www.elsolucionario.net http://www.elsolucionario.net LIBROS UNIVERISTARIOS Y SOLUCIONARIOS DE MUCHOS DE ESTOS LIBROS LOS SOLUCIONARIOS CONTIENEN TODOS LOS EJERCICIOS DEL LIBRO RESUELTOS Y EXPLICADOS DE FORMA CLARA VISITANOS PARA DESARGALOS GRATIS 8.1 Last Section Head Precalculus: Functions and Graphs 4th Edition Mark Dugopolski Southeastern Louisiana University Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.elsolucionario.net i Editor in Chief: Anne Kelly Acquisitions Editor: Kathryn O’Connor Senior Content Editor: Elizabeth Bernardi Editorial Assistant: Judith Garber Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Peggy McMahon Associate Director of Design, USHE North and West: Andrea Nix Cover Designer: Barbara T Atkinson Marketing Manager: 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Dugopolski, Mark Precalculus: functions & graphs / Mark Dugopolski — 4th ed p cm Includes index ISBN-13: 978-0-321-78943-3 ISBN-10: 0-321-78943-1 Precalculus—Textbooks Precalculus—Graphic methods—Textbooks Functions—Textbooks I Title QA39.3.D84 2013 510—dc22 2011012455 Copyright © 2013, 2009, 2005, 2002 Pearson Education, Inc All rights reserved Manufactured in the United States of America This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290 10 1—CRK—14 13 12 11 http://www.pearsonhighered.com www.elsolucionario.net ISBN-10: 0-321-78943-1 ISBN-13: 978-0-321-78943-3 Contents Preface vii Supplements List x Function Gallery xii Equations, Inequalities, and Modeling 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Equations in One Variable Constructing Models to Solve Problems 12 Equations and Graphs in Two Variables 24 Linear Equations in Two Variables 39 Scatter Diagrams and Curve Fitting 52 Complex Numbers 59 Quadratic Equations 65 Linear and Absolute Value Inequalities 82 Chapter Highlights 94 Chapter Review Exercises 96 Chapter Test 101 Concepts of Calculus: Limits 102 Functions and Graphs 2.1 2.2 2.3 2.4 2.5 2.6 103 Functions 104 Graphs of Relations and Functions 117 Families of Functions, Transformations, and Symmetry Operations with Functions 146 Inverse Functions 156 Constructing Functions with Variation 170 Chapter Highlights 178 Chapter Review Exercises 180 Chapter Test 183 Tying It All Together 184 Concepts of Calculus: Instantaneous rate of change Polynomial and Rational Functions 3.1 3.2 3.3 3.4 3.5 131 185 186 Quadratic Functions and Inequalities 187 Zeros of Polynomial Functions 201 The Theory of Equations 212 Miscellaneous Equations 222 Graphs of Polynomial Functions 235 iii www.elsolucionario.net iv Contents 3.6 Rational Functions and Inequalities 249 Chapter Highlights 265 Chapter Review Exercises 267 Chapter Test 270 Tying It All Together 271 Concepts of Calculus: Instantaneous rate of change of the power functions 272 Exponential and Logarithmic Functions 4.1 4.2 4.3 4.4 273 Exponential Functions and Their Applications 274 Logarithmic Functions and Their Applications 289 Rules of Logarithms 302 More Equations and Applications 314 Chapter Highlights 328 Chapter Review Exercises 329 Chapter Test 331 Tying It All Together 332 Concepts of Calculus: The instantaneous rate of change of ƒ1x2 = e x The Trigonometric Functions 5.1 5.2 5.3 5.4 5.5 5.6 335 Angles and Their Measurements 336 The Sine and Cosine Functions 351 The Graphs of the Sine and Cosine Functions 362 The Other Trigonometric Functions and Their Graphs The Inverse Trigonometric Functions 390 Right Triangle Trigonometry 401 Chapter Highlights 415 Chapter Review Exercises 417 Chapter Test 420 Tying It All Together 421 Concepts of Calculus: Evaluating transcendental functions Trigonometric Identities and Conditional Equations 423 6.1 6.2 6.3 6.4 6.5 334 Basic Identities 424 Verifying Identities 433 Sum and Difference Identities 441 Double-Angle and Half-Angle Identities Product and Sum Identities 458 www.elsolucionario.net 450 422 379 Contents 6.6 Conditional Trigonometric Equations Chapter Highlights 481 Chapter Review Exercises 482 Chapter Test 484 Tying It All Together 485 Concepts of Calculus: Area of a circle and p Applications of Trigonometry 7.1 7.2 7.3 7.4 7.5 7.6 7.7 467 486 487 The Law of Sines 488 The Law of Cosines 499 Vectors 509 Trigonometric Form of Complex Numbers 522 Powers and Roots of Complex Numbers 530 Polar Equations 535 Parametric Equations 545 Chapter Highlights 551 Chapter Review Exercises 553 Chapter Test 555 Tying It All Together 557 Concepts of Calculus: Limits and asymptotes 558 Systems of Equations and Inequalities 8.1 8.2 8.3 8.4 8.5 8.6 559 Systems of Linear Equations in Two Variables 560 Systems of Linear Equations in Three Variables 572 Nonlinear Systems of Equations 583 Partial Fractions 592 Inequalities and Systems of Inequalities in Two Variables The Linear Programming Model 609 Chapter Highlights 616 Chapter Review Exercises 618 Chapter Test 619 Tying It All Together 620 Concepts of Calculus: Instantaneous rate of change and partial fractions Matrices and Determinants 9.1 9.2 9.3 9.4 9.5 v 622 Solving Linear Systems Using Matrices 623 Operations with Matrices 637 Multiplication of Matrices 645 Inverses of Matrices 653 Solution of Linear Systems in Two Variables Using Determinants 664 www.elsolucionario.net 601 621 vi Contents 9.6 Solution of Linear Systems in Three Variables Using Determinants 670 Chapter Highlights 678 Chapter Review Exercises Chapter Test 681 Tying It All Together 682 680 10 The Conic Sections 683 10.1 10.2 10.3 10.4 10.5 The Parabola 684 The Ellipse and the Circle 694 The Hyperbola 707 Rotation of Axes 718 Polar Equations of the Conics 726 Chapter 10 Highlights 731 Chapter 10 Review Exercises 733 Chapter 10 Test 736 Tying It All Together 737 Concepts of Calculus: The reflection property of a parabola 738 11 Sequences, Series, and Probability 739 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Sequences 740 Series 750 Geometric Sequences and Series 758 Counting and Permutations 770 Combinations, Labeling, and the Binomial Theorem Probability 786 Mathematical Induction 797 Chapter 11 Highlights 803 Chapter 11 Review Exercises 805 Chapter 11 Test 807 Concepts of Calculus: Limits of sequences 808 Appendix A: Solutions to Try This Exercises 809 Appendix B: Basic Algebra Review 837 B.1 Real Numbers and Their Properties B.2 Exponents and Radicals 843 B.3 Polynomials 852 B.4 Factoring Polynomials 857 B.5 Rational Expressions 861 Credits C-1 Answers A-1 Index of Applications I-1 Index I-6 www.elsolucionario.net 837 777 Preface Making the transition to calculus is more than just finding a new classroom: it’s about being prepared to grasp bigger and more complex mathematical concepts Precalculus: Functions and Graphs is designed to make this transition seamless by focusing on all the skills that will be needed to succeed in calculus and beyond This text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports students’ mathematical needs with a number of tools newly developed for this revision With an eye toward future courses, this text provides students with an excellent opportunity to sharpen their critical thinking skills and introduces them to the usefulness and applicability of mathematics beyond the classroom It is my goal that students will benefit from this approach and find that the mathematics presented here will enrich their classroom experience in this course as well as in future mathematics courses Preparation is the foundation for success, and Precalculus: Functions and Graphs will help you succeed in this course and beyond New to the Fourth Edition For this edition of Precalculus: Functions and Graphs, I have updated explanations, examples, exercises, and art in response to comments from users of the last edition In particular, I have written more than 900 new exercises that are specifically designed to increase student understanding and retention of the concepts that are taught in this text Here are the major changes in this edition ■ ■ ■ ■ ■ ■ ■ ■ Updated real-data in examples, exercises, and chapter openers make the text relevant for today’s students Fill-in-the-blank exercises are now used at the beginning of the section exercise sets to help students learn definitions, rules, and theorems Cumulative review exercises are now used at the end of the section exercise sets to keep current the skills learned in previous sections and chapters These exercises are under the heading “Rethinking.” Tying It All Together exercises have been expanded to include fill-in-theblank exercises that emphasize vocabulary Polynomial and rational inequalities are now solved with one method, the test-point method This change simplifies the procedure and makes it more consistent with techniques used in calculus Limit notation is now introduced and used to describe the asymptotic behavior of exponential, logarithmic, rational, and trigonometric functions Try This exercises have been included after every example in the text These exercises are very similar to the corresponding examples and give the students the opportunity to immediately check their understanding of the example Solutions to all of the Try This exercises are in the appendix of the Student Edition Suggested homework problems are indicated in the Annotated Instructor’s Edition Each section exercise set contains 20 to 24 underlined exercise numbers These exercises can be used as a set of homework exercises for the section Continuing Features With each new edition, all of the features are reviewed to make sure they are providing a positive impact on student success The continuing features of the text are listed here vii www.elsolucionario.net viii Preface Strategies for Success ■ ■ ■ ■ ■ ■ ■ ■ ■ Chapter Opener Each chapter begins with chapter opener text that discusses a real-world situation in which the mathematics of the chapter is used Examples and exercises that relate back to the chapter opener are included in the chapter Foreshadowing Calculus This feature gives a brief indication of the connection between certain algebraic topics and calculus, and identifies topics that will be continued in calculus Summaries of important concepts are included to help students clarify ideas that have multiple parts Strategies contain general guidelines for solving certain types of problems They are designed to help students sharpen their problem-solving skills Procedures are similar to Strategies, but are more specific and more algorithmic Procedures are designed to give students a step-by-step approach for solving a specific type of problem Function Galleries Located throughout the text, these function summaries are also gathered together at the beginning of the text These graphical summaries are designed to help students link the visual aspects of various families of functions to the properties of the functions Historical Notes Located in the margins throughout the text, these brief essays are designed to connect the topics of precalculus to the mathematicians who first studied them and to give precalculus a human face Hints Selected applications include hints that are designed to encourage students to start thinking about the problem at hand A Hint logo HINT is used where a hint is given Graphing Calculator Discussions Optional graphing calculator discussions have been included in the text They are clearly marked by graphing calculator icons so that they can be easily skipped if desired Although the graphing calculator discussions are optional, all students will benefit from reading them In this text, the graphing calculator is used as a tool to support and enhance algebraic conclusions, not to make conclusions Section Exercises and Review ■ ■ ■ ■ ■ For Thought Each exercise set is preceded by a set of ten true/false questions that review the basic concepts in the section, help check student understanding, and offer opportunities for writing and discussion The answers to all For Thought exercises are included in the back of the Student Edition Exercise Sets The exercise sets range from easy to challenging, and are arranged in order of increasing difficulty Those exercises that require a graphing calculator are optional and are marked with an icon Writing/Discussion and Cooperative Learning Exercises These exercises deepen students’ understanding by giving them the opportunity to express mathematical ideas both in writing and to their classmates during small group or team discussions Thinking Outside the Box Found throughout the text, these problems are designed to get students and instructors to some mathematics just for fun I enjoyed solving these problems and hope that you will too The problems can be used for individual or group work They may or may not have anything to with the sections in which they are located and might not even require any techniques discussed in the text So be creative and try Thinking Outside the Box The answers are given in the Annotated Instructor’s Edition only, and complete solutions can be found in the Instructor’s Solutions Manual Pop Quizzes Included at the end of every section of the text, the Pop Quizzes give instructors and students convenient quizzes of to 10 questions that can be used in the classroom to check understanding of the basics The answers appear in the Annotated Instructor’s Edition only www.elsolucionario.net Index directrix, finding, 686 equation of, 684–686 with three points, 577–578 focus, finding, 686 general form, 189 graphing See Graphs, parabola intercepts, 191–192 and line, graphing, 583–584 opening left/right, 688–689 opening to right, 119 opening upward/downward, 189–189 reflection property, 689–690, 738 tangent lines, intersection, 693 vertex, 684 finding, 189–190, 686 Parallel lines defined, 44 equation, finding from graph, 44 vectors, 514 Parallelogram law, 511 Parametric equations, 545–548 basketball, flight of, 547–548 converting polar equation to, 547 defined, 545 line segment, graphing, 545–546 writing, 547 parameter, eliminating, 546 parameter of, 545 Partial fractions, 592–598 decomposition, 594–598 defined, 594 and instantaneous rate of change, 621 repeated linear factor, 595–596 repeated prime quadratic factor, 597–598 single prime quadratic factor, 596–597 Pascal, Blaise, 781 Pascal’s triangle binomial expansion, finding, 781–782 Perfect cube, 849 Perfect nth power, 849 Perfect square, 849 Perfect square trinomial, 859 Perigee, 706 Periodic compounding, 309 Periodic function, 363–368 calculator, misleading, 368 defined, 363 graphing, 365–366 period, changing, 367–368 Permutations, 771–773 defined, 771 of n things taken r at a time, finding, 773 number of, finding, 772–773 Perpendicular lines, 44–46 defined, 44 diagonals of rhombus as, 46 equation, finding, 45 graphing, 45, 45–46 vectors, 514 Phase shifts, 366–367 defined, 366 sine and cosine function graphs, 366–367 Piecewise functions, 121–124 absolute value function, 121–122 defined, 121 graphing, 121–124 Plane degenerate conic, 684 graphing, 573 Planets, mass, finding, 178 Plotting, 25 Points Cartesian coordinate system, 25 polar coordinate system, 535–536 Point-slope form, 40–43 Polar coordinate system, 535–537 defined, 535 distance in, 550–551 points, plotting, 535–536 polar axis, 535 polar-rectangular conversion, 536–537 pole, 535 Polar equations, 537–542, 726–729 graphing See Graphs, polar equations parametric equations, converting to, 547 rectangular equation, converting to, 541–542, 729 Polygons, describing, 609 Polynomial(s), 852–860 addition of, 853 defined, 852 degree of, 852 dividing, 855–856 factoring See Polynomial factoring FOIL method, 854 function notation, 852 Horner’s method, 212 multiplying, 853–855 naming, 853 in one variable, 852 prime, 860 quadratic, 852 ratio of See Rational expressions special products, 854–855 subtraction of, 853 Taylor, 378–379 Polynomial equations, 212–218 all roots, finding, 213–214 bounds on the roots, 216–218 conjugate pairs theorem, 214 Descartes’ rule of signs, 214–216 first-degree See Linear equations n-Root theorem, 213 number of roots, 212–214 second-degree See Quadratic equations sign, variation of, 215 Polynomial factoring, 857–860 ac-method, 858–859 difference and sums of two cubes, 860 factored completely, 860 factor theorem, 204–205 four-term polynomials, 858 greatest common factor (GCF), 857–858 by grouping, 858 higher-degree polynomials, 222–223 special products, 859 trinomials, 859 Polynomial functions, 201–209 defined by polynomial, 187 www.elsolucionario.net I-14 factor theorem, 204–205 graphs of See Graphs, polynomial functions intermediate value theorem, 235 rational zero theorem, 205–209 remainder theorem, 201–202 synthetic division, 202–204 Polynomial inequalities, test points, solving with, 241–242 Population modeling growth and debt, 117 life expectancy and Social Security, 130 logistic growth model, 327 U.S population, 301 Position vector, 511–512 Positive correlation, 54–55 Positive integers, statements involving, 797–798 Positive slope of line, 43 Potassium-argon dating, 318 Power complex numbers, trigonometric form, 530–531 defined, 840 inverse functions, 165 Power rule logarithms, 304–305 power of power rule, 844 power of product rule, 844 power of quotient rule, 844 Prediction interpolation/extrapolation, 54 probability, 786–792 scatter diagrams, 54 Present value, 281 Prime polynomials, 860 Principal square root, 62–63 Probability, 786–792 addition rule, 789–790 coin toss, 787 complementary events, 790–791 mutually exclusive events, 789–790 odds, 791–792 rolling pair of dice, 788 rolling single die, 788 of winning lottery, 788–789 Problem-solving annuity, value of, finding, 766, 770 average speed, finding, 16–17 baseball statistics, finding, 81 basketball, flight of, 547–548 bus tour, break-even point, 229–230 circle, area/circumference formulas, finding, 498–499 construction costs, minimizing, 234–235 crystal ball design, 221 distance, investigating, 550–551 domain/range, determining, 362 exponential and linear models, comparing, 288 eyebrow window, constructing, 440 Horner’s method, 212 hyperbola, eccentricity and focal radii, 717 length/width, finding, 15–16 mean, finding, 758 I-15 Index Problem-solving (continued) mixture problem, 17–18 ocean depth, measuring, 592 orbit, eccentricity, 706 parabola, tangent line intersection, 693 pay day loans, payback, 170 planet mass, finding, 178 poker hands, 786 polygons, describing, 609 probabilities, finding, 797 profit, maximizing, 467 race car, weight distribution, 664 race track design, 200 racing boat design, 145 rectangle, length/width, 15 sequences, converging, 749 sine function, constructing, 351 speed, average, 16–17 steps in, 14–15 taxation, sales tax, finding, 14 state and federal as function of each other, 583 Taylor polynomials, graphing, 378–379 teams, ranking, 652–653 transcendental functions, evaluating, 422 travel time, minimizing, 522, 776 uniform-motion problem, 16–17 viewing angle, maximizing, 400 volume, maximizing, 248 work, time spent, 18–19, 600 See also Models/modeling Product functions, 146–148 of sum and difference, 854 See also Multiplication Product rule complex numbers, 525–527 exponential expressions, 844 logarithms, 302–303 power of product rule, 844 radicals, 848–849 Product-to-sum identities, 459–460 defined, 459–460 product, evaluating, 460 expressing as sum, 459–460 Profit, maximizing, 467 Projectile maximum height, finding, 194 motion, modeling, 432–433, 476–477 Proportionality constant, 170 Pythagoras of Samos, 25 Pythagorean theorem defined, 26, 74 identities, 424, 427 quadratic equations from, 74–75 right triangle, solving, 404 Q Quadrantal angles defined, 337 determining quadrant of angle, 339 Quadrants of coordinate system, 25 quadrantal angles, 337, 339 Quadratic equations, 65–76 completing the square, solving by, 68–69 defined, 66 equations of quadratic type, 226–228 factoring, solving by, 66–68 models with, 73–75 and Pythagorean theorem, 74–75 quadratic formula, 69–71 quadratic regression, calculator use, 75–76 and rational exponents, 227–228 square root property, 67–68 zero factor property, 66 Quadratic formula, 69–71 defined, 70 discriminant, 71–72 solving equations with, 70–71 using, 70–71 Quadratic functions, 187–195 completing the square, 187–188 defined, 187 graphing See Graphs, quadratic functions maximum and minimum, 194–195 quadratic inequalities, 192–193 Quadratic inequalities, 192–193 defined, 192 graphical method to solve, 192–193 test-point method to solve, 193–194 Quadratic mean, 758 Quadratic polynomials defined, 852 functions, graphs of, 243 Quadratic regression, calculator use, 75–76 Quadratic type, equations of, 226–228 Quadrilateral, area of, finding, 493 Quartic polynomial functions, graphs of, 243 Quotient functions, 146–148 real numbers, 838 See also Division Quotient rule complex numbers, 525–527 integral exponents, 844 logarithms, 303–304 power of quotient rule, 844 radicals, 848–849 R Race car, weight distribution, 664 Race track, designing, 200 Racing boat, designing, 145 Radian measure of angles, 340–343 in calculus, 357 coterminal angles, finding, 342–343 defined, 340 finding, 340 radian to degree conversion, 341–342 Radicals, 847–850 defined, 847 evaluating, 847–848 like radicals, 850 parts of, 847 product and quotient rules, 848–849 radical notation, 847–848 rational exponents written as, 848 rationalizing the denominator, 849–850 www.elsolucionario.net same index, operations with, 850 simplified form, 849–850 Radicand, 847 Radioactive dating, 318–319 decay model, exponential functions, 283–284 drug elimination, 319 half-life, 318 logarithmic equations, 318–319 Sauropods in Utah, 318–319 Radius of circle defined, 29, 699 finding, 700 Radius vector, 511–512 Range defined, 107 determining, 362 functions, determining, 108 for sine and cosine functions, 362 Rational approximation, logarithmic equations, 307–308 Rational exponents, 846–848 defined, 846 equations, solving with, 225–226 expressions, evaluating with, 846–847 radicals, 847–850 writing as, 848 roots, 846–847 rules for, 844–845 simplifying expressions with, 847 Rational expressions, 861–865 addition of, 592–593, 865 reversing, 593–594 defined, 861 denominator, building up, 864–865 dividing, 863–864 domain of, 861–862 equations with, 4–6 equivalence, 864–865 least common denominator (LCD), 865 lowest terms, reducing to, 861–862 multiplying, 863 Rational functions, 249–259 defined, 249 domain, 249 graphing See Graphs, rational functions horizontal/vertical asymptotes, 249–255 oblique asymptotes, 252–253 Rational inequalities, 256–258 defined, 256 solving with test-point method, 257–258 Rationalizing the denominator with conjugates, 855 of radicals, 849–850 Rational numbers principle of, 862 as real numbers, 59, 837 See also Rational expressions Rational zero theorem, rational zeros, finding, 205–209 Ratios, trigonometric, 401–402 Ray bearing, 493 defined, 336 Real axis, 522 Real numbers, 837–842 absolute value, 838 Index algebraic expressions, 841–842 arithmetic expressions, 840–841 in calculus, 357 complex, 59–63, 522–533 defined, 59, 837 exponential expressions, 839–840 imaginary, 59 inequalities, 82–83 integers, 837 irrational, 59, 837 opposites, properties of, 838–839 ordered pair, 24 order of operations, 841 properties of, 4–5, 838 rational, 59, 837 real, 59 sine and cosine of, 357 Real part, 60 Reciprocal identities, 424 Rectangle coordinate system based on See Cartesian coordinate system fundamental, 710–713 length/width, finding, 15 minimum/maximum area, 194–195 Rectangular equations defined, 537 polar-rectangular conversion, 536–537 Recursion formulas arithmetic sequences, 746 defined, 743 sequences, 743 Recycling problem, modeling, 24 Reduction formula, 462–464 graphing with, 463 motion of spring, modeling, 464 theorem, 462 Reference angles, 354–357 defined, 355 finding, 355–356 trigonometric functions with, 356–357 Reflection in calculus, 738 defined, 132 ellipse, 694 exponential functions, graphing with, 279 graphing, 132–133 hyperbola, 707 parabola, 689–690, 738 stretching/shrinking of graphs, 133–134 transformations, 132–133 Regression line, line of best fit, 54–55 Relations, 105–108 defined, 105 domain and range of, 107–108 functions as, 105–108 graphs of See Graphs, functions Remainder, 855 Remainder theorem, 201–202 defined, 201 polynomials, evaluating with, 201–202 Resultant, 511 Rhombus, diagonals as perpendicular, 46 Right angles, defined, 337 Right triangles, 401–408 Pythagorean theorem, 26, 74 sides, names of, 402 significant digits, 404–405 solving, 403–404 trigonometric functions of, 402–403 trigonometric ratios, 401–402 Rigid transformations, 136 Rise, slope of line, 39 Root(s) bounds on, 216–218 complex numbers, trigonometric form, 531–533 cube root, 845 of the equation, 2, 33 even and odd roots, 845 extraneous, 6, 223 inverse functions, 165 nth root, 531–532, 845–846 of polynomial equation, 212–214 radicals, 847–850 of unity, 532–533 See also Square root(s) Row matrix, 623 Row operations, augmented matrices, 625–626 Run, slope of line, 39 Runners, oxygen uptake, modeling, 12 S Sales tax, modeling, 14 Sample space, 786–787 Scalar(s), defined, 640 Scalar multiplication, 510–513 matrices, 641 parallelogram law, 511 scalar product, 513 vectors, 510–511 Scalar quantities absolute value, 522–523 defined, 509 Scatter diagrams, 52–55 correlations in, 54–55 defined, 53 linear relationship in, 53–55 line of best fit, 53–55 lines, fitting to data, 54 nonlinear relationship in, 53 prediction using, 54 Secant defined, 379 graphs of See Graphs, secant inverse of, 394–395 Second coordinate, 24 Seconds of degrees, 339–340 decimal degrees, converting from, 339 decimal degrees, converting to, 340 defined, 339 Secret codes, and matrices, 660 Semicircles, graphing, 120–121 Sequences, 740–746 arithmetic sequences, 743–746 converging, 749 defined, 740 factorial notation, 741–742 finite, 740–741 geometric, 758–761 infinite, 740–741 limits of, 808 mean, finding, 753 nth term, 740, 742–743 www.elsolucionario.net I-16 recursion formulas, 743 terms, finding and listing, 740–741 Series, 750–754 arithmetic, 753–754 defined, 750 geometric, 761–764 index of summation, changing, 752–753 mean, 752–753 summation notation, 750–752 Sets composition of functions defined by, 149 empty, intersection, 85 solution See Solution sets union, 85 Shrinking, graph, 133–134 Sign array, 671 Significant digits, 404–405 defined, 404 finding, 405 Signs Descartes’ rule, 214–216 variation of, 215 Simplify algebraic expressions, 842 complex number, 61 defined, 842 with integral exponents, 845 radicals, 849–850 with rational exponents, 847 with real number properties, 842 with trigonometric identities, 236, 246, 425–426, 433–434 trigonometric identities, 436 Sine(s), 351–373 calculator, approximating with, 357–358 as circular function, 357 defined, 351 function graphs See Graphs, sine and cosine functions fundamental identity, 358 inverse of, 390–392, Taylor law of See Sines, law of motion of a spring, modeling, 358–359 of multiple of 30 degrees, 353–354 of multiple of 45 degrees, 352–353 of multiple of 90 degrees, 352 of real number, 357 reference angles, 354–357 signs and quadrants, 352 of sum or difference, 444–447 value, exact, finding, 445 values, approximate, 357–358 Sine equations with double angle, 471 solving, 469–470 Sines, law of, 488–495 ambiguous case (SSA), 490–492 area of triangle, 492–494 height of object from distance, finding, 494–495 proof, 488–489 theorem, 488 two angles and included side (ASA), 489–490 I-17 Index Sine wave cycle of, 363 defined, 363 graphing See Graphs, sine and cosine functions Six sixth roots of unity, 532–533 Slant asymptotes See Oblique asymptotes Slope of line, 39–43 defined, 39 from equation, finding, 41 finding, 39–43 graphing line using, 42–43 interpreting, 47 negative, 43 perpendicular lines, 44–45 point-slope form, 40–43 positive, 43 rise and run, 39 slope-intercept form, 41–43 zero, 43 Soccer teams, ranking, 652–653 Social Security, and life expectancy, 130 Solution set defined, 29, 95 equations with one variable, equations with two variables, 28–29, 560 infinite, 574–576 Sonic boom, and hyperbola, 707 Special products, 854–855 polynomial factoring, 859 rules, using, 854–855 Speed, average, finding, 16–17 Spiral of Archimedes, 539–540, 542 Spring, motion of, modeling, 358–359, 464, 475–476 Square of a difference, 854 functions, graphing, 118, 141 matrix, 623, 626 of a sum, 854 Square root(s) defined, 845 equations, solving with, 223–224 functions, 118–119 and irrational numbers, 837 negative numbers, 62–63 principal square root, 62–63 square-root property, 67–68 Square-root functions, 118–119 defined, 118 graphing, 118–119, 141 Standard form circle, equation of, 29 complex numbers, 60 linear equations, 32, 43 Standard position of angle, 336 Statements, 797–799 defined, 797 and positive integers, 797–798 writing, mathematical induction, 798–799 Straight angles, 336 Stretching, graph, 133–134 Substitution method, system of equations, 561–563 Subtraction complex numbers, 60–61 matrices, 640 polynomials, 853 property of equality, 2–3 rational expressions, 865 Sum functions, 146–148 See also Addition Summation notation defined, 750 index of summation, 750, 752 series, writing in, 751–752 summations, evaluating, 750–751 Sum-to-product identities, 460–461 sum, evaluating, 461 sum or difference, expressing as product, 461 Sunrise, time, modeling, 372 Switch-and-solve method, inverse functions, finding, 162–163, 293–294 Symmetry, 138–139 about origin, 139, 236–237 about y-axis, 138, 236–237 in graph, determining, 139 parabola, axis of, 189, 236–237 Synthetic division, 202–204 defined, 202 polynomials, evaluating with, 203–204 solving, 202–204 System of equations, linear, 560–579 addition method, 563–565 consistent, 561 Cramer’s rule, 665–667 defined, 560 dependent, 561, 563, 565, 575–576 equivalence, 564 graphing See Graphs, system of equations inconsistent, 561, 563, 565, 576–577 independent, 561, 573–574 infinite solution sets, 574–576 matrix inverse, solving with, 658–660 modeling with, 565–567, 578–579 substitution method, 561–563 three variables, 572–579 two variables, 560–567 System of equations, nonlinear, 583–589 defined, 583 elimination of variables, solving by, 583–587 modeling with, 587 parabola and line, graphing, 583–584 solving, 585–587 System of inequalities graphing, 604–605 linear, 604–605 nonlinear, 605 T Tables, functions, identifying from, 106–107 Tangent, 379–382 defined, 379 graphs of See Graphs, tangent inverse of, 394–395 to parabola, 693 of sum or difference, 445 www.elsolucionario.net value, exact, finding, 446 Tangent equations, 470–472 with multiple angles, 471–472 solving, 470–471 Taxation negative income tax model, 52 sales tax, finding with model, 14 state and federal as function of each other, 583 Taylor polynomials, 378–379 Temperature of room, modeling, 371–372 Terminal point, vector, 509 side, angle, 336 Terms of sequences, 740–741 Test-point method linear inequalities, 602–603 polynomial inequality, 241–242 quadratic inequalities, solving, 193–194 rational inequalities, 257–258 * matrix, determinant of, 670–671 Traffic control, modeling, 632, 637 Transcendental functions defined, 274 evaluating, 422 Transformations, 131–137 defined, 131 graphing See Graphs, transformations horizontal, 131–132 multiple, 136–137 nonrigid, 136 reflection, 132–134 rigid, 136 sine and cosine function graphs, 366–367 stretching/shrinking of graphs, 133–134 translations, 131–135 vertical, 135 Translations, 131–135 defined, 131 of ellipses, 698–699 horizontal, 132 reflection, 132–134 to right/left, 131 upward/downward, 135 vertical, 135 Transverse axis, 708 Travel time, minimizing, 522, 776 Tree diagram, 771 Triangles area of See Triangles, area of bearing, 494–495 inequality, 500 oblique, 488 Pascal’s, 781–782 right See Right triangles Triangles, area of bearing in, 493–494 Heron’s formula, 501–504 law of sines, 492–494 quadrilateral, area of, 493 theorem, 492 using only sides, 502 Trigonometric functions, 351–408 compositions of, 396–397 cosecant, 379–380, 385–386 cotangent, 379–380, 383–384 Index domain and range, 362 evaluating, 380–381 inverse functions, 390–397 with reference angles, 356–357 secant, 379–380, 384–385 sine and cosine, 351–373 tangent, 379–382 transcendental, 422 trigonometric identities with, factoring with, 434 Trigonometric identities, 424–467 binomials, multiplying, 434 cofunction, 443–444 compositions, converting to algebraic functions, 427 cosine of difference, 442–443 cosine of sum, 441–442 from definitions, 424 double-angle, 450–451 with equal fractions, 434–435 equation that’s not an identity, proving, 429–430 function values, finding with, 426–427 fundamental, 358 half-angle, 451–454 odd and even, 427–429 one fraction, writing as two, 435 one function in terms of another, 426 product-to-sum, 459–460 proving from equation, 436–437 Pythagorean, 424, 427 reciprocal, 424 reduction formula, 462–464 right-hand side, simplifying, 436 simplifying with, 425–426, 433–434, 446 sine of sum or difference, 444–445 sum-to-product, 460–461 tangent of sum or difference, 445–446 trigonometric functions, factoring with, 434 verifying, 433–437, 454–455 Trigonometric ratios, 401–402 finding, 401–402 theorem, 401 Trigonometry angle of depression, 405–406 angle of elevation, 405–407 complex numbers See Complex numbers, trigonometric form conditional equations, 467–477 early study of, 336 functions See Trigonometric functions identities See Trigonometric identities law of cosines, 499–504 law of sines, 488–495 parametric equations, 545–548 polar coordinate system, 535–537 polar equations, 537–542 right triangles, 401–408 significant digits, 404–405 trigonometric ratios, 401–402 vectors, 509–517 Trinomials coefficient, finding, 783 defined, 852 factoring, 859 perfect square, 859 Triple-angle identity, verifying, 454 * matrix, determinants of, 664–665 U Unbounded intervals, interval notation for, 82 Uniform-motion problem, solving, 16–17 Union, sets, 85 Unit circle defined, 340 sign and cosine, 354 Unit vectors, 515 Upper bound, roots, 216 V Variables dependent/independent, 105 dummy, 109 equations with one See Equations, one variable equations with two See Equations, two variables function notation with, 109–110 functions, 104–105 specified, solving for, 13 value, finding with formula, 13–14 Variations, 170–174 combined, 173–174 constant/proportionality constant, 170–171 direct, 170–171 inverse, 171–172 joint, 172–173 Vectors, 509–517 addition, 511 component form, 512–513 course and ground speed of plane, finding, 517 defined, 509 direction angle, 512 direction of, 509 dot product, 513–514 drift angle, 517 equality, 510 force, finding, 516–517 initial and terminal points, 509 linear combination of, 515 magnitude and direction, finding, 512–513 magnitude of vector quantity, 509 orthogonal, 514 parallel, 514 perpendicular, 514 position vector, 511–512 radius vector, 511–512 resultant, magnitude and direction of, 515–516 scalar multiplication, 510–511 scalar product, 513 sum and difference, 513 two, angle between two, finding, 514 unit, 515 vector quantities, 509 vertical and horizontal components, 511–512 zero, 510 www.elsolucionario.net I-18 Velocity angular, 344–346 linear, 344–346 Vertex of angle, 336 of ellipse, 695 of parabola, finding, 189–190, 686 Vertical component, vectors, 511–512 lines, graphing, 33 line test, 105 Vertical asymptotes, 249–255 defined, 250, 381 graphing, 253–255, 259, 381–382 identifying, 251–252 Vertical translations graphing, 135 sine and cosine function graphs, 367 upward/downward, 135 Viewing angle, maximizing, 400 best view, modeling, 509 Volume, maximizing, 248 W Water quality, modeling, 314 Waveform, guitar note, modeling, 390 Work problems constraints, modeling, 616 defined, 18 working, time spent, 18–19, 600 X x-axis, position on graph, 25 x-intercept defined, 32 parabola, 191–192 polynomial function graph, 237–238 xy-plane, 25 Y y-axis position on graph, 25 symmetry, 138, 236–237 y-intercept defined, 32 graphing line using, 42–43 identifying, slope-intercept form, 41 parabola, 191–192 Z Zero exponent, 844 factor property, 66, 222 and fundamental theorem of algebra, 205 matrix, 639 multiplication property of, 4, 838 rational zero theorem, 205–209 slope of line, 43 solution to equation, 33 vector, 510 Innovative Technology to Help You Succeed MyMathLab can improve any learning environment—whether you are taking a lab-based, hybrid, fully online, or a traditional lecture-style course INTERACTIVE FIGURES Math comes alive with new Interactive Figures in MyMathLab! Your instructor may choose to assign assessment questions that are written to accompany each figure This interaction will lead you to fully understand key mathematical concepts in a hands-on, engaging way A HISTORY OF SUCCESS Results show that you can improve your grade by using the videos, animations, interactive figures, step-by-step examples, and personalized feedback in MyMathLab To see the growing list of case studies for yourself, visit www.mymathlab.com/success-stories www.mymathlab.com www.elsolucionario.net The Right Tools to Drive Your Success The power to succeed is in your hands Stay focused on what’s important, get the support you need, and practice the concepts regularly Strategically placed learning aids throughout the book give you guidance right when you need it Strategies and Procedures appear throughout the text to sharpen your problem-solving skills Strategies contain general guidelines while Procedures provide a step-by-step approach for solving specific types of problems STRATEGY Solving Exponential and Logarithmic Equations If the equation involves a single logarithm or a single exponential expression, then use the definition of logarithm: y = log a1x2 if and only if a y = x Use the one-to-one properties when applicable: a) if a M = a N, then M = N PROCEDURE b) if log a1M2 = log a1N2, then M = N Graphing a Rational Function To graph a rational function in lowest terms: Page 317 Determine the asymptotes and draw them as dashed lines Foreshadowing Calculus gives a brief indication of the connection between certain algebraic topics and calculus, and identifies topics that you will see again in the future Check for symmetry Find any intercepts Plot several selected points to determine how the graph approaches the asymptotes Draw curves through the selected points, approaching the asymptotes Page 253 ■ Foreshadowing Calculus There are several ways to define the number e and the function f1x2 = e x in calculus Calculus can deepen your understanding of many of the concepts that you study in algebra Function Galleries show entire families of functions and their graphs, helping you make the connection between their visual and mathematical properties Page 282 FUNCTION gallery Exponential and Logarithmic Functions Tying It All Together exercises cover the present and prior chapters so that you can integrate multiple concepts and skills Exponential: ƒ1x2 = a x , domain 1؊H, H2, range 10, H AVAILABLE IN – –1 h(x) = ⎞–1⎞ ⎠2⎠ g(x) = x y y y 6 5 4 3 2 1 x Increasing on 1؊H, H2 y-intercept 10, 12 TYING IT ALL together x –2 –1 j(x) = e x 1 x –2 –1 Decreasing on 1؊H, H2 y-intercept 10, 12 x Increasing on 1؊H , H y-intercept 10, 12 Chapters 1–4 Page 295 Solve each equation 2 # log1x - 32 = log142 log 21x - 32 = 4 x - = 1x - = x - = x - 4x = - x - = 4x 1x - 322 = 10 x = CONCEPTS OF x - = 11 log1x - 32 + log142 = log1x2 12 x - 4x + x + = calculus Sketch the graph of each function 14 y = 1x - 22 15 y = 17 y = log 21x - 22 13 y = x 18 y = x - 19 y = 2x 21 y = e 22 y = - x 23 y = 2 x x 16 y = x -2 20 y = log12 x2 24 y = The instantaneous rate of change of ƒ(x) ؍ex x - Page 332 Concepts of Calculus discussions preview important topics of calculus, and may be used as a writing or collaborative learning assignment In the Concepts of Calculus for Chapter 2, we defined the average rate of change of a function f on the interval 3x, x + h4 as ƒ1x + h2 - ƒ1x2 h We then defined the instantaneous rate of change of the function as the limit as h approaches zero of the average rate of change The notation ƒЈ1x2, read “ƒ prime of x,” is used for the instantaneous rate of change So ƒЈ1x2 = lim hS0 ƒ1x + h2 - ƒ1x2 h In the following exercises we will find the instantaneous rate of change of the base e exponential function It will be necessary here to find a limit using a table of values as we did in the Concepts of Calculus for Chapter Page 334 Continued next page www.elsolucionario.net The Right Tools to Drive Your Success End-of-section exercises and review features help you check your progress and make sure you’re mastering the material AVAILABLE IN FOR thought log 1001102 = ln1e 2.4512 = 2.451 For any positive real number x, e ln1x2 = x For any base a, where a and a ϶ 1, log a102 = If ƒ1x2 = log 31x2, then ƒ-11x2 = x log110 32 + log110 52 = log110 82 10 log110002 = 1000 10 log 21322 - log 2182 = log 2142 The domain of ƒ1x2 = ln1x2 is 1-ϱ, ϱ2 EXERCISES For Thought are true/false questions that review the basic concepts from the section, helping you check your understanding before starting on the exercises Answers are available in the back of the book True or False? Explain The first coordinate of an ordered pair in an exponential function is a logarithm 4.2 Fill in the blank NEW! Fill-in-the-blank exercises now appear at the beginning of the section exercise sets to help you learn definitions, rules, and theorems Sketch the graph of each function, and state the domain and range of each function The inverse of an exponential function is a(n) function 33 y = log 31x2 34 y = log 41x2 A base 10 logarithm is a(n) logarithm 35 ƒ1x2 = log 51x2 36 g1x2 = log 81x2 Base e logarithm is a(n) logarithm 37 y = log 1>21x2 38 y = log 1>41x2 The function ƒ1x2 = log a1x2 is if a and if a 39 h1x2 = log 1>51x2 40 s1x2 = log 1>101x2 41 ƒ1x2 = ln1x - 12 42 ƒ1x2 = log 31x + 22 The y-axis is a(n) for the graph of ƒ1x2 = log a1x2 The of the function ƒ1x2 = log a1x2 is 10, ϱ2 The of functions consists of all functions of the form ƒ1x2 = b # log a1x - h2 + k 43 ƒ1x2 = - + log1x + 22 44 ƒ1x2 = - log1x + 62 45 ƒ1x2 = - log1x - 12 46 ƒ1x2 = - # log 21x + 22 Rethinking exercises appear at the end of the section exercise sets for a cumulative review of skills learned in previous sections and chapters Use a graph or a table to find each limit The property of logarithms indicates that if log a1m2 = log a1n2, then m = n Determine the number that can be used in place of the question mark to make the equation true 47 lim log 31x2 48 lim+ log 31x2 49 lim+ log 1>21x2 50 lim log 1>21x2 xSϱ xS0 xS0 xSϱ 51 lim+ ln1x2 52 lim ln1x2 10 = 16 11 = 81 53 lim log1x2 54 lim+ log1x2 12 ? = 13 16 ? = 14 16 ? = 16 Write the equation of each graph in its final position ? 15 a b = 125 1 ? 16 a b = 125 = 64 ? ? ? 18 log 21162 19 log 311>812 20 log 3112 21 log 16122 22 log 161162 23 log 1>511252 24 log 1>511>1252 25 log10.12 26 log110 62 27 log112 29 ln1e2 30 ln102 xSϱ xSϱ xS0 55 The graph of y = ln1x2 is translated three units to the right and then four units downward Find the indicated value of the logarithmic functions 17 log 21642 xS0 28 log1102 31 ln1e - 52 AVAILABLE IN RETHINKING 149 Find the domain and range of the function ƒ1x2 = - x - + 150 Solve x - = 45x - 151 Evaluate 12 * 10 -92315 * 10 322 without a calculator Write the answer in scientific notation 152 A pond contains 2000 fish of which 10% are bass How many bass must be added so that 20% of the fish in the pond are bass? 56 The graph of y = log1x2 is translated five units to the left and then seven units upward 57 The graph of y = log 21x2 is translated five units to the right, reflected in the x-axis, and then translated one unit downward 153 Find all real and imaginary solutions to x + 13x = 4x 58 The graph of y = log 31x2 is translated four units upward, six units to the left, and then reflected in the x-axis 154 The cost of installing an oak floor varies jointly with the length and width of the room If the cost is $875.60 for a room that is feet by 11 feet, what is the cost for a room that is 10 feet by 14 feet? 32 ln1e 92 Page 297 Page 300 AVAILABLE IN POP QUIZ 4.1 What is ƒ142 if ƒ1x2 = - x ? What is a, if ƒ1a2 = and ƒ1x2 = -x? Is ƒ1x2 = -x increasing or decreasing? If $1000 earns 4% annual interest compounded quarterly, then what is the amount after 20 years? Find the domain and range for y = e x - + What is the horizontal asymptote for y = x - 1? Find the amount in the last problem if the interest is compounded continuously Solve 11>42 = 64 x LINKING concepts Page 288 Pop Quizzes at the end of every section offer eight to ten questions to confirm that you understand the material before you continue reading For Individual or Group Explorations Comparing Exponential and Linear Models The function ƒ1t2 = 300e 0.5t gives the number of bacteria present in a culture t hours after the start of an experiment in which the bacteria are growing continuously at a rate of 50% per hour Let A 3a, b4 represent the average rate of change of ƒ on the time interval 3a, b4 a) Fill in the following table Interval a, b4 ƒ 1a2 A3 a, b4 A3 a, b4 , ƒ 1a2 33.00, 3.054 37.50, 7.514 38.623, 8.6244 b) Linking Concepts are multipart exercises that require the use of concepts from previous sections to illustrate the links among various concepts What are the units for the quantity A 3a, b4 ? c) What can you conjecture about the ratio A 3a, b4 >ƒ1a2? d) Test your conjecture on a few more intervals of various lengths Explain how the length of the interval affects the ratio e) State your conjecture in terms of variation (Section 2.6) What is the constant of proportionality? f) Suppose the bacteria were growing in a linear manner, say ƒ1t2 = 800t + 300 Make a table like the given table and make a conjecture about the ratio A 3a, b4>ƒ1a2 g) Explain how your conclusions about average rate of change can be used to justify an exponential model as better than a linear model for modeling growth of a bacteria (or human) population Page 288 www.elsolucionario.net This page intentionally left blank www.elsolucionario.net ^/Geometry Square Rectangle Area - LW Perimeter = 2L + 2W Area = s Perimeter — 4s Triangle Right Triangle Area = \bh Area = \ab Pythagorean theorem: Parallelogram Trapezoid Circle Area = bh Area = {h(b^ + b2) Area = irr2 Circumference = 2nr Right Circular Cone Right Circular Cylinder Sphere Volume = j7rr2h Lateral surface area = irrJr2 + h2 Volume = 7rr2h Lateral surface area = 27irh Volume — jTjr3 Surface area = 47rr2 English-Metric Conversion Metric Abbreviations Length mm millimeter cm centimeter dm decimeter IB meter dam dekameter hm hectometer km kilometer Volume mL milliliter cL centiliter dL deciliter L liter daL dekaliter hL hectoliter kL kiloliter Weight mg milligram centigram eg dg decigram gram g dag dekagram hg hectogram kg kilogram Length lin = 2.540 cm ft = 30.48 cm l y d = 0.9144 m mi = 1.609km Length cm = 0.3937 in cm = 0.03281 ft m -1.0936yd l k m = 0.6215 mi www.elsolucionario.net Volume (U.S.) pt - 0.4732 L qt = 0.9464 L I f f al - 3.785 L Weight oz = 28.35 g Ib = 453.6 g Ib - 0.4536 kg Weight Volume (U.S.) 1L - 2.2233 pt lg = 0.0353 oz 1L - 1.0567qt lg = 0.002205 Ib 1L = 0.2642 gal k g := 2.205 Ib Algebra Subsets of the Real Numbers Exponents Natural numbers — {1, 2, 3, .} Whole numbers = {0, 1, 2, 3, .} an = a • a Integers = { -3, -2, -1, 0, 1, 2, 3, } a (n factors of a) a°= Rational = I — a and b are integers with b + \ Irrational = {x\x is not rational} Properties of the Real Numbers For all real numbers a, b, and c a + b and ab are real numbers a + b = b + a\a*b = b*a (a + b) + c = a + (b + c); , ' ,, (ab)c = a(bc) a(b + c) = aft + ac\ ' , /7 a(b — c) — ab — ac a + — a\\ • a = a a + (-a) = 0; a • - = a Closure Commutative Radicals Associative Distnbutive Identity (a * 0) a •0 = Inverse Multiplication property of Factoring a2 + 2aft + b2 = (a + ft)2 a2 - 2aft + ft2 - (a - ft)2 a2 - ft2 = (a + ft)(a - ft) a3 - ft3 - (a - b)(a2 + ab + ft2) a3 + ft3 - (a + ft)(a2 - aft 4- ft Absolute Value v^c2 = |jc| for any real jc \x\ = k o x = k or x = -k Rational Expressions (k > 0) |JE| < k O -fc < ^ < k (k > 0) |jt| > it k (k > 0) (The symbol means "if and only if.") Quadratic Formula The solutions to ax2 + bx + c = with a + are Interval Notation (a, ft) - {^|a < ^ < ft} (a,ft]- {x\a and a ± Straight Line Slope-intercept form: y — mx + b Slope: m ^-intercept: (0, b) Point-slope form: y — y{ = m(x — xv) Standard form: Ax + By = C Horizontal: y = k Vertical: x = k www.elsolucionario.net Algebra Parabola Arithmetic Sequence y = a(x- h}2 4- k (a + 0) Vertex: (A, k) Axis of symmetry: x = h a l di + d, av + 2d, al + 3d, Formula for nth term: an = al + (n — l)d Sum of n terms: Focus: ( Directrix: y = k — p Geometric Sequence Circle Center: (h, k) Radius: r Center (0, 0) Radius: r a l f a^r, a{r2, alr3, Formula for nth term: a Sum of n terms when r =£ 1: Ellipse Sum of all terms when |r| < 1: Center: (0, 0) Major axis: horizontal 2 Foci: (±c, 0), where c = a - £ Counting Formulas r2 Factorial notation: «! = • • (n — 1) * n Permutation: P(n, r) Center: (0, 0) Major axis: vertical 2 Foci: (0, ±c), where c = a — b Hyperbola x2 y2_ = Combination: C(n, n Binomial Expansion a b Center: (0, 0) Vertices: (±a, 0) Foci: (±c, 0), where c = a2 + &2 (a + 6)2 - a2 + 206 + fc2 (a + ^)3 - a3 + 3a26 + 3«/?2 + b3 (a + ^)4 = a4 + 4a36 + 6a2£2 + 4a/73 + ^4 Asymptotes: y Center: (0, 0) Vertices: (0, ±a) Foci: (0, ±c), where c2 ~ a2 + b2 Asymptotes: y www.elsolucionario.net Trigonometry Exact Values of Trigonometric Functions Trigonometric Functions If the angle a (in standard position) intersects the unit circle at (jc, y), then JC degrees sin jc 0 0° sin a = y cos a = x sec a esc a tan a 30° cot a 45° 60° Trigonometric Ratios 90° If (jc, y) is any point other than the origin on the terminal side of a and r = V*2 + J >then JC radians COSJC 77 "4 V2 V3 V2 77 I V3 2 77 "2 77 ? tanjc V3 V3 Basic Identities sin a cos a esc a sec a tan a cot a Right Triangle Trigonometry tan* COt JC sin jc CSCJC cos jc: sec jc If a is an acute angle of a right triangle, then Pythagorean Identities sin a cos a = tan a sin2 jc + cos2 x = 1 + cot2 x = esc2 x tan2 jc + = sec2 x esc a sec a: cot a Odd Identities Special Right Triangles sin(—*) = — sin(;c) csc(-z) = —csc(^c) tan(-;c) = — tan(;c) cot(—jc) = — cot(x) Even Identities cos(—x) = cos(jc) www.elsolucionario.net sec(— jc) — sec(jc) Trigonometry Product-to-Sum Identities Cofunction Identities sinl cos u cos( sin u sin A cos B sin(A - B)] tan! cot u cotf tan u sin A sin B - cos(A + B)] seel esc u esc I sec u cos A sin B sin(A - B)] cos A cos B Cosine of a Sum or Difference cos(a + ft) = cos a cos ft — sin a sin /3 Sum-to-Product Identities cos(a - ft) = cos a cos ft H- sin a sin ft sin x + sin y = sinl - Sine of a Sum or Difference sin x — sin y = cost sin(a + ft) = sin a cos ft + cos a sin ft sin(a ~ ft) = sin a cos /3 - cos a sin ft cos(A + B)] cos x + cos y = cos I cos x — cos y = —2 sir Tangent of a Sum or Difference tan(a + ft} = tan a + tan ft *-1 - tan a tan tan(a - ft) = tan a - tan *-1 + tan a tan ft Reduction Formula If a is an angle in standard position whose terminal side contains (a, b), then for any real number x Oblique Triangle Double-Angle Identities sin 2x = sin x cos x cos 2x = cos2 x - sin2 x — cos2 * — = — sin2 jc tan 2* = Law of Sines In any triangle, Half-Angle Identities Law of Cosines a2 = b2 + c2 - 26c cos a ^2 = a2 + c2 - 2ac cos j3 c2 - a2 +fc2- 2a^ cos y www.elsolucionario.net ... Cataloging-in-Publication Data Dugopolski, Mark Precalculus: functions & graphs / Mark Dugopolski — 4th ed p cm Includes index ISBN-13: 978-0-321-78943-3 ISBN-10: 0-321-78943-1 Precalculus? ??Textbooks Precalculus? ??Graphic... Limits 102 Functions and Graphs 2.1 2.2 2.3 2.4 2.5 2.6 103 Functions 104 Graphs of Relations and Functions 117 Families of Functions, Transformations, and Symmetry Operations with Functions 146... foundation for success, and Precalculus: Functions and Graphs will help you succeed in this course and beyond New to the Fourth Edition For this edition of Precalculus: Functions and Graphs, I have updated