Building Linear Models from Data 162 %SBXBOE*OUFSQSFU4DBUUFS%JBHSBNTr%JTUJOHVJTICFUXFFO-JOFBSBOE /POMJOFBS3FMBUJPOTr6TFB(SBQIJOH6UJMJUZUP'JOEUIF-JOFPG#FTU'JU 2.3 Quadratic Functions and Their Zeros 169 'JOEUIF;FSPTPGB2VBESBUJD'VODUJPOCZ'BDUPSJOHr'JOEUIF;FSPTPGB 2VBESBUJD'VODUJPO6TJOHUIF4RVBSF3PPU.FUIPEr'JOEUIF;FSPTPGB 2VBESBUJD'VODUJPOCZ$PNQMFUJOHUIF4RVBSFr'JOEUIF;FSPTPGB2VBESBUJD 'VODUJPO6TJOHUIF2VBESBUJD'PSNVMBr'JOEUIF1PJOUPG*OUFSTFDUJPOPG 5XP'VODUJPOTr4PMWF&RVBUJPOT5IBU"SF2VBESBUJDJO'PSN 2.4 Properties of Quadratic Functions 180 (SBQIB2VBESBUJD'VODUJPO6TJOH5SBOTGPSNBUJPOTr*EFOUJGZUIF7FSUFY BOE"YJTPG4ZNNFUSZPGB2VBESBUJD'VODUJPOr(SBQIB2VBESBUJD 'VODUJPO6TJOH*UT7FSUFY "YJT BOE*OUFSDFQUTr'JOEB2VBESBUJD 'VODUJPO(JWFO*UT7FSUFYBOE0OF0UIFS1PJOUr'JOEUIF.BYJNVNPS Minimum Value of a Quadratic Function 2.5 Inequalities Involving Quadratic Functions 192 Solve Inequalities Involving a Quadratic Function 2.6 Building Quadratic Models from Verbal Descriptions and from Data 196 #VJME2VBESBUJD.PEFMTGSPN7FSCBM%FTDSJQUJPOTr#VJME2VBESBUJD Models from Data 2.7 Complex Zeros of a Quadratic Function 207 'JOEUIF$PNQMFY;FSPTPGB2VBESBUJD'VODUJPO 2.8 Equations and Inequalities Involving the Absolute Value Function 210 4PMWF"CTPMVUF7BMVF&RVBUJPOTr4PMWF"CTPMVUF7BMVF*OFRVBMJUJFT Chapter Review 216 Chapter Test 219 www.com CONTENTS 7 Cumulative Review 220 Chapter Projects 221 3 Polynomial and Rational Functions 223 3.1 Polynomial Functions and Models 224 *EFOUJGZ1PMZOPNJBM'VODUJPOTBOE5IFJS%FHSFFr(SBQI1PMZOPNJBM 'VODUJPOT6TJOH5SBOTGPSNBUJPOTr*EFOUJGZUIF3FBM;FSPTPGB1PMZOPNJBM 'VODUJPOBOE5IFJS.VMUJQMJDJUZr"OBMZ[FUIF(SBQIPGB1PMZOPNJBM 'VODUJPOr#VJME$VCJD.PEFMTGSPN%BUB 3.2 The Real Zeros of a Polynomial Function 244 6TFUIF3FNBJOEFSBOE'BDUPS5IFPSFNTr6TF%FTDBSUFT3VMFPG4JHOTUP %FUFSNJOFUIF/VNCFSPG1PTJUJWFBOEUIF/VNCFSPG/FHBUJWF3FBM;FSPT PGB1PMZOPNJBM'VODUJPOr6TFUIF3BUJPOBM;FSPT5IFPSFNUP-JTUUIF 1PUFOUJBM3BUJPOBM;FSPTPGB1PMZOPNJBM'VODUJPOr'JOEUIF3FBM;FSPTPG B1PMZOPNJBM'VODUJPOr4PMWF1PMZOPNJBM&RVBUJPOTr6TFUIF5IFPSFNGPS #PVOETPO;FSPTr6TFUIF*OUFSNFEJBUF7BMVF5IFPSFN 3.3 Complex Zeros; Fundamental Theorem of Algebra 258 6TFUIF$POKVHBUF1BJST5IFPSFNr'JOEB1PMZOPNJBM'VODUJPOXJUI 4QFDJGJFE;FSPTr'JOEUIF$PNQMFY;FSPTPGB1PMZOPNJBM'VODUJPO 3.4 Properties of Rational Functions 264 'JOEUIF%PNBJOPGB3BUJPOBM'VODUJPOr'JOEUIF7FSUJDBM"TZNQUPUFT PGB3BUJPOBM'VODUJPOr'JOEUIF)PSJ[POUBMPS0CMJRVF"TZNQUPUFPGB Rational Function 3.5 The Graph of a Rational Function 275 "OBMZ[FUIF(SBQIPGB3BUJPOBM'VODUJPOr4PMWF"QQMJFE1SPCMFNT Involving Rational Functions 3.6 Polynomial and Rational Inequalities 290 4PMWF1PMZOPNJBM*OFRVBMJUJFTr4PMWF3BUJPOBM*OFRVBMJUJFT Chapter Review 298 Chapter Test 302 Cumulative Review 302 Chapter Projects 303 4 Exponential and Logarithmic Functions 305 4.1 Composite Functions 306 'PSNB$PNQPTJUF'VODUJPOr'JOEUIF%PNBJOPGB$PNQPTJUF'VODUJPO 4.2 One-to-One Functions; Inverse Functions 314 %FUFSNJOF8IFUIFSB'VODUJPO*T0OFUP0OFr%FUFSNJOFUIF*OWFSTFPGB 'VODUJPO%FGJOFECZB.BQPSB4FUPG0SEFSFE1BJSTr0CUBJOUIF(SBQIPG UIF*OWFSTF'VODUJPOGSPNUIF(SBQIPGUIF'VODUJPOr'JOEUIF*OWFSTFPGB Function Defined by an Equation 4.3 Exponential Functions 326 &WBMVBUF&YQPOFOUJBM'VODUJPOTr(SBQI&YQPOFOUJBM'VODUJPOTr%FGJOF the Number er4PMWF&YQPOFOUJBM&RVBUJPOT 4.4 Logarithmic Functions 343 Change Exponential Statements to Logarithmic Statements and Logarithmic 4UBUFNFOUTUP&YQPOFOUJBM4UBUFNFOUTr&WBMVBUF-PHBSJUINJD&YQSFTTJPOT r%FUFSNJOFUIF%PNBJOPGB-PHBSJUINJD'VODUJPOr(SBQI-PHBSJUINJD 'VODUJPOTr4PMWF-PHBSJUINJD&RVBUJPOT www.5 Properties of Logarithms 356 8PSLXJUI1SPQFSUJFTPG-PHBSJUINTr8SJUFB-PHBSJUINJD&YQSFTTJPOBT B4VNPS%JGGFSFODFPG-PHBSJUINTr8SJUFB-PHBSJUINJD&YQSFTTJPOBTB 4JOHMF-PHBSJUINr&WBMVBUFB-PHBSJUIN8IPTF#BTF*T/FJUIFS/PSe r(SBQIB-PHBSJUINJD'VODUJPO8IPTF#BTF*T/FJUIFS/PSe 4.6 Logarithmic and Exponential Equations 365 4PMWF-PHBSJUINJD&RVBUJPOTr4PMWF&YQPOFOUJBM&RVBUJPOTr4PMWF Logarithmic and Exponential Equations Using a Graphing Utility 4.7 Financial Models 371 %FUFSNJOFUIF'VUVSF7BMVFPGB-VNQ4VNPG.POFZr$BMDVMBUF&GGFDUJWF 3BUFTPG3FUVSOr%FUFSNJOFUIF1SFTFOU7BMVFPGB-VNQ4VNPG.POFZ r%FUFSNJOFUIF3BUFPG*OUFSFTUPSUIF5JNF3FRVJSFEUP%PVCMFB-VNQ Sum of Money 4.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 381 'JOE&RVBUJPOTPG1PQVMBUJPOT5IBU0CFZUIF-BXPG6OJOIJCJUFE(SPXUI r'JOE&RVBUJPOTPG1PQVMBUJPOT5IBU0CFZUIF-BXPG%FDBZr6TF /FXUPOT-BXPG$PPMJOHr6TF-PHJTUJD.9 Building Exponential, Logarithmic, and Logistic Models from Data 391 #VJMEBO&YQPOFOUJBM.PEFMGSPN%BUBr#VJMEB-PHBSJUINJD.PEFMGSPN %BUBr#VJMEB-PHJTUJD.PEFMGSPN%BUB Chapter Review 399 Chapter Test 404 Cumulative Review 405 Chapter Projects 406 5 Trigonometric Functions 407 5.1 Angles and Their Measures 408 Convert between Decimals and Degrees, Minutes, Seconds Measures for "OHMFTr'JOEUIF-FOHUIJGBO"SDPGB$JSDMFr$POWFSUGSPN%FHSFFT UP3BEJBOTBOEGSPN3BEJBOTUP%FHSFFTr'JOEUIF"SFBPGB4FDUPSPGB $JSDMFr'JOEUIF-JOFBS4QFFEPGBO0CKFDU5SBWFMJOHJO$JSDVMBS.2 Trigonometric Functions: Unit Circle Approach 422 Find the Exact Values of the Trigonometric Functions Using a Point on the 6OJU$JSDMFr'JOEUIF&YBDU7BMVFTPGUIF5SJHPOPNFUSJD'VODUJPOTPG 2VBESBOUBM"OHMFTr'JOEUIF&YBDU7BMVFTPGUIF5SJHPOPNFUSJD Functions of p/4 = 45°r'JOEUIF&YBDU7BMVFTPGUIF5SJHPOPNFUSJD Functions of p/6 = 30° and p/3 = 60°r'JOEUIF&YBDU7BMVFTPGUIF Trigonometric Functions for Integer Multiples of p/6 = 30°, p/4 = 45°, and p/3 = 60°r6TFB$BMDVMBUPSUP"QQSPYJNBUFUIF7BMVFPGB5SJHPOPNFUSJD 'VODUJPOr6TFB$JSDMFPG3BEJVTr to Evaluate the Trigonometric Functions 5.3 Properties of the Trigonometric Functions 439 Determine the Domain and the Range of the Trigonometric Functions r%FUFSNJOFUIF1FSJPEPGUIF5SJHPOPNFUSJD'VODUJPOTr%FUFSNJOFUIF4JHOT PGUIF5SJHPOPNFUSJD'VODUJPOTJOB(JWFO2VBESBOUr'JOEUIF7BMVFTPG UIF5SJHPOPNFUSJD'VODUJPOT6TJOH'VOEBNFOUBM*EFOUJUJFTr'JOEUIF&YBDU 7BMVFTPGUIF5SJHPOPNFUSJD'VODUJPOTPGBO"OHMF(JWFO0OFPGUIF 'VODUJPOTBOEUIF2VBESBOUPGUIF"OHMFr6TF&WFO0EE1SPQFSUJFTUP Find the Exact Values of the Trigonometric Functions 5.4 Graphs of the Sine and Cosine Functions 452 Graph Functions of the Form y = A sin (vx) Using Transformations r(SBQI'VODUJPOTPGUIF'PSNy = A cos (vx) Using Transformations www.com CONTENTS 9 r%FUFSNJOFUIF"NQMJUVEFBOE1FSJPEPG4JOVTPJEBM'VODUJPOTr(SBQI 4JOVTPJEBM'VODUJPOT6TJOH,FZ1PJOUTr'JOEBO&RVBUJPOGPSB4JOVTPJEBM Graph 5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 467 Graph Functions of the Form y = A tan(vx) + B and y = A cot (vx) + B r(SBQI'VODUJPOTPGUIF'PSNy = A csc (vx) + B and y = A sec (vx) + B 5.6 Phase Shift; Sinusoidal Curve Fitting 475 Graph Sinusoidal Functions of the Form y = A sin (vx - f) + B r#VJME4JOVTPJEBM.PEFMTGSPN%BUB Chapter Review 486 Chapter Test 492 Cumulative Review 492 Chapter Projects 493 6 Analytic Trigonometry 495 6.1 The Inverse Sine, Cosine, and Tangent Functions 496 'JOEUIF&YBDU7BMVFPGBO*OWFSTF4JOF'VODUJPOr'JOEBO"QQSPYJNBUF 7BMVFPGBO*OWFSTF4JOF'VODUJPOr6TF1SPQFSUJFTPG*OWFSTF'VODUJPOT UP'JOE&YBDU7BMVFTPG$FSUBJO$PNQPTJUF'VODUJPOTr'JOEUIF*OWFSTF 'VODUJPOPGB5SJHPOPNFUSJD'VODUJPOr4PMWF&RVBUJPOT*OWPMWJOH*OWFSTF Trigonometric Functions 6.2 The Inverse Trigonometric Functions (Continued) 508 Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, BOE5BOHFOU'VODUJPOTr%FGJOFUIF*OWFSTF4FDBOU $PTFDBOU BOE $PUBOHFOU'VODUJPOTr6TFB$BMDVMBUPSUP&WBMVBUFTFD -1 x, csc -1 x, and cot -1 xr8SJUFB5SJHPOPNFUSJD&YQSFTTJPOBTBO"MHFCSBJD&YQSFTTJPO 6.3 Trigonometric Equations 514 4PMWF&RVBUJPOT*OWPMWJOHB4JOHMF5SJHPOPNFUSJD'VODUJPOr4PMWF 5SJHPOPNFUSJD&RVBUJPOT6TJOHB$BMDVMBUPSr4PMWF5SJHPOPNFUSJD &RVBUJPOT2VBESBUJDJO'PSNr4PMWF5SJHPOPNFUSJD&RVBUJPOT6TJOH 'VOEBNFOUBM*EFOUJUJFTr4PMWF5SJHPOPNFUSJD&RVBUJPOT6TJOHB Graphing Utility 6.4 Trigonometric Identities 523 6TF"MHFCSBUP4JNQMJGZ5SJHPOPNFUSJD&YQSFTTJPOTr&TUBCMJTI*EFOUJUJFT 6.5 Sum and Difference Formulas 531 6TF4VNBOE%JGGFSFODF'PSNVMBTUP'JOE&YBDU7BMVFTr6TF4VNBOE %JGGFSFODF'PSNVMBTUP&TUBCMJTI*EFOUJUJFTr6TF4VNBOE%JGGFSFODF 'PSNVMBT*OWPMWJOH*OWFSTF5SJHPOPNFUSJD'VODUJPOTr4PMWF Trigonometric Equations Linear in Sine and Cosine 6.6 Double-angle and Half-angle Formulas 543 6TF%PVCMFBOHMF'PSNVMBTUP'JOE&YBDU7BMVFTr6TF%PVCMFBOHMF 'PSNVMBTUP&TUBCMJTI*EFOUJUJFTr6TF)BMGBOHMF'PSNVMBTUP'JOE&YBDU Values 6.7 Product-to-Sum and Sum-to-Product Formulas 553 &YQSFTT1SPEVDUTBT4VNTr&YQSFTT4VNTBT1SPEVDUT Chapter Review 557 Chapter Test 561 Cumulative Review 561 Chapter Projects 562 www.com 10 CONTENTS 7 Applications of Trigonometric Functions 563 7.1 Right Triangle Trigonometry; Applications 564 Find the Value of Trigonometric Functions of Acute Angles Using Right 5SJBOHMFTr6TFUIF$PNQMFNFOUBSZ"OHMF5IFPSFNr4PMWF3JHIU5SJBOHMFT r4PMWF"QQMJFE1SPCMFNT 7.2 The Law of Sines 576 4PMWF4""PS"4"5SJBOHMFTr4PMWF44"5SJBOHMFTr4PMWF"QQMJFE Problems 7.3 The Law of Cosines 587 4PMWF4"45SJBOHMFTr4PMWF4445SJBOHMFTr4PMWF"QQMJFE1SPCMFNT 7.4 Area of a Triangle 593 'JOEUIF"SFBPG4"45SJBOHMFTr'JOEUIF"SFBPG4445SJBOHMFT 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves 600 #VJMEB.PEFMGPSBO0CKFDUJO4JNQMF)BSNPOJD.PUJPOr"OBMZ[F4JNQMF )BSNPOJD.PUJPOr"OBMZ[FBO0CKFDUJO%BNQFE.PUJPOr(SBQIUIF4VN of Two Functions Chapter Review 609 Chapter Test 612 Cumulative Review 613 Chapter Projects 613 8 Polar Coordinates; Vectors 615 8.1 Polar Coordinates 616 1MPU1PJOUT6TJOH1PMBS$PPSEJOBUFTr$POWFSUGSPN1PMBS$PPSEJOBUFTUP 3FDUBOHVMBS$PPSEJOBUFTr$POWFSUGSPN3FDUBOHVMBS$PPSEJOBUFTUP1PMBS $PPSEJOBUFTr5SBOTGPSN&RVBUJPOTCFUXFFO1PMBSBOE3FDUBOHVMBS'PSNT 8.2 Polar Equations and Graphs 625 Identify and Graph Polar Equations by Converting to Rectangular &RVBUJPOTr5FTU1PMBS&RVBUJPOTGPS4ZNNFUSZr(SBQI1PMBS&RVBUJPOT by Plotting Points 8.3 The Complex Plane; De Moivre’s Theorem 640 1MPU1PJOUTJOUIF$PNQMFY1MBOFr$POWFSUB$PNQMFY/VNCFSCFUXFFO 3FDUBOHVMBS'PSNBOE1PMBS'PSNr'JOE1SPEVDUTBOE2VPUJFOUTPG $PNQMFY/VNCFSTJO1PMBS'PSNr6TF%F.PJWSFT5IFPSFNr'JOE Complex Roots 8.4 Vectors 648 (SBQI7FDUPSTr'JOEB1PTJUJPO7FDUPSr"EEBOE4VCUSBDU7FDUPST "MHFCSBJDBMMZr'JOEB4DBMBS.VMUJQMFBOEUIF.BHOJUVEFPGB7FDUPSr'JOE B6OJU7FDUPSr'JOEB7FDUPSGSPN*UT%JSFDUJPOBOE.PEFM with Vectors 8.5 The Dot Product 662 'JOEUIF%PU1SPEVDUPG5XP7FDUPSTr'JOEUIF"OHMFCFUXFFO5XP 7FDUPSTr%FUFSNJOF8IFUIFS5XP7FDUPST"SF1BSBMMFMr%FUFSNJOF 8IFUIFS5XP7FDUPST"SF0SUIPHPOBMr%FDPNQPTFB7FDUPSJOUP5XP 0SUIPHPOBM7FDUPSTr$PNQVUF8PSL 8.6 Vectors in Space 670 'JOEUIF%JTUBODFCFUXFFO5XP1PJOUTJO4QBDFr'JOE1PTJUJPO 7FDUPSTJO4QBDFr1FSGPSN0QFSBUJPOTPO7FDUPSTr'JOEUIF%PU 1SPEVDUr'JOEUIF"OHMFCFUXFFO5XP7FDUPSTr'JOEUIF%JSFDUJPO Angles of a Vector www.7 The Cross Product 679 'JOEUIF$SPTT1SPEVDUPG5XP7FDUPSTr,OPX"MHFCSBJD1SPQFSUJFTPGUIF $SPTT1SPEVDUr,OPX(FPNFUSJD1SPQFSUJFTPGUIF$SPTT1SPEVDUr'JOEB 7FDUPS0SUIPHPOBMUP5XP(JWFO7FDUPSTr'JOEUIF"SFBPGB Parallelogram Chapter Review 685 Chapter Test 688 Cumulative Review 689 Chapter Projects 689 9 Analytic Geometry 691 9.1 Conics 692 Know the Names of the Conics 9.2 The Parabola 693 "OBMZ[F1BSBCPMBTXJUI7FSUFYBUUIF0SJHJOr"OBMZ[F1BSBCPMBTXJUI Vertex at (h, k r4PMWF"QQMJFE1SPCMFNT*OWPMWJOH1BSBCPMBT 9.3 The Ellipse 701 "OBMZ[F&MMJQTFTXJUI$FOUFSBUUIF0SJHJOr"OBMZ[F&MMJQTFTXJUI$FOUFSBU (h, k r4PMWF"QQMJFE1SPCMFNT*OWPMWJOH&MMJQTFT 9.4 The Hyperbola 711 "OBMZ[F)ZQFSCPMBTXJUI$FOUFSBUUIF0SJHJOr'JOE"TZNQUPUFTPG )ZQFSCPMBr"OBMZ[F)ZQFSCPMBTXJUI$FOUFSBU h, k r4PMWF"QQMJFE Problems Involving Hyperbolas 9.5 Rotation of Axes; General Form of a Conic 724 *EFOUJGZB$POJDr6TFB3PUBUJPOPG"YFTUP5SBOTGPSN&RVBUJPOT r"OBMZ[FBO&RVBUJPO6TJOH3PUBUJPOPG"YFTr*EFOUJGZ$POJDTXJUIPVUB Rotation of Axes 9.6 Polar Equations of Conics 731 "OBMZ[FBOE(SBQI1PMBS&RVBUJPOTPG$POJDTr$POWFSUUIF1PMBS&RVBUJPO of a Conic to a Rectangular Equation 9.7 Plane Curves and Parametric Equations 737 (SBQI1BSBNFUSJD&RVBUJPOTr'JOEB3FDUBOHVMBS&RVBUJPOGPSB$VSWF %FGJOFE1BSBNFUSJDBMMZr6TF5JNFBTB1BSBNFUFSJO1BSBNFUSJD &RVBUJPOTr'JOE1BSBNFUSJD&RVBUJPOTGPS$VSWFT%FGJOFECZ3FDUBOHVMBS Equations Chapter Review 749 Chapter Test 752 Cumulative Review 752 Chapter Projects 753 10 Systems of Equations and Inequalities 754 10.1 Systems of Linear Equations: Substitution and Elimination 755 4PMWF4ZTUFNTPG&RVBUJPOTCZ4VCTUJUVUJPOr4PMWF4ZTUFNTPG&RVBUJPOT CZ&MJNJOBUJPOr*EFOUJGZ*ODPOTJTUFOU4ZTUFNTPG&RVBUJPOT$POUBJOJOH 5XP7BSJBCMFTr&YQSFTTUIF4PMVUJPOPGB4ZTUFNPG%FQFOEFOU&RVBUJPOT $POUBJOJOH5XP7BSJBCMFTr4PMWF4ZTUFNTPG5ISFF&RVBUJPOT$POUBJOJOH 5ISFF7BSJBCMFTr*EFOUJGZ*ODPOTJTUFOU4ZTUFNTPG&RVBUJPOT$POUBJOJOH 5ISFF7BSJBCMFTr&YQSFTTUIF4PMVUJPOPGB4ZTUFNPG%FQFOEFOU&RVBUJPOT Containing Three Variables www.2 Systems of Linear Equations: Matrices 770 8SJUFUIF"VHNFOUFE.BUSJYPGB4ZTUFNPG-JOFBS&RVBUJPOTr8SJUFUIF 4ZTUFNPG&RVBUJPOTGSPNUIF"VHNFOUFE.BUSJYr1FSGPSN3PX0QFSBUJPOT POB.BUSJYr4PMWFB4ZTUFNPG-JOFBS&RVBUJPOT6TJOH.3 Systems of Linear Equations: Determinants 784 &WBMVBUFCZ%FUFSNJOBOUTr6TF$SBNFST3VMFUP4PMWFB4ZTUFNPG 5XP&RVBUJPOT$POUBJOJOH5XP7BSJBCMFTr&WBMVBUFCZ%FUFSNJOBOUT r6TF$SBNFST3VMFUP4PMWFB4ZTUFNPG5ISFF&RVBUJPOT$POUBJOJOH 5ISFF7BSJBCMFTr,OPX1SPQFSUJFTPG%FUFSNJOBOUT 10.4 Matrix Algebra 794 'JOEUIF4VNBOE%JGGFSFODFPG5XP.BUSJDFTr'JOE4DBMBS.BUSJYr'JOEUIF1SPEVDUPG5XP.BUSJDFTr'JOEUIF*OWFSTFPGB.BUSJY r4PMWFB4ZTUFNPG-JOFBS&RVBUJPOT6TJOHBO*OWFSTF.5 Partial Fraction Decomposition 813 Decompose P/Q, Where Q)BT0OMZ/POSFQFBUFE-JOFBS'BDUPST r%FDPNQPTFP/Q, Where Q)BT3FQFBUFE-JOFBS'BDUPSTr%FDPNQPTF P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor r%FDPNQPTFP/Q, Where Q Has a Repeated Irreducible Quadratic Factor 10.6 Systems of Nonlinear Equations 821 4PMWFB4ZTUFNPG/POMJOFBS&RVBUJPOT6TJOH4VCTUJUVUJPOr4PMWFB4ZTUFN of Nonlinear Equations Using Elimination 10.7 Systems of Inequalities 830 (SBQIBO*OFRVBMJUZr(SBQIB4ZTUFNPG*OFRVBMJUJFT 10.8 Linear Programming 838 4FUVQB-JOFBS1SPHSBNNJOH1SPCMFNr4PMWFB-JOFBS1SPHSBNNJOH1SPCMFN Chapter Review 845 Chapter Test 848 Cumulative Review 849 Chapter Projects 850 11 Sequences; Induction; the Binomial Theorem 851 11.1 Sequences 852 8SJUFUIF'JSTU4FWFSBM5FSNTPGB4FRVFODFr8SJUFUIF5FSNTPGB 4FRVFODF%FGJOFECZB3FDVSTJWF'PSNVMBr6TF4VNNBUJPO/PUBUJPO r'JOEUIF4VNPGB4FRVFODF 11.2 Arithmetic Sequences 862 %FUFSNJOF8IFUIFSB4FRVFODF*T"SJUINFUJDr'JOEB'PSNVMBGPSBO "SJUINFUJD4FRVFODFr'JOEUIF4VNPGBO"SJUINFUJD4FRVFODF 11.3 Geometric Sequences; Geometric Series 868 %FUFSNJOF8IFUIFSB4FRVFODF*T(FPNFUSJDr'JOEB'PSNVMBGPSB (FPNFUSJD4FRVFODFr'JOEUIF4VNPGB(FPNFUSJD4FRVFODFr%FUFSNJOF 8IFUIFSB(FPNFUSJD4FSJFT$POWFSHFTPS%JWFSHFTr4PMWF"OOVJUZ1SPCMFNT 11.4 Mathematical Induction 879 Prove Statements Using Mathematical Induction 11.5 The Binomial Theorem 883 Evaluate a b r6TFUIF#JOPNJBM5IFPSFN n j Chapter Review 890 Chapter Test 892 Cumulative Review 892 Chapter Projects 893 www.com CONTENTS 13 12 Counting and Probability 894 12.1 Counting 895 'JOE"MMUIF4VCTFUTPGB4FUr$PVOUUIF/VNCFSPG&MFNFOUTJOB4FU r4PMWF$PVOUJOH1SPCMFNT6TJOHUIF.2 Permutations and Combinations 900 Solve Counting Problems Using Permutations Involving n%JTUJODU0CKFDUT r4PMWF$PVOUJOH1SPCMFNT6TJOH$PNCJOBUJPOTr4PMWF$PVOUJOH1SPCMFNT Using Permutations Involving n/POEJTUJODU0CKFDUT 12.3 Probability 909 $POTUSVDU1SPCBCJMJUZ.PEFMTr$PNQVUF1SPCBCJMJUJFTPG&RVBMMZ-JLFMZ 0VUDPNFTr'JOE1SPCBCJMJUJFTPGUIF6OJPOPG5XP&WFOUTr6TFUIF Complement Rule to Find Probabilities Chapter Review 919 Chapter Test 921 Cumulative Review 922 Chapter Projects 922 13 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 923 13.1 Finding Limits Using Tables and Graphs 924 'JOEB-JNJU6TJOHB5BCMFr'JOEB-JNJU6TJOHB(SBQI 13.2 Algebra Techniques for Finding Limits 929 'JOEUIF-JNJUPGB4VN B%JGGFSFODF BOEB1SPEVDUr'JOEUIF-JNJUPG B1PMZOPNJBMr'JOEUIF-JNJUPGB1PXFSPSB3PPUr'JOEUIF-JNJUPGB 2VPUJFOUr'JOEUIF-JNJUPGBO"WFSBHF3BUFPG$IBOHF 13.
