A Brief Guide to Getting the Most from this Book Read the Book Feature Description Benefit Section-Opening Scenarios Every section opens with a scenario presenting a unique application of algebra or trigonometry in your life outside the classroom Realizing that algebra and trigonometry are everywhere will help motivate your learning (See page 106.) Detailed Worked-Out Examples Examples are clearly written and provide step-by-step solutions No steps are omitted, and each step is thoroughly explained to the right of the mathematics The blue annotations will help you understand the solutions by providing the reason why every algebraic or trigonometric step is true (See page 674.) Applications Using Real-World Data Interesting applications from nearly every discipline, supported by up-to-date real-world data, are included in every section Ever wondered how you’ll use algebra and trigonometry? This feature will show you how algebra and trigonometry can solve real problems (See page 265.) Great Question! Answers to students’ questions offer suggestions for problem solving, point out common errors to avoid, and provide informal hints and suggestions By seeing common mistakes, you’ll be able to avoid them This feature should help you not to feel anxious or threatened when asking questions in class (See page 109.) Brief Reviews NEW to this edition Brief Reviews cover skills you already learned but may have forgotten Having these refresher boxes easily accessible will help ease anxiety about skills you may have forgotten (See page 478.) Achieving Success NEW to this edition Achieving Success boxes offer strategies for persistence and success in college mathematics courses Follow these suggestions to help achieve your full academic potential in college mathematics (See page 586.) Explanatory Voice Balloons Voice balloons help to demystify algebra and trigonometry They translate mathematical language into plain English, clarify problem-solving procedures, and present alternative ways of understanding Does math ever look foreign to you? This feature often translates math into everyday English (See page 201.) Learning Objectives Every section begins with a list of objectives Each objective is restated in the margin where the objective is covered The objectives focus your reading by emphasizing what is most important and where to find it (See page 633.) Technology The screens displayed in the technology boxes show how graphing utilities verify and visualize algebraic and trigonometric results Even if you are not using a graphing utility in the course, this feature will help you understand different approaches to problem solving (See page 110.) Work the Problems Feature Description Benefit Check Point Examples Each example is followed by a matched problem, called a Check Point, that offers you the opportunity to work a similar exercise The answers to the Check Points are provided in the answer section You learn best by doing You’ll solidify your understanding of worked examples if you try a similar problem right away to be sure you understand what you’ve just read (See page 739.) Concept and Vocabulary Checks These short-answer questions, mainly fill-in-the-blank and true/false items, assess your understanding of the definitions and concepts presented in each section It is difficult to learn algebra and trigonometry without knowing their special language These exercises test your understanding of the vocabulary and concepts (See page 229.) Extensive and Varied Exercise Sets An abundant collection of exercises is included in an Exercise Set at the end of each section Exercises are organized within several categories Your instructor will usually provide guidance on which exercises to work The exercises in the first category, Practice Exercises, follow the same order as the section’s worked examples The parallel order of the Practice Exercises lets you refer to the worked examples and use them as models for solving these problems (See page 406.) Practice Plus Problems This category of exercises contains more challenging problems that often require you to combine several skills or concepts It is important to dig in and develop your problem-solving skills Practice Plus Exercises provide you with ample opportunity to so (See page 407.) Retaining the Concepts NEW to this edition Beginning with Chapter 2, each Exercise Set contains review exercises under the header “Retaining the Concepts.” These exercises improve your understanding of the topics and help maintain mastery of the material (See page 234.) Preview Problems Each Exercise Set concludes with three problems to help you prepare for the next section These exercises let you review previously covered material that you’ll need to be successful for the forthcoming section Some of these problems will get you thinking about concepts you’ll soon encounter (See page 660.) Review for Quizzes and Tests Feature Description Benefit Mid-Chapter Check Points At approximately the midway point in the chapter, an integrated set of review exercises allows you to review the skills and concepts you learned separately over several sections By combining exercises from the first half of the chapter, the Mid-Chapter Check Points give a comprehensive review before you move on to the material in the remainder of the chapter (See page 776.) Chapter Review Grids Each chapter contains a review chart that summarizes the definitions and concepts in every section of the chapter Examples that illustrate these key concepts are also referenced in the chart Review this chart and you’ll know the most important material in the chapter! (See page 815.) Chapter Review Exercises A comprehensive collection of review exercises for each of the chapter’s sections follows the grid Practice makes perfect These exercises contain the most significant problems for each of the chapter’s sections (See page 209.) Chapter Tests Each chapter contains a practice test with approximately 25 problems that cover the important concepts in the chapter Take the practice test, check your answers, and then watch the Chapter Test Prep Videos to see worked-out solutions for any exercises you miss You can use the chapter test to determine whether you have mastered the material covered in the chapter (See page 213.) Chapter Test Prep Videos These videos contain worked-out solutions to every exercise in each chapter test and can be found in MyMathLab and on YouTube The videos let you review any exercises you miss on the chapter test Objective Videos NEW to this edition These fresh, interactive videos walk you through the concepts from every objective of the text The videos provide you with active learning at your own pace Cumulative Review Exercises Beginning with Chapter 2, each chapter concludes with a comprehensive collection of mixed cumulative review exercises These exercises combine problems from previous chapters and the present chapter, providing an ongoing cumulative review Ever forget what you’ve learned? These exercises ensure that you are not forgetting anything as you move forward (See page 667.) Algebra and Trigonometry Robert Blitzer Miami Dade College th Edition Director, Portfolio Management: Anne Kelly Executive Marketing Manager: Peggy Lucas Courseware Portfolio Manager: Dawn Murrin Marketing Assistant: Adiranna Valencia Portfolio Management Administrator: Joseph Colella Senior Author Support/Technology Specialist: Joe Vetere Content Producer: Kathleen A Manley Production Coordination: Francesca Monaco/codeMantra Managing Producer: Karen Wernholm Text Design and Composition: codeMantra Producer: Erica Lange Illustrations: Scientific Illustrators Manager, Courseware QA: Mary Durnwald Photo Research and Permission Clearance: Cenveo Publisher Services Manager, Content Development: Kristina Evans Cover Design: Studio Montage Product Marketing Manager: Claire Kozar Cover Image: Ray_of_Light/Shutterstock Marketing Assistant: Jennifer Myers Copyright © 2018, 2014, 2010 Pearson Education, Inc All Rights Reserved Printed 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 otherwise For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/ Acknowledgments of third-party content appear on page C1, which constitutes an extension of this copyright page PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks owned by Pearson Education, Inc or its affiliates in the U.S and/or other countries Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its affiliates, authors, licensees or distributors Library of Congress Cataloging-in-Publication Data Names: Blitzer, Robert Title: Algebra and trigonometry / Robert Blitzer, Miami Dade College Description: Sixth edition | Hoboken, NJ : Pearson Prentice Hall, [2018] | Includes answers to selected exercises | Includes subject index Identifiers: LCCN 2016042563 | ISBN 9780134463216 Subjects: LCSH: Algebra—Textbooks | Trigonometry—Textbooks Classification: LCC QA152.3.B63 2018 | DDC 512/.13—dc23 LC record available at https://lccn.loc.gov/2016042563 ISBN 13: 978-0-13-446321-6 ISBN 10: 0-13-446321-8 Contents Preface vii Mid-Chapter Check Point 171 Acknowledgments x 1.6 Other Types of Equations 173 Dynamic Resources xii 1.7 Linear Inequalities and Absolute Value Inequalities 189 To the Student xv Summary, Review, and Test 206 About the Author xvi Review Exercises 209 Applications Index xvii Chapter Test 213 P Prerequisites: Fundamental Concepts of Algebra P.1 Algebraic Expressions, Mathematical Models, and Real Numbers 2 P.2 Exponents and Scientific Notation 20 P.3 Radicals and Rational Exponents 35 P.4 Polynomials 51 Mid-Chapter Check Point 63 P.5 Factoring Polynomials 64 P.6 Rational Expressions 76 Summary, Review, and Test 89 Review Exercises 90 Chapter P Test 92 Equations and Inequalities 93 1.1 Graphs and Graphing Utilities 94 1.2 Linear Equations and Rational Equations 106 1.3 Models and Applications 124 1.4 Complex Numbers 139 1.5 Quadratic Equations 148 iii iv Contents Mid-Chapter Check Point 410 3.5 Rational Functions and Their Graphs 411 3.6 Polynomial and Rational Inequalities 431 3.7 Modeling Using Variation 444 Summary, Review, and Test 454 Review Exercises 456 Chapter Test 460 Cumulative Review Exercises (Chapters 1–3) 461 Exponential and Logarithmic Functions 463 4.1 Exponential Functions 464 4.2 Logarithmic Functions 478 4.3 Properties of Logarithms 493 Mid-Chapter Check Point 503 4.4 Exponential and Logarithmic Equations 504 4.5 Exponential Growth and Decay; Modeling Data 519 Summary, Review, and Test 533 Review Exercises 535 Chapter Test 539 Functions and Graphs 215 2.1 Basics of Functions and Their Graphs 216 2.2 More on Functions and Their Graphs 235 2.3 Linear Functions and Slope 255 2.4 More on Slope 271 Mid-Chapter Check Point 281 2.5 Transformations of Functions 282 2.6 Combinations of Functions; Composite Functions 298 2.7 Inverse Functions 313 2.8 Distance and Midpoint Formulas; Circles 325 Summary, Review, and Test 334 Review Exercises 337 Chapter Test 341 Cumulative Review Exercises (Chapters 1–2) 343 Polynomial and Rational Functions 345 3.1 Quadratic Functions 346 3.2 Polynomial Functions and Their Graphs 364 3.3 Dividing Polynomials; Remainder and Factor Theorems 382 3.4 Zeros of Polynomial Functions 395 Cumulative Review Exercises (Chapters 1–4) 540 Contents v 5.1 Angles and Radian Measure 542 7.1 The Law of Sines 732 5.2 Right Triangle Trigonometry 559 7.2 The Law of Cosines 744 5.3 Trigonometric Functions of Any Angle 576 7.3 Polar Coordinates 753 5.4 Trigonometric Functions of Real Numbers; Periodic Functions 589 7.4 Graphs of Polar Equations 765 Mid-Chapter Check Point 597 5.5 Graphs of Sine and Cosine Functions 599 7.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 777 5.6 Graphs of Other Trigonometric Functions 620 7.6 Vectors 790 5.7 Inverse Trigonometric Functions 633 7.7 The Dot Product 805 5.8 Applications of Trigonometric Functions 649 Summary, Review, and Test 815 Summary, Review, and Test 660 Review Exercises 818 Review Exercises 663 Chapter Test 820 Chapter Test 666 Cumulative Review Exercises (Chapters 1–7) 821 Trigonometric Functions 541 Cumulative Review Exercises (Chapters 1–5) 667 Analytic Trigonometry 669 6.1 Verifying Trigonometric Identities 670 6.2 Sum and Difference Formulas 681 6.3 Double-Angle, Power-Reducing, and Half-Angle Formulas 692 Mid-Chapter Check Point 703 6.4 Product-to-Sum and Sum-to-Product Formulas 704 6.5 Trigonometric Equations 713 Summary, Review, and Test 726 Review Exercises 727 Chapter Test 729 Cumulative Review Exercises (Chapters 1–6) 729 Additional Topics in Trigonometry 731 Mid-Chapter Check Point 776 vi Contents Systems of Equations and Inequalities 823 8.1 Systems of Linear Equations in Two Variables 824 8.2 Systems of Linear Equations in Three Variables 843 8.3 Partial Fractions 851 8.4 Systems of Nonlinear Equations in Two Variables 862 Mid-Chapter Check Point 872 8.5 Systems of Inequalities 873 8.6 Linear Programming 885 Summary, Review, and Test 893 Review Exercises 895 Chapter Test 898 Cumulative Review Exercises (Chapters 1–8) 898 Matrices and Determinants 901 9.1 Matrix Solutions to Linear Systems 902 9.2 Inconsistent and Dependent Systems and Their Applications 916 9.3 Matrix Operations and Their Applications 925 Mid-Chapter Check Point 940 9.4 Multiplicative Inverses of Matrices and Matrix Equations 941 9.5 Determinants and Cramer’s Rule 955 Summary, Review, and Test 968 Review Exercises 969 Chapter Test 971 Cumulative Review Exercises (Chapters 1–9) 972 10 Conic Sections and Analytic Geometry 973 10.1 The Ellipse 974 10.2 The Hyperbola 989 10.3 The Parabola 1005 Mid-Chapter Check Point 1019 10.4 Rotation of Axes 1021 10.5 Parametric Equations 1032 10.6 Conic Sections in Polar Coordinates 1042 Summary, Review, and Test 1052 Review Exercises 1055 Chapter 10 Test 1057 Cumulative Review Exercises (Chapters 1–10) 1058 11 Sequences, Induction, and Probability 1059 11.1 Sequences and Summation Notation 1060 11.2 Arithmetic Sequences 1071 11.3 Geometric Sequences and Series 1082 Mid-Chapter Check Point 1097 11.4 Mathematical Induction 1098 11.5 The Binomial Theorem 1107 11.6 Counting Principles, Permutations, and Combinations 1115 11.7 Probability 1126 Summary, Review, and Test 1141 Review Exercises 1143 Chapter 11 Test 1146 Cumulative Review Exercises (Chapters 1–11) 1147 Appendix: Where Did That Come From? Selected Proofs 1149 Answers to Selected Exercises AA1 Subject Index I1 Photo Credits C1 Preface I’ve written Algebra and Trigonometry, Sixth Edition, to help diverse students, with different backgrounds and future goals, to succeed The book has three fundamental goals: To help students acquire a solid foundation in algebra and trigonometry, preparing them for other courses such as calculus, business calculus, and finite mathematics To show students how algebra and trigonometry can model and solve authentic real-world problems To enable students to develop problem-solving skills, while fostering critical thinking, within an interesting setting One major obstacle in the way of achieving these goals is the fact that very few students actually read their textbook This has been a regular source of frustration for me and for my colleagues in the classroom Anecdotal evidence gathered over years highlights two basic reasons that students not take advantage of their textbook: • “I’ll never use this information.” • “I can’t follow the explanations.” I’ve written every page of the Sixth Edition with the intent of eliminating these two objections The ideas and tools I’ve used to so are described for the student in “A Brief Guide to Getting the Most from This Book,” which appears at the front of the book What’s New in the Sixth Edition? • New Applications and Real-World Data The Sixth Edition contains 63 worked-out examples and exercises based on new data sets, and 36 examples and exercises based on data updated from the Fifth Edition Many of the new applications involve topics relevant to college students, including student-loan debt (Chapter P, Mid-Chapter Check Point, Exercise 31), grade inflation (Exercise Set 1.2, Exercises 97–98), median earnings, by final degree earned (Exercise Set 1.3, Exercises 3–4), excuses for not meeting deadlines (Chapter Summary, Exercise 36), political orientation of college freshmen (Chapter Summary, Exercise 53), sleep hours of college students (Exercise Set 8.1, Exercise 74), and the number of hours college students study per week, by major (Exercise Set 8.2, Exercises 33–34) • Brief Reviews Beginning with Chapter 1, the Brief Review boxes that appear throughout the book summarize mathematical skills, many of which are course prerequisites, that students have learned, but which many students need to review This feature appears whenever a particular skill is first needed and eliminates the need for you to reteach that skill For more detail, students are referred to the appropriate section and objective in a previous chapter where the topic is fully developed • Achieving Success The Achieving Success boxes, appearing at the end of many sections in Chapters through 8, offer strategies for persistence and success in college mathematics courses • Retaining the Concepts Beginning with Chapter 2, Section 2.1, each Exercise Set contains three or four review exercises under the header “Retaining the Concepts.” These exercises are intended for students to review previously covered objectives in order to improve their understanding of the topics and to help maintain their mastery of the material If students are not certain how to solve a review exercise, they can turn to the section and worked example given in parentheses at the end of each exercise The Sixth Edition contains 216 new exercises in the “Retaining the Concepts” category • New Blitzer Bonus Videos with Assessment Many of the Blitzer Bonus features throughout the textbook have been turned into animated videos that are built into the MyMathLab course These videos help students make visual connections to algebra and trigonometry and the world around them Assignable exercises have been created within the MyMathLab course to assess conceptual understanding and mastery These videos and exercises can be turned into a media assignment within the Blitzer MyMathLab course • Updated Learning Guide Organized by the textbook’s learning objectives, this updated Learning Guide helps students make the most of their textbook for test preparation Projects are now included to give students an opportunity to discover and reinforce the concepts in an active learning environment and are ideal for group work in class • Updated Graphing Calculator Screens All screens have been updated using the TI-84 Plus C What Content and Organizational Changes Have Been Made to the Sixth Edition? • Section P.1 (Algebraic Expressions, Mathematical Models, and Real Numbers) follows an example on the cost of attending college (Example 2) with a new Blitzer Bonus, “Is College Worthwhile?” • Section P.6 (Rational Expressions) uses the least common denominator to combine rational expressions with different denominators, including expressions having no common factors in their denominators • Section 1.1 (Graphing and Graphing Utilities) contains a new example of a graph with more than one x-intercept (Example 5(d)) vii Subject Index I3 Ellipsis, Empirical probability, 1126–1127 formula, 1126 with real-world data, 1126 Empty set, 6, 113, 116, 196, 880, 917 End behavior, 366–369, 374, 375 Endeavor space shuttle, 1013 English sentences and inequalities, 192 Equality of complex numbers, 140 Equal matrices, 926 Equals sign, 3, 874 Equal vectors, 791–792 Equation(s), See also Linear equations; Polynomial equations absolute value, 182–184 conditional, 116 dependent, 831, 847, 918, 919 exponential, 504–508, 510–513 functions as, 220–221 graphing, 95–106 graphing, using point-plotting method, 95–97 identifying even or odd functions from, 243–245 inconsistent, 116 inverse variation, 448 logarithmic, 508–513 matrix, solving, 929–930, 948–950 polar, 758–759, 765 quadratic in form, 180–182 rational, 112–114 solving, 107 trigonometric, 713–726 in two variables, 95 types of, 115–117 of variation, 444 Equation of line general form, 262–264, 274 horizontal lines, 261–262, 264 parallel to given line, 271–272 point-slope form of, 257–259, 264, 272, 273 slope-intercept form of, 259–262, 264, 272, 273 various forms of, finding, 264 vertical lines, 262, 264 Equation of the inverse, finding, 316–318 Equilibrium, 803 Equilibrium position, 654 Equilibrium price, 838 Equilibrium quantity, 838 Equivalent equations, 107, 108 Equivalent inequalities, 192 Euler, Leonhard, 488 Evaluating an algebraic expression, Evaluating the function, 221–223, 245–246 Even function, 241–245 cosine function as, 609 definition of, 241 secant function as, 626 y-axis symmetry and, 241–242 Even multiplicity, zero of, 371 Even-odd identities, 670 Even roots, 43, 300 Event(s) defined, 1128 empirical probabilities assigned to, 1126 independent, 1135–1137 mutually exclusive, 1132–1133 non-mutually exclusive, 1133–1135 theoretical probability of, 1128–1131 Even trigonometric function, 592–593 Exact value(s) of composite functions with inverse trigonometric functions, 641–645 cosine of sum to find, using, 686 of cos-1 x, 637–638 difference formula for cosines to find, 683 double-angle formulas to find, 693–694 half-angle formula to find, 697 sine of sum to find, using, 685, 687 of sin-1 x, 635–636 of tan-1 x, 639–640 using even and odd functions to find, 593 using periodic properties to find, 593–594 Expanding a logarithmic expression, 493–497 and power rule, 495–496 and product rule, 493–494 properties for, 496 and quotient rule, 494 Expanding binomials, 1108–1109 Expanding the summation notation, 1065–1066 Expansion by minors, 959, 960–961 Experiment, 1128 Exponential equations applications of, 510–513 defined, 504 natural logarithms used in solving, 506–508 solving, 504–508 Exponential expression, dividing, quotient rule for, 21–22 multiplying, product rule for, 20–21 simplifying, 26–28, 46–47 Exponential form, 479 changing from, to logarithmic form, 480 changing from logarithmic form to, 479 location of base and exponent in, 479 Exponential functions, 464–477 characteristics of, 467 defined, 464 evaluating, 465 examples of, 464 expressing model in base e, 528 graphing, 465–467, 481–482 modeling data with, 524–527 natural, 470 transformations involving, 468–469 Exponential growth, 468, 1083–1084 Exponential growth and decay models, 519–523 Exponential notation, Exponential regression option, graphing utility, 526 Exponents, 2, 3, 20–35 fractional, factoring involving, 73 and large numbers, 28 negative, 23–24, 28–30, 73 negative integers as, 23–24 power rule for, 24 products-to-powers rule for, 24–25 properties of, 46 quotient-to-power rule for, 25–26 rational, 44–47, 178–180 zero, 22 Extraneous solutions, 176 F Face cards, 1129 Factorial notation, 1063–1064 Factorials, from through 20, 1063 Factoring algebraic expressions containing fractional and negative exponents, 73 difference of two squares, 69–70 to find domain of function, 300 with fractions, 154 to identify vertical asymptotes of rational function, 416 over the set of integers, 64 perfect square trinomials, 70 polynomial equations solved by, 173–175 polynomials, 64–76 polynomials, strategy for, 71–73 quadratic equations solved by, 148–150, 161, 162 separating different functions in trigonometric equation using, 717–718 sum and difference of two cubes, 71 trinomial in two variables, 68–69 trinomial whose leading coefficient is not 1, 67–68 trinomial with leading coefficient of 1, 66–67 verifying an identity using, 673–674 Factoring by grouping, 65 Factoring completely, 64, 69–70 Factoring out Greatest Common Factor, 64–65, 73 Factors in denominator, partial fraction decomposition and, 848–856 of polynomials, 386 of term, 13 Factor Theorem, 390–391, 401 Fermat, Pierre de, 94, 1098, 1106 Fermat’s Last Theorem, 1098, 1106 Ferrari (student of Cardano), 395 Fibonacci (Leonardo of Pisa), 1060 Fibonacci numbers, on piano keyboard, 1060 Fibonacci sequence, 1060 Finite sequences, 1061 First terms, in binomial, 55 Fixed cost, 832, 833 Flu, modeling spread of, 523–524 Focus (foci) of ellipse, 976 of hyperbola, 989, 991–992, 994, 995 of parabola, 1006–1012 Focus-directrix definitions of conic sections, 1043 FOIL method and factoring by grouping, 65 and factoring trinomials, 68, 69 for multiplying complex numbers, 141, 142, 781 for multiplying polynomials, 54–55 for multiplying polynomials in two variables, 59 for multiplying sum and difference of two terms, 55–56 and special products, 58 and square of binomial, 56, 57 Forces, computing work done by, 811–812 Force vector, 799–800 Formula(s), 3–4, 93 amount of rotation, 1023 for the angle between two vectors, 807 for area, perimeter, and volume, 132 Binomial Theorem, 1109 for combinations, 1120–1122 compound interest, 473, 512, 520, 1090 distance, 325–327, 975 double-angle, 692–695, 699 empirical probability, 1126 exponential growth and decay, 520 for general term of arithmetic sequence, 1073–1077 for general term of geometric sequence, 1083–1085 graphs modeled by, 106, 118–119 half-angle, 696–698, 699 linear speed, 554 and mathematical models, 3–5 midpoint, 326–327 permutations, 1117–1119 power-reducing, 695–696, 699 product-to-sum, 705 quadratic, 157 radian measure, 545 recursion, 1062–1063 rotation of axes, 1021–1023 solving for variable in, 133–134 solving problems modeled by, 183–184 special-product, 55–58 sum and difference, 681–692, 699 sum of first n terms of an arithmetic sequence, 1076–1078 sum of first n terms of geometric sequence, 1086–1089 sum-to-product, 706–707 Tartaglia’s secret, 395 value of an annuity, 1088–1090 variation, 444 Fourier, John, 655, 710 Fraction(s) complex (See Complex rational expressions) with factorials, evaluating, 1064 linear equations with, 111 partial, 847–858 partial fraction decomposition, 847–858 solving linear inequality containing, 195–196 writing repeating decimal as, 1092 Fractional equation See Rational equations Fractional exponents, factoring algebraic expressions containing, 73 Fractional expressions, verifying an identity by combining, 674–675 Free-falling object, modeling position of, 438–440 Frequency, 655 of object in simple harmonic motion, 655, 656 of sine wave, 708 Function(s), 215–249 See also Exponential functions; Logarithmic functions; Polynomial functions; Quadratic functions; Rational functions algebra of, 300–304 analyzing graphs of, 225–226 average cost, 424–425 average rate of change, 275–278 business, 832–834 combinations of, 298–312 composite, 304–308 constant, 236–237, 262, 365 cost, 424–425, 832, 833 decomposing, 308–309 defined, 219 determining domain, 302–303 difference quotients of, 247–248 domain of, 226–228, 298–300, 302–303 as equations, 220–221 evaluating, 221–223 even, 241–245, 592–593 graphing, 223–229, 235–249, 283 graphs of common, recognizing, 283 greatest integer, 247 identifying intercepts from graph of, 229 increasing and decreasing, 235–237 inverse, 313–324, 478, 479 linear, 223, 255–269 objective, 886, 889–890 odd, 241–245, 592–593 one-to-one, 319, 478 parametric equations for, finding, 1036–1037 periodic, 593–594 piecewise, 245–247 profit, 834 quadratic, 345–363, 365, 847 reciprocal, 412–414 relations as, 218–220 relative maximum or relative minimum of, 237–238 revenue, 832–834 step, 247 sum of (addition of), 301–303 transformation of, 282–297 vertical line test for, 224 zeros of, 229 Function machine, 221 Function notation, 221–223 Fundamental Counting Principle, 1115–1116 applications of, 1115–1117 defined, 1115 Fundamental Theorem of Algebra, 401–402 Fundamental trigonometric identities, 670–681 defined, 670 to verify other identities, using, 670–678 G Galileo, 94 Galois, Evariste, 396 Games of chance, theoretical probability in, 1128–1131 Gauss, Carl Friedrich, 401, 790, 906, 910 Gaussian elimination, 906–910, 916–922 applied to dependent systems, 918–920 applied to inconsistent systems, 917–918 applied to systems with more variables than equations, 919 applied to systems without unique solutions, 916–922 with back-substitution, 906–908 solving problems using, 920–922 Gauss-Jordan elimination, 910–911, 946, 947 General form of equation of a line, 262–264 of equation of circle, 330–331 perpendicular lines, 274 of polynomial equation, 173 of quadratic equation, 148, 156, 157 using intercepts to graph, 263–264 General second-degree equation, 1021–1022 graphing, 1026 transformed to standard equations of conic sections, 1023–1028 General term of arithmetic sequence, 1073–1077 of geometric sequence, 1083–1085 of sequence, 1060 Geometric figures, formulas for area, perimeter, and volume, 132 Geometric population growth, 1084 Geometric sequences, 1081–1097 applications with, 1090 common ratio of, 1082–1083, 1086–1087 defined, 1082 general term of, 1083–1085 Ponzi schemes and, 1085 sum of first n terms of, 1086–1089 writing terms of, 1083 Geometric series, 1090–1093 infinite, 1090–1093 Gilbert, Dennis, 969 Glenn, John, 1042, 1051 Global warming, 215, 264–266 Golden rectangle, 50 Graphing calculators/graphing utilities, 97 adding and subtracting matrices, 927 angle of elevation on, 570, 571 binomial coefficients computed with, 1108, 1109 I4 Subject Index Graphing calculators/graphing utilities (cont.) change-of-base property to graph logarithmic functions on, 500 checking partial fraction decomposition on, 858 checking solution of polynomial equation on, 175 checking solution of radical equation on, 176 checking solutions of trigonometric equation on, 720 checking tables and graphs on, 223 to check real solutions of quadratic equation, 150 circles graphed on, 328, 331 combinations on, 1122 common logarithms evaluated on, 485 computations with scientific notation on, 30 converting from decimal to scientific notation on, 30 determinant of matrix evaluated on, 960 determining domain of function on, 303 “dot” mode, 416 ellipse graphed on, 977 equations graphed on, 97–100 e to various powers evaluated on, 470 evaluating trigonometric functions using, 568 exponential functions evaluated on, 465 exponential regression option, 526 factorials found on, 1063 functions evaluated on, 222 graphing polar equation using, 766 inverse trigonometric functions on, 641 keystroke sequences for rational exponents, 46 linear inequality application on, 201 linear regression feature, 526 linear system with unique solution on, 949 logarithmic regression option, 525 logistic growth function on, 524 matrix multiplication on, 932 maximum function feature, 358 minimum function feature, 847 modeling data with, 525 models for scatter plots obtained on, 266 multiplicative inverse of matrix on, 943, 948 numeric or graphic check on, 110, 117, 129, 198 parabola on, 1008, 1009, 1011, 1012 parametric mode and radian mode of, 1037 permutations on, 1118 plane curve represented by parametric equations on, 1034 polar mode, 1046 polar-to-rectangular point conversion on, 757 polynomial equations, 175 position function for free-falling object on, 440 providing evidence of identity on, 671 quadratic formula on, 163 quadratic regression program, 432 rational functions graphed with, 416 rectangular-to-polar point conversion on, 758 reduced row-echelon form on, 911 row-echelon form on, 908, 910 scientific notation on, 32 SHADE feature, 875 SINe REGression feature, 613 solutions to linear equations in one variable on, 110 solutions to rational equations on, 115 solving systems on, 827 square roots on, 45 statistical menu on, 432 sum of first n terms of arithmetic sequence on, 1077 sum of first n terms of geometric sequence on, 1087 sum of sequence on, 1066, 1077 TABLE feature on, 117, 174, 176, 357, 511, 527, 671, 717, 858 TABLE SETUP function, 98 terms of sequences, 1061–1062 verifying model of periodic behavior with, 615 verifying observations on, 687 for verifying solutions of absolute value equation, 183 of equation quadratic in form, 181 of exponential equation, 507 of linear inequality, 195 of logarithmic equation, 508 of polynomial function on, 403 of polynomial inequality, 433, 435, 436 of quadratic equation, 158 of radical equation on, 178 of rational inequality, 438 of scalar multiplication, 928 of trigonometric equation quadratic in form on, 717 ZERO feature, 370, 373 zeros of polynomial function on, 370 ZOOM SQUARE setting, 328, 331, 977 Graphs/graphing asymptotes of, horizontal and vertical, 466, 467, 482–484 circles, 329–331 of complex numbers, 778 continuous, 365 of cosecant function, 625–628 of cosine function, 608–612, 628 of cotangent function, 623–625, 628 ellipse, 977–981 equations, 95–106 even functions, 241–245 exponential functions, 465–467, 481–482 functions, 223–229, 235–249, 283 general second-degree equation, 1026 greatest integer function, 247 and horizontal line test, 319 horizontal shifts, 285–287, 291–293, 468, 483 horizontal stretching and shrinking, 290–291, 468, 483 hyperbolas, 993–999 information obtained from, 225–226 interpreting information given by, 100–102 inverse functions, 319321, 640 lemniscates, 772 limaỗons, 769 linear inequalities in two variables, 873–877 lines, 100–102 logarithmic functions, 481–484 modeling periodic behavior, 613–616 nonlinear inequalities in two variables, 876–877 odd functions, 241–245 one-to-one function, 319 parabolas, 1007–1012 piecewise function, 245–247 plane curves described by parametric equations, 1033–1034 polar equations, 765–775, 1045–1049 polynomial functions, 366–369, 373–376 quadratic functions, 346–353 rational functions, 418–423 real numbers, reciprocal function, 413 rectangular equation of curve defined parametrically, 1034–1037 reflection about the x-axis, 287–288, 291, 292 reflection about the y-axis, 288, 291 reflections of, 287–288, 291, 319, 468, 483, 484 relative maximum or relative minimum on, 237–238 rose curves, 770–771 secant function, 625–627 sequences, 1062 sequences of transformations, 291–293 sine function, 599–608, 628 smooth, 365 systems of linear equations in two variables solved by, 825, 832 systems of linear inequalities, 878–881 tangent function, 620–623, 628 of transformations of functions, 282–297 using intercepts, 98–100, 263–264 vertical lines, 262 vertical shifts, 284–287, 291–293, 612–613 vertical stretching and shrinking, 289–293, 468, 483 Gravitation, Newton’s formula for, 450 Gravity model, 453 Greater than or equal to symbol, 9, 190 Greater than symbol, 9, 190, 198 Greatest Common Factor (GCF), 64 common binomial factor, 65 factoring out, 64–65, 73 Greatest integer function, 247 Greatest perfect square factor, 38, 40 Ground speed, 803 Grouping/grouping method, factoring by, 65 Growing entity, 520 Growth models, logistic, 523–524 Growth rate, for world population, 519 H Half-angle formulas, 696–699 Half-life, 522 Half-planes, 874 Halley’s Comet, 984, 999, 1012 Hamermesh, Daniel, 214 Harding, Warren G., 704 Harmonic motion, simple, 653–656 Harvey, William, 94 Height of children, model for, 513 Heron’s formula, 748, 1150–1151 Higher-order determinants, 963–964 Horizontal asymptote(s) definition of, 416 of exponential function, 482–483 locating, 417 in logistic growth model, 523 of rational functions, 416–417, 419–421 x-axis as, 466, 467 Horizontal component of vector, 793 Horizontal lines, 261–262, 264 Horizontal line test applying, 319 for inverse functions, 318–319 Horizontal shifts, 285–287, 291–293 of exponential function, 468 to left, 285, 286 of logarithmic function, 483 to right, 285 Horizontal stretching or shrinking, 290–291 of exponential function, 468 of logarithmic function, 483 Hubble Space Telescope, 1005, 1013, 1014 Humor, 93, 106–107, 118–119 Hyperbolas, 974, 989–1004 applications with, 999–1000 asymptotes of, 992–995, 997, 999, 1023, 1153–1154 definition of, 989 eccentricity for, 1043 focus-directrix definition of, 1043 graphing, 993–999 identifying, 1021, 1029 polar equation of, graphing, 1048–1049 standard form of equation of, 989–992, 996, 1023 Hyperbolic cosine function, 478 Hyperbolic sine function, 478 Hypocycloid, 1040 Hypotenuse of right triangle, 164, 559 Pythagorean Theorem to find length of, 561, 562 I i (imaginary unit), 139–140, 144–145 IBM, 955 Identity, 115 Identity(ies), 564–566, 713 cofunction, 567 for cosine of difference of two angles, 681–683 double-angle formulas, 692–695, 699 even-odd, 670 fundamental trigonometric, 670–681 half-angle formulas, 696–698, 699 principal trigonometric, 692–699 product-to-sum formulas, 705 Pythagorean, 566, 670, 720 quotient, 564–565, 670 reciprocal, 564, 565, 670 solving trigonometric equations using, 718–721 sum-to-product formulas, 706–707 Identity function, graph of, 283 Identity property of addition, 12 of multiplication, 12 scalar, 929 Imaginary axis, 778 Imaginary numbers, 139, 159–161 Imaginary part, of complex number, 140 Incidence ratio, 429 Inconsistent equation, 116 Inconsistent systems, 830–831, 847 determinants used for identifying, 963 geometric possibilities for, 918 matrices used for identifying, 917–918 Increasing function, 236–238 Independent events, 1135–1137 Independent variable, 220 Index, 42 reducing, 47 of summation, 1065, 1066 Individual Retirement Account, 1090 Inequalities absolute value, 198–200 compound, 197 and English sentences, 192 equivalent, 192 with infinitely many solutions, 196 isolating x in middle of, 197 with no solution, 196 polynomial, 431–436, 438–440 properties of, 193 rational, 436–438 solution set of, 189 solving, 189, 192–196 triangle, 11 with unusual solution sets, 196 Inequality symbols, 9, 189, 193, 874, 875 Infinite geometric series, 1090–1093 sum of, 1090–1093 Infinite interval, 190 Infinite sequence, 1061 Infinity symbol, 190 Initial point of directed line segment, 790 of vector at origin (position vector), 794 Initial side of angle, 542–543 Inside terms, in binomial, 55 Integers, Intercepts for graphing linear equations, 263–264 graphing using, 98–100, 263–264 identifying, 99–100 identifying, from function’s graph, 229 Interest compound, 472–473, 512, 520, 1089 simple, 130–131, 133–134 simple, dual investments with, 130–131 Interest rates, doubling time and, 512 Intermediate Value Theorem, 372–373 Intersection(s) of intervals, 191–192 of sets, 5–6, 299 Interval notation, 189–191, 193, 196, 197, 199, 200, 226–228 Intervals closed, 190 infinite, 190 intersections and unions of, 191–192 of nonrestricted cosine function, 640 open, 189, 236, 237 on real number line, 190 satisfying polynomial inequality, 432–434 satisfying rational inequality, 438 on which function increases, decreases, or is constant, 236–237 Inverse additive, 13, 927 equation of the, finding, 316–318 multiplicative (reciprocal), 13, 79 Inverse cosine function, 636–638 defined, 637 exact values of, 637–638 inverse properties, 642 Inverse function(s), 313–324, 478, 633 defined, 315 of exponential function (See Logarithmic functions) finding, 316–318 graphing, 319–321 horizontal line test for, 318–319 inverse of domain-restricted function, finding, 320 notation, 314, 315 verifying, 315 Inverse property(ies) of addition, 12 of logarithms, 481 of multiplication, 12 using, 487 Inverse sine, 570 Inverse sine function, 633–636 defined, 634 exact values of, finding, 635–636 inverse properties, 642 notation, 634 simplifying expression involving, 645 Inverse tangent function, 638–640 defined, 638 exact values of, 639–640 inverse properties, 642 Subject Index I5 Inverse trigonometric functions, 633–649 exact values of composite functions with, 641–645 inverse properties, 642 using calculator to evaluate, 641 Inverse variation, 447–449 in combined variation, 449–450 equations, 448 problem solving, 448–449 Inverted cycloid, 1038 Invertible square matrix, 944, 948 Investments choosing between, 472–473 college as, and compound interest, 471–473, 512, 1089 and simple interest, 130–131, 133–134 Irrational number, as exponent in exponential function, 465 natural base e, 470–471, 488, 528 as solutions to quadratic equations, 159, 161 Isolating x in the middle of inequality, 197 Isosceles right triangle, 169, 562 J Jeter, Derek, 1040 Job offers, 1082 Joint variation, 450–451 Jordan, Wilhelm, 910 K Kahl, Joseph, 969 Kepler, Johannes, 983, 1049 Kidney stone disintegration, 973, 984 Kim, Scott, 254 Kurzweil, Ray, 468 L Laffer, Arthur, 394 Large numbers and exponents, 28 names of, 28 Last terms, in binomial, 55 Latus rectum of parabola, 1008, 1009, 1011, 1012 Law of Cosines, 744–752, 1150–1151 applications of, 747–748 defined, 745 derivation, 744–746 dot product formula derived using, 807 oblique triangles solved using, 745–747 Law of Sines, 732–743, 746, 747 ambiguous case, solving triangle in, 735–738 applications of, 738–739 area of oblique triangle, finding, 738 defined, 732 derivation of, 732–733 solving oblique triangle using, 733–735 Leading coefficient, 52, 365 Leading Coefficient Test, 366–369 Leaning Tower of Pisa, 439 Learning curve, 122 Least common denominator (LCD), 80 finding, 81–84 in solving linear equation involving fractions, 111 in solving rational equations, 112, 113 Legs, of right triangle, 164 Lemniscates, 772 Length of circular arc, 553 Less than or equal to symbol, 9, 190 Less than symbol, 9, 190, 198 Light, theory of, 1038 Like radicals, 3940 Like terms, 1314, 53, 54 Limaỗon, 769 Limitations, inequalities to describe, 886–888 Line(s) equations of, 264 (See also Equation of line) parallel, 116, 197, 271–272, 830 perpendicular, 272–274 regression, 255 secant, 275–276, 278 slope of, 255–257 tangent, to circle, 334 Linear combination of vectors, 793 Linear equations, 106–111 algebraic word problems solved using, 124–133 applications of, 118–119 defined, 107 with fractions, 111 intercepts used for graphing, 263–264 in one variable, 107–111 solving, 107–111 in three variables, 843 Linear Factorization Theorem, 402–403 Linear factors partial fraction decomposition with distinct, 848–850 partial fraction decomposition with repeated, 850–852 Linear functions, 223, 255–269, 365 arithmetic sequences as, 1072 constant function, 262, 365 data modeled with, 264–266 graphing in slope-intercept form, 259 modeling data with, 524, 526–527 Linear inequalities containing fractions, solving, 195–196 problem solving with, 200–201 properties of, 193 solving, 189, 192–196 systems of, 877–881, 885–893 with unusual solution sets, 196–197 Linear inequalities in one variable, 189, 192–196 Linear inequalities in two variables, 873–877 graphing, 873–877 Linear numerators, partial fraction decomposition with, 852–853 Linear programming, 885–893 constraints in, 886–888 objective functions in, 886, 889–890 problem solving with, 888–890 Linear regression feature, graphing utility, 526 Linear speed, 554, 555 Linear systems See Systems of linear equations Line graphs, 100–102 Line segments, directed, 791–792 Line segments, midpoint of, 326–327 Lissajous Curve, 1040 Logarithmic equations, 508–513 applications of, 510–513 defined, 508 one-to-one property of logarithms to solve, 510 product rule used for solving, 509 quotient rule used for solving, 511 solving, 508–511 Logarithmic expressions condensing, 497–498 expanding, 493–497 Logarithmic form, 479 changing from exponential form to, 480 changing to exponential form from, 479 equations in, 479 location of base and exponent in, 479 Logarithmic functions, 478–491 with base b, 479 change-of-base property to graph, 500 characteristics of, 483 common, 485–486 definition of, 478–479 domain of, 484 graphs of, 481–484 modeling data with, 524–527 natural, 486–488 transformations involving, 483–484 Logarithmic notation, 479 Logarithmic properties, 493–500, 1149–1150 change-of-base property, 498–500, 1149–1150 involving one, 480 power rule, 495–496, 497, 498 product rule, 493–494, 496, 497, 498, 509, 1149 quotient rule, 494, 496, 497, 498, 510 using, 480–481 Logarithmic REGression option, graphing utility, 525 Logarithms common, 485–486, 499 evaluating, 480 inverse properties of, 481 natural, 486–488, 499, 500, 506–508 one-to-one property of, solving logarithmic equations using, 510 Logistic growth model, 523–524 Long division, polynomial, 382–386 Lower limit of summation, 1065, 1066 M Mach, Ernst, 702 Magnitude, 790 of directed line segment, 791 equal vectors with same direction and, 791–792 scalars involving, 790 of single point, 793 of vector in rectangular coordinates, finding, 794 writing vector in terms of its direction and, 798–799 Main diagonal, 902 Major axis, of ellipse, 975, 977–981 Malthus, Thomas, 1084 Mandelbrot set, 778, 786 Mansfield, Jayne, 823 Marx, Groucho, 1124 Mathematical induction, 1098–1106 domino analogy illustrating, 1099 principle of, 1098–1101 proving statements about positive integers using, 1101–1104 steps in proof by, 1099 Mathematical models/modeling, 4–5, 101–102 and formulas, 3–5 solving applied problems using, 118–119 with systems of linear inequalities, 877–878 word problems, 124–133 Mathematics, universality of, 1111 Matrix (matrices), 901–972 augmented, 902–911, 916–922, 946, 947 coded, 950, 951 coding, 950, 951 coefficient, 948 column, 948 constant, 948 defined, 902 determinant of * 2, 955–956 determinant of * 3, 958–961 elements of, 902, 930, 931 equal, 926 inconsistent and dependent systems identified with, 916–919 linear systems solved using, 902–916 multiplicative inverse of, 941–954 nonsquare, 944 notation for, 925–926 of order m * n, 925 square, 925, 941–949 zero, 927 Matrix equations, solving, 929–930 using inverse of matrix, 948–950 Matrix operations, 925–940 addition, 926–927 applications, 934–936 multiplication, 930–934 scalar multiplication, 928–930 solving matrix equations involving, 929–930 subtraction, 926–927 Matrix row operations, 903–905, 906–908 Maximized quantity, objective function describing, 886, 889–890 Maximum, relative, 237–238 Maximum point on cosine curve, 627 in graph of cosine function, 611, 612 in graph of sine function, 600–602, 605, 607, 610 on sine curve, 626 Maximum value of quadratic functions, 353–356, 358 Midpoint formula, 326–327 Minimized quantity, objective function describing, 886 Minimum, relative, 237–238 Minimum function feature, on graphing utility, 847 Minimum point on cosine curve, 627 in graph of cosine function, 611, 612 in graph of sine function, 600–602, 605, 607, 610 on sine curve, 626 Minimum value of quadratic functions, 353–357 Minor, 959, 964 expansion by, 959–961 Minor axis, of ellipse, 975, 980–981 Minuit, Peter, 476 Mixtures, problems involving, 834–836 Model breakdown, Modeling chaos theory and, 731 data with exponential and logarithmic functions, 524–527 music, 655 periodic behavior, 613–616 simple harmonic motion, 653–656 Modeling, with variation, 444–454 Modulus of complex number, 780 Monomials, 52 adding, 53 multiplying, 53 multiplying polynomial that is not monomial and, 53 Monteverdi, 94 Morphing, 282 Moving object, parametric representation of, 1037 Multiple angles, trigonometric equations involving, 715–716 Multiple representation of points, 755 Multiplication associative property of, 12 of binomial and trinomial, 53–54 commutative property of, 12 of complex numbers, 141–142, 781–782 of conjugates, 41–42 distributive property of, over addition, 12 identity property of, 12 inverse property of, 12 matrix, 930–934 of monomial and polynomial, 53 of monomials, 53 with numbers in scientific notation, 30 of numerator and denominator by same factor, verifying an identity by, 675 of polynomials, 53–56, 60 of polynomials in two variables, 59 product rule (See Product rule) of radical expressions, 43–44 of rational expressions, 78–79 of real numbers, 12 scalar, 792, 796, 797, 928–930 of sum and difference of two terms, 55–56, 58 Multiplication property of inequality, 193 Multiplicative identity, 12 Multiplicative identity matrix, 941–942 Multiplicative inverse (or reciprocal), 13, 79 Multiplicative inverse of matrix, 941–954 applications to coding, 950–951 of n * n matrices with n greater than 2, 945–948 quick method for finding, 944–945 solving systems of equations using, 948–950 of a square matrix, 941–949 Multiplicities of zeros, x-intercepts and, 371–372, 374, 375 Multiplier effect, and tax rebates, 1092 Music sinusoidal sound, 708 sound quality and amusia, 681, 683 Mutually exclusive events, 1132–1133 N National debt, 20, 28, 31–32 Natural base (e), 470–471, 488 in continuous compounding formula, 473 evaluating functions with, 470 expressing exponential model in, 528 Natural exponential function, 470 Natural logarithmic functions, 486–488 Natural logarithms, 486–488 changing base to, 500 exponential equations solved using, 506–508 introducing, 499 properties of, 487 Natural numbers, n compounding periods per year, 473 Negative angles, 543, 549 Negative-exponent rule, 23–24 Negative exponents, 23–24, 28–30, 73 Negative integers, as exponents, 23–24 Negative life events, responding to, 106–107, 118–119 Negative multiplication property of inequality, 193 Negative numbers, multiplying a vector by, 792 principal square root of, 144–145 properties of, 14–15 square root of, 37 square root of, as multiples of i, 139 square root of, operations with, 144–145 Negative real zeros, 403, 404 Negative reciprocal, 273 I6 Subject Index Negative slope, 256, 257 Negative square root, 36 Newton, Isaac, 450, 488 n factorial (n!), 488 Nonlinear inequality in two variables, graphing, 876–877 Nonlinear systems, 862–872 applications with, 867–868 recognizing, 862 solving by addition method, 865–867 solving by substitution, 862–865 Nonnegative square root, 36 Nonsingular square matrix, 944 Nonsquare matrix, 944 Nonsquare systems, 919 nth-order determinant, 963 nth partial sum, 1076, 1086 nth roots even and odd, 43 of real numbers, 42 solving radical equations containing, 175 Null set, Numbers, sets of, irrational numbers, rational numbers, real numbers, 7–8 Numerator(s), 13 negative exponents in, 24 partial fraction decomposition with constant, 851, 859 partial fraction decomposition with linear, 852–853, 858–859 Numerical coefficient, 13 O Objective functions, in linear programming, 886, 889–890 Oblique triangle, 732 abbreviating known measurements in, 733 area of, finding, 738 Law of Cosines to solve, 745–747 Law of Sines to solve, 733–735 Obtuse angle, 543 Odd function, 241–245 cosecant function as, 625 cotangent function as, 624 definition of, 242 and origin symmetry, 241 sine function as, 592, 600 tangent function as, 620, 621 Odd multiplicity, zero of, 371 Odd roots, 43 Odd trigonometric function, 592–593 to find exact values, 593 Ohm’s law, 147 One radian, 544 One-to-one correspondence, One-to-one functions, 319, 478 Open dots, 225 Open intervals, 189, 236, 237 Opposites (additive inverses), 13 Orbits, planetary, 1049 Order, distinguishing between combination and permutation and, 1120 Ordered pairs, 94 as solutions of systems, 824–825 Ordered triples, as solution of system of linear equation in three variables, 843–844, 918 Order of operations, Orientation, 1033 Origin, 8, 94 graphing ellipse centered at, 977–979 graphing ellipse not centered at, 979–981 graphing hyperbolas centered at, 993–996 graphing hyperbolas not centered at, 996–999 graphing parabolas with vertices at, 1007–1009 graphing parabolas with vertices not at, 1009–1012 Origin symmetry, 241, 621 Or probabilities with events that are not mutually exclusive, 1133–1135 with mutually exclusive events, 1132–1133 with real-world data, 1134–1135 Orthogonal vectors, 808 dot product and, 808 vector as sum of two, 810–811 Oscillatory motion, modeling, 653 diminishing motion with increasing time, 654 simple harmonic motion, 653–656 Outside terms, in binomial, 55 P Parabolas, 346–352, 974, 1005–1019 applications with, 1012–1014 and axis of symmetry, 347, 1006, 1007, 1010 definition of, 1005–1006 downward opening, 346–348, 351, 352, 1007, 1009, 1010 eccentricity for, 1043 finding parametric equations for, 1037 focus-directrix definition of, 1043 in form f(x) = a(x - h)2 + k, 347–350 in form f(x) = ax2 + bx + c, 351–352 graphing, 1007–1012 identifying, 1021, 1029 latus rectum of, 1008, 1009, 1011, 1012 leftward opening, 1007, 1010 polar equation of, graphing, 1047 rightward opening, 1007, 1010 standard form of equation of, 1006–1012 translations of, 1009–1012 upward opening, 346–349, 351, 352, 1007, 1010 Parallel lines, 116, 197, 271–272 inconsistent system and, 830 and slope, 271–272 Parallel vectors, 808 Parameter, 1032, 1034–1035 Parametric equations, 1032–1042 advantages over rectangular equations, 1037–1038 defined, 1033 for function y = f(x), finding, 1036 plane curves and, 1032–1037 Parentheses and distributive property, 14–15 in interval notation, 190 and simplifying algebraic expressions, 14 Partial fraction, 847–858 Partial fraction decomposition, 847–858 with distinct linear factors, 848–850 idea behind, 851–852 with prime, nonrepeated quadratic factors, 852–854 with prime, repeated quadratic factor, 858–860 with repeated linear factors, 850–852 steps in, 850 Pascal, Blaise, 94, 1111 Pascal’s triangle, 1111 Per, rate of change described using, 277 Perfect nth power, 43 Perfect square, 37, 38, 40 Perfect square trinomials, 70, 152–154 factoring, 70 with fractions, factoring, 154 Perimeter, formulas for, 132, 133 Period, 593 of cosecant function, 625 of cosine function, 608, 609–612 of cotangent function, 624 of secant function, 626 of simple harmonic motion, 654, 656 of sine function, 599–606, 716 of tangent function, 620, 621, 715 Periodic functions, 593–594 definition of, 593–594 modeling periodic behavior, 613–616 Permutations, 1117–1120 combinations compared to, 1118–1120 defined, 1118 notation, 1118 of n things taken r at a time, 1118 Perpendicular lines, 272–274 Phase shift, 606, 611 Phi (f), Photography, digital, 934–935 Picture cards, 1129 Piecewise functions, 245–247 Pixels, 934, 935 Plane curves, 1032–1037 defined, 1033 described by parametric equations, graphing, 1033–1034 finding parametric equations, 1036–1037 Planetary motion, modeling, 1049 Plotting points graphing functions by, 223–224 in polar coordinate system, 753–754 in rectangular system, 94–95 Point conversion polar-to-rectangular, 756–757 rectangular-to-polar, 757–758 Point-plotting method, 765 graphing plane curves described by parametric equations, 1033–1034 graphing polar equation by, 765–766 Point-plotting method, graphing equation using, 95–97 Points, plotting See Plotting points Point-slope form of equation of line, 257–259, 264, 272 parallel lines, 272 perpendicular lines, 273 writing, 259 Polar axis, 753–754 as axis of symmetry, 1044, 1046, 1047 symmetry with respect to, 767, 768, 770, 771, 1044, 1046, 1047 Polar coordinates, 753–754 circles in, 766 conic sections in, 1042–1052 equation conversion from rectangular coordinates to, 758–759 equation conversion to rectangular coordinates from, 759–761 multiple sets of, for given point, 755 plotting points with, 754 point conversion from rectangular coordinates to, 757–758 point conversion to rectangular coordinates from, 756–757 relations between rectangular coordinates and, 756 sign of r and point’s location in, 753 tests for symmetry in, 767 Polar coordinate system, 753 multiple representation of points in, 755 plotting points in, 753–754 Polar equation, 758–759, 765 of conics, 1043–1049 conversion to rectangular equation, 759–761 converting rectangular equation to, 758–759 graphing, 1045–1049 graphs of, 765–775 for planetary orbits, 1049 standard forms of, 1044 Polar form of complex number, 779–786, 1151 defined, 780 powers of complex numbers, 782–784 product of two complex numbers, 781–782 quotient of two complex numbers, 782 roots of complex numbers, 784–786 Polar grids, 765 Pole, 753–754 symmetry with respect to, 767, 768, 770, 771 Polk, James K., 704 Polynomial(s), 51–63 adding, 53 defined, 51 degree of, 52, 58 dividing, 382–395 dividing by those containing more than one term, 382–386 dividing using synthetic division, 387–389 factoring, 64–76 factoring completely, 64 long division of, 382–386 prime (irreducible over the integers), 64, 69 standard form of, 52 strategy for factoring, 66, 71–73 subtracting, 53 in two variables, 58–59 vocabulary of, 51–52 in x, definition of, 52 Polynomial equations, 173–175 degree of, 173 Factor Theorem to solve, 390–391 in general form, 173 properties of, 401 and roots of, 401–402 solving by factoring, 173–175 solving for roots of, 399–401 Tartaglia’s formula giving root for third degree, 395 Polynomial functions, 363–381 definition of, 365 of degree n, 365 end behavior of, 366–369, 374 even-degree, 366, 368, 369 with given zeros, finding, 402–403 graphs of, 366–369, 373–376 Intermediate Value Theorem for polynomials, 372–373 multiplicities of zeros of, 371–372, 374, 375 odd-degree, 366, 367 quadratic functions, 345–363, 365, 847 rational zeros of, 396–397 Remainder Theorem used for evaluating, 389 turning points of, 373, 374, 376 zeros of, 369–371, 374, 375, 395–409 Polynomial inequality, 431–436 definition of, 432 solving, 432–436 solving problems modeled by, 438–440 Polynomial multiplication, 53–56, 60 FOIL method used in, 54–55 multiplying monomial and polynomial that is not monomial, 53 polynomials in two variables, 59 when neither is monomial, 53–54 Ponzi schemes, 1085 Population growth geometric, 1084 U.S., modeling, 520–521 world, 519 (See also World population) Position function for a free-falling object near Earth’s surface, 438–440 Position vector, 794 Positive angles, 543 degree and radian measures of selected, 549 in terms of revolutions of angle’s terminal side around origin, 549 Positive multiplication property of inequality, 193 Positive numbers, Positive real zeros, 403, 404 Positive slope, 256, 257 Power-reducing formulas, 695–696, 699 Power rule, 24, 495–498 Powers of complex numbers in polar form, 782–784 Price, equilibrium, 838 Prime polynomials, 64, 69 Prince, Richard E., 1021 Principal, 130, 133–134, 472 Principal nth root of a real number, definition of, 42 Principal square root, 144 of a2, 37 definition of, 36 of negative number, 144–145 Probability, 1125–1140 combinations and, 1130–1131 empirical, 1126–1127 of event not occurring, 1131–1132 or probabilities with events that are not mutually exclusive, 1133–1135 or probabilities with mutually exclusive events, 1132–1133 and probabilities with independent events, 1135–1137 theoretical, 1128–1131 Problem solving involving maximizing/minimizing quadratic functions, 354–359 involving rational functions, 424–425 with linear inequalities, 200–201 with linear programming, 888–890 mixtures, 834–836 quadratic formula used for, 163–164 with scientific notation, 31–32 word problems, 124–133 Product(s) of functions, 301, 302 minimizing, 357 of rational expressions, 78–79 special, 58 special-product formula, 55–58 of sum and difference of two terms, 56, 58 of two binomials, 54–55 of two complex numbers in polar form, 781–782 of two matrices, defined, 931 Product rule, 20–21, 493, 496–498, 1149 for radicals, 43–44 for solving logarithmic equations, 509 for square roots, 37–38 using, 493–494 Products-to-powers rule, 24–25 Product-to-sum formulas, 705 Profit function, 834 gain or loss, 834 Projectiles, 346 Pujols, Albert, 1032, 1059 Pure imaginary number, 140 Pythagorean identities, 566, 670, 720 Pythagorean Theorem, 325, 561 Subject Index I7 Q Quadrantal angle, 543, 576–577 degree and radian measures of, 586 trigonometric functions of, 578–579, 586 Quadrants, 94 signs of trigonometric functions and, 579–580, 586 in which angle lies, finding, 579 Quadratic equations, 148–171, 712 applications of, 163 defined, 148 determining most efficient method for solving, 161–162 discriminant of, 160–161 in general form, 148, 156, 158 irrational solutions to, 159–161 rational solutions to, 160, 161 solving by completing the square, 152–156 solving by factoring, 148–150, 161, 162 solving by square root property, 151–152, 162 solving using quadratic formula, 156–160, 162 Quadratic factors in denominator of rational expression prime, nonrepeated, 852–854 prime, repeated, 858–860 Quadratic formula, 156–160 deriving, 156 for problem solving, 163–164 quadratic equations solved with, 156–160, 162 trigonometric equation solved using calculator and, 722 Quadratic functions, 345–363, 365, 847 applications of, 354–359 defined, 346 graphs of, 346–353 minimum and maximum values of, 353–358 obtaining information about, from its equation, 353–354 standard, graph of, 283 in standard form, 347–350 Quadratic in form, solving equations, 180–182 trigonometric equations, 716–717 Quantity(ies) equilibrium, 838 scalar, 790 Quarterly compounding of interest, 472 Quotient(s), 13 of complex numbers in polar form, 782 difference, 247–248 of functions, 301, 302 of two complex numbers in polar form, 1151 of two rational expressions, 79 Quotient identities, 564–565, 670 Quotient rule, 21–22, 494, 496–498 for radicals, 43–44 for solving logarithmic equations, 494, 510 for square roots, 38–39 using, 494 Quotients-to-powers rule, 25–26 R Radian(s), 544–545, 549, 586 converting between degrees and, 545–546 definition of, 544 Radical(s) like, 39–40 and nth roots, 42 Radical equations, 175–178 involving rational exponents, 178–180 solving, 175–178 solving those containing nth roots, 175 with two radicals, 177–178 Radical expressions, 36 adding and subtracting, 39–40 combining those requiring simplification, 40 dividing, 43–44 product rule for, 43–44 quotient rule for, 43–44 simplifying, 37, 40, 43–44 simplifying using rational exponents, 47 Radical sign, 36 Radical symbols, 36 Radicand, 36, 42 Radius, of circle, 327, 329 Radius of circle, 554 Range, 219 of cosecant function, 625 of cosine function, 591–592, 609 of cotangent function, 624 identifying, from function’s graph, 226–228 of relation, 217–218 of secant function, 626 of sine function, 591–592, 600 of tangent function, 621 Rate of change average, 275–278 slope as, 274–275 Rational equations defined, 112 solving, 112–114 Rational exponents, 44–47 defining, 44 with numerators other than one, 45–46 radical expressions simplified using, 47 reducing the index of, 47 simplifying expressions with, 46–47 solving equations with, 178–180 Rational expressions, 76–89 addition of those with common denominators, 80 addition of those with different denominators, 80–84 complex, 84–86 defined, 77 dividing, 79 domain of, excluding numbers from, 77 multiplying, 78–79 partial fraction decompositions for, 847–858 simplifying, 77–78 subtraction of those with different denominators, 80–84 subtraction of those with same denominators, 80 Rational functions, 411–430 applications, 424–425 domain of, 411–412 graphs of, 418–423 horizontal asymptote of, 416–417, 419–421 inverse variation equation as, 448 reciprocal function and, 412–414 slant asymptotes of, 422–423 vertical asymptotes of, 414–416, 419–421 Rational inequalities, 436 solving, 436–438 solving problems modeled by, 438–440 Rationalizing denominators, 40–42, 561 containing one term, 40–41 containing two terms, 42 Rational numbers, as solutions to quadratic equations, 160, 161 Rational Zero Theorem, 396–398 Ray, 542 Ray, Greg, 1124 Real axis, 778 Real number line, 8–9 distance between two points on, 11 intervals on, 190 locating square root on, Real numbers adding, 11–12 dividing, 13 graphing, multiplying, 12 ordering, principal nth root of, 42 properties of, 11–13 set of, 7–8 subsets of, 7–8 subtracting, 13 trigonometric functions of, 589–592 Real part, of complex number, 140 Reciprocal (or multiplicative inverse), 13, 79, 273 Reciprocal function, 412–414 defined, 412 graph of, 413 Reciprocal identities, 564, 565, 670 Rectangle(s) area and perimeter formulas for, 132, 133 golden, 50 Rectangular coordinates ellipse on, 975 equation conversion from polar to, 759–761 equation conversion to polar coordinates from, 758–759 point conversion from polar to, 756–757 point conversion to polar coordinates from, 757–758 relations between polar coordinates and, 756 vectors in, 793–795 Rectangular coordinate system, 94 circle in, 325 distance between two points on, 325 graphing equations in, 95–100 points plotted in, 94–95 Rectangular equation of curve defined parametrically, finding and graphing, 1034–1037 Rectangular form, complex number in, 779, 780 Rectangular solid, volume formula for, 132 Recursion formulas, 1062–1063 Reduced row-echelon form, 910, 911 Reference angles, 580–586 for angles greater than 360° (2p) or less than -360° (-2p), finding, 581–582 definition of, 580 evaluating trigonometric functions using, 582–586 finding, 581–582 Reflecting light, and parabolas, 1013–1014 Reflections of graphs, 287–288, 291 about x-axis, 287–288, 291, 292 about y-axis, 288, 291 of exponential function, 468 of logarithmic function, 483, 484 one-to-one functions, 319 Regression line, 255 Relations, 216–218 defined, 217 as functions, 218–220 Relative maximum, 237–238 Relative minimum, 237–238 Relativity, Einstein’s theory of, 47 Remainder Theorem, 389–390 Repeated factorization, 69–70 Repeating decimals, written as fractions, 1092 Repetitive behavior of sine, cosine, and tangent functions, 594 Representative numbers test value, solving polynomial inequalities at, 432–434 test value, solving rational inequality at, 438 Resultant vector, 792, 799–800 Revenue, 832 Revenue function, 832–834 Richter scale, 478, 486 Right angle, 543 Right triangle, 164, 169, 325 isosceles, 562 names of sides of, 559 solving a, 649–652 solving problem using two, 651–652 special, 586 Right triangle trigonometry, 559–575 applications of, 569–571 fundamental identities, 564–566 Rise, 255 Rolling motion, 1038 Root, domain of function containing even, 300 Roots of complex numbers in polar form, 784–786 Roots of equations, 107 extraneous, 176 polynomial equation, 369, 399–401 Rose curves, 770–771 Roster method, Rotational motion, 1038 Rotation of axes, 1021–1031 equations of rotated conics in standard form, writing, 1023–1028 formulas, using, 1021–1023 identifying conic sections without, 1014–1015, 1029 Row-echelon form, 904, 908, 910 Row equivalent, 904 Row operations, 903–905 on augmented matrix, 904–908 Royal flush, 1122 Run, 255 Rutherford, Ernest, 1002 S St Mary’s Cathedral (San Francisco), 989 Saint-Vincent, Grégoire de, 488 Salaries comparing, 1082 lifetime computation, 1088–1089 Sample space, 1128, 1129 Satisfying the equation, 95, 107 Satisfying the inequality, 189, 873 Savings, and doubling time, 512 Scalar, 790, 793 dot product of two vectors as, 806 Scalar identity property, 929 Scalar multiple, 792, 928 Scalar multiplication, 792, 797, 928–930 with vector in terms of i and j, 796 Scatter plots, 255, 265, 266, 524–526 Scientific calculators angle of elevation on, 570, 571 combinations on, 1122 common logarithms evaluated on, 485 computations with scientific notation on, 30 converting from decimal to scientific notation on, 30 evaluating e to various powers, 470 evaluating trigonometric functions using, 568 exponential functions evaluated on, 465 factorials found with, 1063 inverse trigonometric functions on, 641 keystroke sequences for rational exponents, 46 quadratic formula on, 163 Scientific notation, 28–32 computations with, 30–31 converting from decimal notation to, 29–30 converting to decimal notation from, 28–29 defined, 28 problem solving with, 31–32 Secant (sec), 559 of 45°, evaluating, 563 as cofunction of cosecant, 567 defined, 560 evaluating, 561 Secant curve characteristics of, 626 cosine curve to obtain, 627 Secant function as even function, 592 graph of, 625–627 reference angle to evaluate, 585 Secant line, 275–276, 278 Second-degree equation, general, 1021–1022 Second-degree polynomial equation, 148 See also Quadratic equations Second-order determinants, evaluating, 955–956 Seinfeld, Jerry, 1124 Semiannual compounding of interest, 472 Sense of the inequality, changing, 193 Sequences, 1060–1097 arithmetic, 1071–1081 defined, 1061 defined using recursion formulas, 1062 factorial notation, 1063–1064 Fibonacci, 1060 finding particular terms of, from general term, 1061 finite, 1061 geometric, 1081–1097 infinite, 1061 summation notation, 1064–1067 of transformations, 291–293 Series, geometric, 1090–1093 infinite, 1090–1093 Set(s), 5–6 empty, 880, 917 empty (null), 6, 114, 116, 197 intersection of, 5–6, 299 of irrational numbers, of numbers, of rational numbers, of real numbers, 7–8 union of, 6, 299 Set-builder notation, 5, 189, 190, 194, 196, 197, 199 for or probabilities with mutually exclusive events, 1134 to represent function’s domain and range, 227, 228 Sherpa, Ang Rita, 559 Shrinking graphs horizontal, 290–291, 468, 483 vertical, 289–290, 292 Signs Descartes’s Rule of, 403–405 of trigonometric functions, 579–580, 586 Simple harmonic motion, 653–656 analyzing, 655–656 diminishing motion with increasing time, 654 finding equation for object in, 654–655 frequency of object in, 655, 656 resisting damage of, 656 I8 Subject Index Simple interest, 130–131, 133–134 Simplifying algebraic expressions, 13–15 Simplifying complex numbers, 140 Simplifying complex rational expressions by dividing, 84 by multiplying by one, 84–86 Simplifying exponential expressions, 26–28 common errors in, 28 with rational exponents, 46 Simplifying radical expressions, 37, 40, 43–44, 47 Simplifying rational expressions, 77–78 Simplifying square roots, 38, 39 Sine (sin), 559 of 45°, evaluating, 563 of 30° and 60°, evaluating, 563–564 as cofunction of cosine, 567 defined, 560 double-angle formula for, 692, 693, 699 evaluating, 561 half-angle formula for, 696, 699 inverse, 570 Law of Sines, 732–743, 746, 747 power-reducing formula for, 699 product-to-sum formulas for, 705 rotation of axes and, 1021–1022 sum and difference formulas for, 684–687, 699 sum-to-product formulas for, 706–707 verifying an identity by changing to, 671–676 Sine curve, 600 to obtain cosecant curve, 626–627 stretching and shrinking, 601 vertical shifts of, 612–613 Sine function amplitude of, 601–606 domain, 591–592, 600 domain of restricted, vs interval of nonrestricted, 640 in form y = A sin(Bx - C), 606–608 in form y = A sin Bx, 604–606 graph of, 599–608, 628 inverse, 633–636, 640, 645 inverse properties, 642 key points in graphing, 600–608, 610 modeling periodic behavior, 613–616 as odd function, 592, 600 period, 599–600, 601–606, 716 properties of, 593, 600 of quadrantal angles, 578–579 range, 591–592, 600 reference angle to evaluate, 583 repetitive behavior of, 594 solving right triangles using, 651 variations of y = sin x, graphing, 600–608 vertical shifts of sine curve, 612–613 Sine waves, 708 Singular square matrix, 944 Sinusoidal functions, modeling musical sounds with, 655 Sinusoidal graphs, 609, 612–613 Sinusoidal sounds, 708 Slant asymptotes, 422–423 Slope as average rate of change, 275–278 defined, 255, 274 interpreting, 274–275 of line, 255–257 negative, 256, 257 notation for, 256 and parallel lines, 271–272 and perpendicular lines, 272–274 point-slope form of the equation of a line, 257–259, 264, 273 positive, 256, 257 as rate of change, 274–275 undefined, 256, 257 zero, 256, 257 Slope-intercept form of equation of line, 259–262, 264 linear functions graphed in, 259 modeling data with, 265–266 parallel lines, 272 perpendicular lines, 273 Smooth graphs, 365 Solution(s), 713 of equation in two variables, 95 extraneous, 176 of inequality, 189 of inequality in two variables, 873 of linear equation in one variable, 107 of nonlinear system in two variables, 862 of polynomial equation, 369, 399–401 of system of linear equations, 824–825 of system of linear equations in three variables, 843–844, 918 of system of linear inequalities, 877 of trigonometric equation, finding, 713–715 Solution set, 107 inequalities with unusual, 196–197 of inequality, 189 of linear equation in one variable, 107, 110 of nonlinear system in two variables, 862 of system of linear equations in three variables, 843 of system of linear inequalities, 878 Solving a formula for a variable, 133–134 Solving an equation, 107 Solving an inequality, 189, 192–196 Solving an oblique triangle, 733–735 Solving a right triangle, 649–652 Solving linear equations, 107–111 Sonic boom, hyperbolic shape of, 999 Sound quality and amusia, 681, 683 Sounds, sinusoidal, 708 Space, photographs sent back from, 935 Spaceguard Survey (NASA), 1003 Space Telescope Science Institute, 1013 Special-product formula, 55–58 Special products, using, 58 Speed angular, 554–555 ground, 803 linear, 554–555 Speed of light, 47 Sphere, volume formula for, 132 Square area and perimeter formulas for, 132 perfect, 37 Square matrix, 925 invertible or nonsingular, 944, 948 multiplicative identity matrix of order n, 941–942 multiplicative inverse of, 941–949 singular, 944 Square of binomial difference, 57, 58 Square of binomial sum, 56–58, 60 Square root function, graph of, 283 Square root property defined, 151 quadratic equations solved by, 151–152, 162 Square roots, 7, 36–37 adding and subtracting, 39–40 evaluating, 36–37 locating on real number line, product rule for, 37–38 quotient rule for, 38–39 simplified, 38, 39 Square root signs, 36 Square roots of negative numbers, 37 as multiples of i, 139 multiplying, 144–145 Standard cubic function, graph of, 283 Standard form complex numbers in, 140 of equation of circle, 327–330 of equation of ellipse, 975–979, 1153 of equation of hyperbola, 989–992, 996, 1023 of equation of parabola, 1006–1012 of polynomial, 52 of quadratic function, graphing, 347–350 transforming rotated conics to, 1023–1028 Standard position, angle in, 542, 543 drawing, 547–549 Standard quadratic function, graph of, 283 Standard viewing rectangle, 97 Statuary Hall (U.S Capitol Building), 983, 987 Step functions, 247 Straight angle, 543 Stretching graphs horizontal, 290–291, 468, 483 vertical, 289, 293 Subscripts, 52 Subsets of real numbers, 7–8 Substitution for eliminating variables, 825–827, 844, 862–865 nonlinear systems solved by, 862–865 systems of linear equations in two variables solved by, 825–827 Subtraction of complex numbers, 140–141 definition of, 13 of like radicals, 39–40 matrix, 926 of polynomials, 53 of polynomials in two variables, 59 of radical expressions, 39–40 of rational expressions with different denominators, 80–84 of rational expressions with same denominators, 80 of real numbers, 13 of square roots, 39–40 vector, 795–796 Sum binomial, square of, 56–58, 60 of cubes, factoring, 71 and difference of two terms, product of, 55–56, 58 of first n terms of arithmetic sequence, 1076–1078 of first n terms of geometric sequence, 1086–1089 of functions, 301–303 of infinite geometric series, 1090–1093 Sum and difference formulas, 681–692, 699 for cosines, 681–687, 699 for sines, 684–687, 699 for tangents, 687–688, 699 Summation notation, 1064–1067 properties of sums, 1067 using, 1065–1066 writing sums in, 1067 Sum-to-product formulas, 706–707 Supercomputers, 908 Switch-and-solve strategy, 316 Symbols approximation, for binomial coefficients, 1107 for elements in sets, empty set, 196 greater than, 9, 190, 198 greater than or equal to, 9, 190 inequality, 9, 189, 192, 874, 875 infinity, 190 less than, 9, 190, 198 less than or equal to, 9, 190 negative square root, 36 radical sign, 36 for set of real numbers, sigma, in adding terms of sequence, 1065 square root, 36 for subsets of real numbers, Symmetry axis of, 346, 347, 1006, 1007, 1010, 1044, 1046, 1047 definitions/tests for, 239 even and odd functions and, 241–245 graphing a polar equation using, 767–772 with respect to polar axis, 767, 768, 770, 771, 1044, 1046, 1047 with respect to the origin, 238, 239, 241, 621 with respect to x-axis, 238, 239 with respect to y-axis, 238, 239, 241–242, 374, 767, 768, 770, 771, 1044 tests for, in polar coordinates, 767 Synthesizers, 650, 655 Synthetic division, 387–390 Systems of equations See Systems of linear equations Systems of inequalities See Systems of linear inequalities Systems of linear equations, 824 matrix solutions to, 902–916 multiplicative inverses of matrices to solve, 948–950 problem solving and business applications using, 832–834 Systems of linear equations in three variables, 843–851 Gaussian elimination applied to, 916–922 inconsistent and dependent systems, 847 problem solving with, 847 solution of, 843–844 solving by eliminating variables, 844–847 solving by using matrices, 902–916 solving those with missing terms, 846 solving using determinants and Cramer’s rule, 961–963 Systems of linear equations in two variables, 824–840 determining if ordered pair is solution of, 824–825 with infinitely many solutions, 830, 831–832 with no solutions, 830–831 number of solutions to, 830–832 solving by addition method, 827–830 solving by graphing, 825, 832 solving by substitution method, 825–827 solving by using determinants and Cramer’s rule, 956–958 Systems of linear inequalities applications of, 885–890 graphing, 878–881 Systems of nonlinear equations in two variables, 862–872 applications with, 867–868 recognizing, 862 solving by addition method, 865–867 solving by substitution method, 862–865 T Tables, creating with graphing utility, 98, 174, 176, 357, 511, 527, 858 of solutions of equations in two variables, 98 Tangent (tan), 559 of 45°, evaluating, 563, 564 of 30° and 60°, evaluating, 564 as cofunction of cotangent, 567 defined, 560 double-angle formula for, 692–694, 699 evaluating, 561 half-angle formula for, 696, 698, 699 power-reducing formula for, 699 sum and difference formulas for, 687–688 Tangent curve, characteristics of, 621 Tangent function domain, 621 domain of restricted, vs interval of nonrestricted, 640 finding bearing using, 653 graph of, 620–623, 628 inverse, 638–640 inverse properties, 642 as odd function, 592, 620, 621 period, 620, 621, 715 periodic properties of, 594 of quadrantal angles, 578–579 range, 621 reference angle to evaluate, 583–585 repetitive behavior of, 594 solving right triangles using, 650–652 vertical asymptote of, 620–622 Tangent line, 334 Tartaglia’s formula giving root for third-degree polynomial equation, 395 Tax-deferred savings plans, 1090 Tax rebates, and multiplier effect, 1092 Tele-immersion, 901 Telephone numbers, running out of, 1117 Temperature, atmospheric carbon dioxide concentration and global, 215, 264–266 Terminal point of directed line segment, 790 Terminal side of angle, 542–543 angle lies in quadrant of, 543 angles formed by revolutions of, 547–549 Terminating decimals, Terms of algebraic expressions, 13 constant, 52 finding, in binomial expansion, 1110 of geometric sequence, 1083 like, 13–14 multiplying sum and difference of, 55–56, 58 outside/last/inside/first, in binomial, 55 of sequence, written from general term, 1061 of sequences involving factorials, finding, 1063 Test point graphing linear inequalities without using, 875–876 graphing linear inequality in two variables, 873–875 graphing nonlinear inequalities in two variables, 876 Test value, 432–434, 438 Theoretical probability, 1128–1131 computing, 1128–1131 computing without listing event and sample space, 1130 Third-order determinants defined, 958, 959 evaluating, 959–961 Tidal cycle, modeling, 615 Total economic impact, 1096 Transformations of computer graphics image, 935–936 Subject Index I9 Transformations of functions, 282–297 exponential functions, 468–469 horizontal shifts, 285–287, 291–293, 468, 469, 483 horizontal stretching and shrinking, 290–291, 468, 483 logarithmic functions, 483–484 rational functions, 418 recognizing graphs of common functions and, 283 reflections of graphs, 287–288, 291, 292, 319, 320, 468, 483, 484 sequences of, 291–293 vertical shifts, 284–287, 291–293, 468, 483 vertical stretching and shrinking, 289–293, 468, 483 Translations of ellipses, 979–981 of hyperbolas, 996–999 of parabolas, 1009–1012 Transverse axis, of hyperbola, 989, 990, 996 Trapezoid, area and perimeter formulas for, 132 Tree diagram, 1115 Triangle area and perimeter formulas for, 132 area of, 1150–1151 Heron’s formula for area of, 748 isosceles right, 169 oblique, 732–735, 738, 745–747 Pascal’s, 1111 right, 164, 325 solving a right, 649–652 Triangle inequality, 11 Trigonometric equations, 713–726 calculator to solve, using, 721–722 defined, 713 factoring to separate two different functions in, 717–718 finding all solutions of, 713–714 identities to solve, using, 718–721 involving single trigonometric function, 714 with multiple angles, 715–716 quadratic in form, 716–717 Trigonometric functions, 559–562 of 45°, evaluating, 563, 564, 586 of 30° and 60°, evaluating, 563–564, 586 of any angle, 576–588 applications of, 649–660 bearings, 652–653 cofunction identities, 567 and complements, 566–567 definitions of, in terms of unit circle, 589–592 equations involving single, 714 evaluating, 577, 579–580 even and odd, 592–593 finding values of, 590–591 function values for some special angles, 562–564, 586 fundamental identities, recognizing and using, 564–566 graphs of, 628 inverse, 633–649 modeling periodic phenomena with, 593 names and abbreviations of, 559 of quadrantal angles, 578–579, 586 of real numbers, 589–592 reducing power of, 695–696 reference angles to evaluate, 582–586 right triangle definitions of, 560 signs of, 579–580, 586 simple harmonic motion, 653–656 solving right triangles, 649–652 using calculator to evaluate, 567–568 using right triangles to evaluate, 562 Trigonometric identities, 564–566 eliminating the parameter and, 1035 even-odd identities, 670 principal, 692–699 Pythagorean identities, 566, 670, 720 quotient identities, 564–565, 670 reciprocal identities, 564, 565, 670 verifying, 670–681 Trigonometry, defined, 559 Trinomials, 52, 114 factoring those whose leading coefficient is not 1, 67–68 factoring those with leading coefficient of 1, 66–67 multiplying binomial and, 53–54 perfect square, 70, 153 in two variables, factoring, 68–69 Turning points, 373, 374, 376 U Undefined slope, 256, 257 Union of sets, 6, 299 defined, solution sets, 882 Unions of intervals, 191–192 Unit circle, 589–594 Unit distance, United Nations Intergovernmental Panel on Climate Change, 265 United States population, modeling growth of, 520–521 Unit vectors, 793, 797 finding, in same direction as given nonzero vector, 797–798 i and j, 793–795 Upper limit of summation, 1065, 1066 U.S Census Bureau, 1084 V Value of an annuity, 1088–1090 of second-order determinant, 955 Variable cost, 833 Variables, dependent, 220 independent, 220 solving for, in formulas, 133–134 Variables, eliminating addition method of, 827–830, 844–846, 865–867 solving linear system in three variables by, 844–847 solving linear systems in two variables by, 825–830, 845, 846 solving nonlinear systems by, 862–867 substitution method of, 825–827, 844, 862–865 Variation combined, 449–450 constant of, 444, 447 direct, 444–447 equations of, 444 formulas, 444 inverse, 447–449 joint, 450–451 modeling using, 444–454 Variation problems, solving, 445–447 Vector(s), 790–805 adding and subtracting, in terms of i and j, 795–796 angle between two, 807 applications, 799–800 components of, 793 difference of two, 793 directed line segments and geometric, 791–793 dot product of, 806–807 equal, using magnitude and direction to show, 791–792 force, 799–800 orthogonal, 808 parallel, 808 position, 794 projection of vector onto another vector, 809–810 properties of vector addition and scalar multiplication, 797 in rectangular coordinate system, 793–795 relationships between, 791 resultant, 792, 799–800 scalar multiplication with, 792, 796, 797 as sum of two orthogonal vectors, 810–811 unit, 793–795, 797–798 velocity, 799, 800 writing, in terms of magnitude and direction, 798–799 zero, 797 Vector components of force, 809 Vector components of v, 810 Vector projection of v onto w, 809–810 Velocity vector, 799, 800 Verifying an identity, 670–671 changing to sines and cosines for, 671–676 combining fractional expressions for, 674–675 difference formula for cosines for, 684 double-angle formula for cosines for, 694–695 guidelines for verifying trigonometric identities, 677–678 half-angle formula for, 698 multiplying numerator and denominator by same factor for, 675 sum and difference formulas for tangents for, 688 sum-to-product formulas to, 707 using factoring, 673–674 using fundamental identities to, 670–681 using two techniques, 673–674 working with both sides separately in, 676–677 Verlander, Justin, 1059 Vertex (vertices), 888, 889 of angle, 542 of ellipse, 975 of hyperbola, 989–992, 994–998 of parabola, 346–352, 1006, 1010–1012 Vertical asymptotes of cosecant function, 625 of cotangent function, 624–625 defined, 414 locating, 415 of logarithmic function, 482, 483, 484 of rational function, 414–416, 419–421 of secant function, 626 of tangent function, 620–622 y-axis as, 483, 484 Vertical component of vector, 793 Vertical lines, 264 equations of, 262 graphs of, 262 Vertical line test, for functions, 224 Vertical shifts (transformations), 284–287, 291–293 combining horizontal shifts and, 286–287 downward, 284–285 of exponential function, 468 of logarithmic function, 483 of sinusoidal graphs, 612–613 upward, 284 Vertical stretching and shrinking, 289–293 of exponential function, 468 of logarithmic function, 483 Viewing rectangle on graphing utility, 97 understanding, 97–98 Volume, formulas for, 132 W Wadlow, Robert, 452 Walters, Barbara, 1106 Washburn, Brad, 571 Watson, James, 1038 Whispering gallery, 983 Whole numbers, Wiles, Andrew, 1098, 1106 Wilson, Michael, 187 Witch of Agnesi, 1040 Wolf population, 470–471 Word problems, solving, 124–133 Work, 805 definition of, 811 dot product to compute, 811–812 World population future of, 528 growth in, 519 modeling data about, 526–527 rewriting model in base e, 528 Wright, Steven, 1124 X x-axis, 94 as horizontal asymptote, 466, 467 reflection about, 287–288, 291, 292 x-coordinate, 94, 99 horizontal shifts and, 285–287 horizontal stretching and shrinking and, 290–291 x-intercept, 98, 99, 263, 264 on cosine curve, 627 of cotangent function, 624, 625 of function, 229 graphing quadratic function in standard form, 348–349, 351–352 in graph of cosine function, 612 in graph of sine function, 600, 602, 603, 605, 607, 610 multiplicity and, 371, 374, 375 on sine curve, 626 solving polynomial inequalities and, 432 of tangent function, 621–623 Y y-axis, 94 as axis of symmetry, 1044 even function symmetric with respect to, 241–242 reflection about, 288, 291 symmetry with respect to, 767, 768, 770, 771, 1044 as vertical asymptote, 483, 484 y-coordinate, 94, 99 vertical shifts and, 283–284 vertical stretching and shrinking and, 289–290 y-intercept, 98, 99, 259, 263, 264 of function, 229 graphing polynomial function and, 374–376 graphing quadratic function in standard form, 349, 351–352 graphing using slope and, 259–261 Z Zero-exponent rule, 22 Zero factorial (0!), 1063 Zero matrix, 927 Zero-Product Principle, 149, 162 Zero slope, 256, 257 Zeros of a function, 229 Zeros of polynomial functions, 369–371, 374, 375, 395–409 Descartes’s Rule of Signs and, 403–405 finding, 396–399 Intermediate Value Theorem and, 372–373 kinds of, 396 Linear Factorization Theorem, 402–403 multiplicities of, 371–372, 374, 375 rational, 396–397 Zero vector, 797 Zero with multiplicity k, 371 Zoom in/zoom 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Blitzer Chapter 2 p 215 Robert F Blitzer p 216 Creative Collection Tolbert Photo/Alamy Stock Photo p 216 Everett Collection Inc./Alamy Stock Photo p 216 Hyperstar/Alamy Stock Photo p 216 NIBSC/Science Source p 216 Peter Brooker/Presselect/Alamy Stock Photo p 216 Robert F Blitzer p 216 ZUMA Press/Alamy Stock Photo p 217 Robert F Blitzer p 222 Screenshots from Texas Instruments Courtesy of Texas Instruments p 225 Robert F Blitzer p 232 Robert F Blitzer p 233 National Center for Health Statistics p 233 U.S Bureau of Labor Statistics p 235 Dorling Kindersley Ltd p 235 U.S Department of Transportation p 235 (left) Oleksiy Maksymenko Photography/Alamy Stock Photo p 235 (margin) U.S Department of Transportation p 253 Robert F Blitzer p 253 Robert F Blitzer p 255 Image Source/AGE Fotostock p 255 Robert F Blitzer p 257 Robert F Blitzer p 257 Soo hee kim/ Shutterstock p 265 National Oceanic and Atmospheric Administration p 268 National Center for Health Statistics p 268 U.S Census Bureau p 269 Robert F Blitzer p 269 Robert F Blitzer p 269 Robert F Blitzer p 271 FloridaStock/Shutterstock p 271 Marino Bocelli/Shutterstock p 271 Olaf Speier/Shutterstock p 271 U.S Census Bureau p 279 Robert F Blitzer p 279 Robert F Blitzer p 279 Robert F Blitzer p 282 Everett Collection p 287 Gene Ahrens/SuperStock p 296 Robert F Blitzer p 296 Robert F Blitzer p 298 United States Department of Health & Human Services p 298 Zoonar GmbH/Alamy Stock Photo p 309 Robert F Blitzer p 311 U.S Census Bureau p 313 Archives du 7eme Art/Photos 12/Alamy Stock Photo p 313 Robert F Blitzer p 313 (right) Everett Collection p 322 Robert F Blitzer p 323 Robert F Blitzer p 325 Millard H Sharp/Science Source p 333 Robert F Blitzer p 339 Robert F Blitzer p 339 U.S Census Bureau p 342 U.S Census Bureau p 951 Everett Collection Chapter 3 p 450 Ty Allison/Getty Images p 345 Cultura RM/ Alamy Stock Photo p 346 Gunnar Assmy/Shutterstock p 346 Gunnar Assmy/Shutterstock p 346 Scott Kim p 346 (bl) Pascal Rondeau/Staff/ Getty Images p 359 Source: Herbert Benson, Your Makimum Mind, Random House, 1987 p 362 Allstar Picture Library/Alamy Stock Photo p 362 Allstar Picture Library/Alamy Stock Photo p 362 Allstar Picture Library/Alamy Stock Photo p 362 Amanda M Parks/Newscom p 362 Debbie Van Story/Newscom p 362 Dee Cercone/Everett Collection/ Alamy Stock Photo p 362 Everett Collection Inc/Alamy Stock Photo p 362 Jen Lowery/Splash News/Newscom p 362 Newscom p 362 Pictorial Press Ltd/Alamy Stock Photo p 362 Robert F Blitzer p 362 Roth Stock/Everett Collection Inc/Alamy Stock Photo p 362 Wenn Ltd/Alamy Stock Photo p 364 DC5/Dominic Chan/WENN/ Newscom p 364 Department of Health and Human Services p 379 Robert F Blitzer p 380 Robert F Blitzer p 380 U.S Energy Information Administration p 382 Nature’s Images/Science Source p 395 Middle Temple Library/Science Source p 403 La Geometrie p 403 Library of Congress p 411 ZUMA Press Inc/Alamy Stock Photo p 413 Nathan Regener p 429 Robert F Blitzer p 429 Robert F Blitzer p 431 National Highway Traffic Safety Administration p 431 © 1995 Warren Miller/ The New Yorker Collection/Cartoonbank p 442 National Highway Traffic Safety Administration p 444 Cosmo Condina North America/ Alamy Stock Photo p 444 Joseph Sohm/Shutterstock p 445 NBCUniversal/Getty Images p 446 Exactostock-1527/SuperStock p 450 David Madison/Getty Images p 452 Ullstein Bild/The Image Works p 457 Robert F Blitzer p 459 National Highway Traffic Safety Administration p 459 U.S Census Bureau p 460 Robert F Blitzer Chapter 4 p 463 Mega Pixel/Shutterstock p 464 Hero Images/ Getty Images p 464 Robert F Blitzer p 468 Rolffimages/Global Connection/Dreamstime LLC p 468 Seth Wenig/AP Images p 470 U.S Fish and Wildlife Service p 471 Getty Images p 476 Robert F Blitzer p 476 U.S Census Bureau p 477 Robert F Blitzer p 478 Newscom p 487 Robert F Blitzer p 491 Robert F Blitzer p 493 Blend Images/Alamy Stock Photo p 504 Benn Mitchell/Stockbyte/Getty Images p 516 Robert F Blitzer p 518 Jamaway/Alamy Stock Photo p 518 Robert F Blitzer p 519 Chlorophylle/Fotolia p 520 U.S Census Bureau p 522 French Ministry of Culture and Communication p 525 Robert F Blitzer p 525 Robert F Blitzer p 525 Science Photo Library/SuperStock p 526 U.S Census Bureau p 526 U.S Census Bureau p 528 U.S Census Bureau p 529 Robert F Blitzer p 529 U.S Census Bureau p 530 C1 C2 Photo Credits Robert F Blitzer p 531 Robert F Blitzer p 531 Robert F Blitzer p 531 Robert F Blitzer p 531 Robert F Blitzer p 532 U.S Census Bureau p 537 Robert F Blitzer p 538 Robert F Blitzer p 538 Robert F Blitzer p 538 U.S Census Bureau p 538 U.S Census Bureau p 539 Robert F Blitzer Chapter 5 p 541 BlueSkyImage/Shutterstock p 542 Bjanka Kadic/AGE Fotostock p 554 Sarah Beard Buckley/Moment/ Getty Images p 559 Koonyongyut/iStock/Getty Images p 571 AGE Fotostock/SuperStock p 574 Paulbriden/Fotolia p 575 Pixtal/ Superstock p 576 Laszlo Podor/Alamy Stock Photo p 576 Laszlo Podor/ Alamy Stock Photo p 589 Kathy Collins/Photographer’s Choice/Getty Images p 597 G L FRENCH/ClassicStock/Alamy Stock Photo p 599 Johan Larson/Shutterstock p 620 Rbv/123RF p 633 Atlaspix/ Alamy Stock Photo p 649 Image Source Plus/Alamy Stock Photo p 656 Paul Sakuma/AP Images p 667 Robert F Blitzer Chapter 6 p 669 AF Archive/Alamy Stock Photo p 669 Charles Sykes/AP Images p 669 Everett Collection p 669 Everett Collection p 669 Everett Collection Historical/Alamy Stock Photo p 670 Martin Bond/Alamy Stock Photo p 681 Elizabeth Livermore/Moment/ Getty Images p 692 Jari Hindstroem/Shutterstock p 704 Archive Pics/ Alamy Stock Photo p 704 Everett Collection Historical/Alamy Stock Photo p 713 Witold Krasowski/Fotolia Chapter 7 p 731 Friedrich Saurer/ImageBROKER/Alamy Stock Photo p 732 George H.H Huey/Alamy Stock Photo p 744 Antonello Lanzellotto/AGF RM/AGE Fotostcok p 765 Maurizio La Pira/Splash News/Newscom p 777 PATRICK CATUDAL/Shutterstock p 790 Jeffrey Greenberg/Science Source p 790 Pearson Education, Inc p 793 Pearson Education, Inc p 805 A.RICARDO/Shutterstock p 811 Chev Wilkinson/Cultura/Getty Images p 814 Laurence Griffiths/Getty Images p 814 Tose/Shutterstock p 814 YURI CORTEZ/Staff/AFP/Getty Images Chapter 8 p 591 Terry J Alcorn/Getty Images p 823 Big Cheese Special/Alamy Stock Photo p 824 Photographee.eu/Fotolia p 832 Roberto Machado Noa/Contributor/LightRocket/Getty Images p 836 Pearson Education, Inc p 839 Robert F Blitzer p 839 Robert F Blitzer p 839 U.S Census Bureau p 840 Bureau of Justice Statistics p 840 Robert F Blitzer p 840 Robert F Blitzer p 841 Robert F Blitzer p 841 Robert F Blitzer p 843 lzf/Fotolia p 849 Robert F Blitzer p 850 Robert F Blitzer p 850 Robert F Blitzer p 851 Maks Narodenko/Shutterstock p 862 Dave King/Dorling Kindersley Ltd p 871 U.S Justice Department p 877 U.S Department of Health and Human Services p 878 U.S Department of Health and Human Services p 883 Centers for Disease Control and Prevention p 883 Centers for Disease Control and Prevention p 883 U.S Department of Health and Human Services p 885 AP Images p 887 United States Navy p 895 Department of Health and Human Services p 896 Based on Centers for Disease Control and Prevention Chapter 9 p 901 TVD&SUN Creative Services Ltd p 902 U.S Bureau of Labor Statistics p 914 Robert F Blitzer p 915 The White House p 916 Perfectlab/Fotolia p 925 Pannawat/ Fotolia p 933 Science Source p 935 NASA p 939 HENSLIN, JAMES M., SOCIOLOGY: A DOWN-TO-EARTH APPROACH (SUBSCRIPTION), 12th Ed., ©2014 Reprinted and Electronically reproduced by permission of Pearson Education, Inc., New York, NY p 941 Popperfoto/Getty Images p 951 Heritage Image Partnership Ltd/ Alamy Stock Photo p 955 SSPL/The Image Works p 969 HENSLIN, JAMES M., SOCIOLOGY: A DOWN-TO-EARTH APPROACH (SUBSCRIPTION), 12th Ed., ©2014 Reprinted and Electronically reproduced by permission of Pearson Education, Inc., New York, NY Chapter 10 p 973 Stocktrek Corporation/SuperStock p 974 Kevin Fleming/Corbis Documentary/VCG/Getty Images p 984 NASA p 989 Gunter Marx/AT/Alamy Stock Photo p 1005 ESA/STScI/Arizona State University/NASA p 1013 MarcelClemens/ Shutterstock p 1021 Richard E Prince Fiberglass, steel, paint, graphite; 51 * 18 * 14 in Equinox Gallery, Vancouver, Canada p 1032 Chris Carlson/AP Images p 1038 GIPhotoStock/Science Source p 1042 NASA Chapter 11 p 1059 Louis Lopez/Cal Sport Media/Newscom p 1060 Cobalt88/Shutterstock p 1070 Pearson Education, Inc p 1071 Imtmphoto/Fotolia p 1075 U.S Census Bureau p 1082 Monkey Business Images/Shutterstock p 1084 Jokerpro/Shutterstock p 1088 Bureau of Printing and Engraving p 1088 Bureau of Printing and Engraving p 1088 Bureau of Printing and Engraving p 1088 Bureau of Printing and Engraving p 1088 Bureau of Printing and Engraving p 1088 Bureau of Printing and Engraving p 1098 Charles Rex Arbogast/ AP Images p 1115 Blend Images/SuperStock p 1116 Monkey Business/ Fotolia p 1117 Bloom Design/Shutterstock p 1119 AF Archive/Alamy Stock Photo p 1119 Marka/Alamy Stock Photo p 1126 Jupiterimages/ Getty Images p 1137 Bettmann/Contributor/Getty Images p 1144 Anna Omelchenko/Shutterstock p 1148 Robert F Blitzer p 1107 Rudolph Schild/Science Source Definitions, Rules, and Formulas The Real Numbers Natural Numbers: {1, 2, 3, } Interval Notation, Set-Builder Notation, and Graphs (a, b) = {x|a x b} Whole Numbers: {0, 1, 2, 3, } [a, b) = {x|a … x b} Integers: { , - 3, - 2, - 1, 0, 1, 2, 3, } Rational Numbers: ba ͉ a and b are integers, b ≠ 06 (a, b] = {x|a x … b} Irrational Numbers: {x|x is real and not rational} [a, b] = {x|a … x … b} Properties of Addition and Multiplication Commutative: a + b = b + a; ab = ba ( - ∞, b) = {x|x b} Associative: (a + b) + c = a + (b + c); (ab)c = a(bc) ( - ∞, b] = {x|x … b} Distributive: a(b + c) = ab + ac; a(b - c) = ab - ac Identity: a + = a; a # = a Inverse: a + ( - a) = 0; a # a b a b a b a a ( - ∞, ∞) = {x|x is a real number} = {x|x ∊ R} ( - 1)( - a) = a; a # = 0; ( - a)(b) = (a)( - b) = - ab; ( - a)( - b) = ab Slope Formula Exponents slope (m) = Definitions of Rational Exponents a b [a, ∞) = {x|x Ú a} Multiplication Properties: ( - 1)a = - a; b b (a, ∞) = {x|x a} = (a ≠ 0) a m m an = 2a 2 a n = 12a m or 2am 3 a- n = n n n change in y change in x = m y2 - y1 x2 - x1 Equations of Lines Slope-intercept form: y = mx + b m is the line’s slope and b is its y-intercept an Properties of Rational Exponents If m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then General form: Ax + By + C = Point-slope form: y - y1 = m(x - x1) m is the line’s slope and (x1, y1) is a fixed point on the line bm bm # bn = bm + n 2 = bm - n 3 (bm)n = bmn bn Horizontal line parallel to the x-axis: y = b a a (ab)n = anbn 5 a b = n b b Vertical line parallel to the y-axis: x = a Radicals x = b n n n Absolute Value n If 2a and 2b are real numbers, then If |x| c, then x - c or x c (c 0) The product rule: 2a # 2b = 2ab n n The quotient rule: 2a n 2b = n Special Factorizations Difference of two squares: A2 - B2 = (A + B)(A - B) n a Ab Algebra’s Common Graphs Identity Function y f(x) = x –1 Perfect square trinomials: A2 + 2AB + B2 = (A + B)2 A2 - 2AB + B2 = (A - B)2 Sum of two cubes: A3 + B3 = (A + B)(A2 - AB + B2) Difference of two cubes: A3 - B3 = (A - B)(A2 + AB + B2) Standard Quadratic Function y Absolute Value Function y –2 f(x) = ∣x∣ x –2 –1 1 Square Root Function y 1 if x Ú if x If |x| c, then - c x c (c 0) n n x -x If |x| = c, then x = c or x = - c (c 0) n If n is even, then 2an = |a| If n is odd, then 2an = a , (x1 ≠ x2) x –2 –1 f(x) = x2 f(x) = x x –2 –1 –1 –1 –1 –1 –2 –2 –2 –2 x Standard Cubic Function y 2 f(x) = x3 –2 –1 Greatest Integer Function y Cube Root Function y –2 –1 –1 1 x f(x) = x Reciprocal Function y x –2 –1 f(x) = x –2 –1 x 2 x –1 –1 –2 f(x) = int(x) –2 –2 –2 Transformations In each case, c represents a positive real number Vertical translations Horizontal translations Function Draw the graph of f and: y = f(x) + c e y = f(x) - c Shift f upward c units Shift f downward c units y = f(x - c) y = f(x + c) Shift f to the right c units Shift f to the left c units y = - f(x) y = f( - x) Reflect f about the x-axis Reflect f about the y-axis y = cf(x); c y = cf(x); c Vertically stretch f, multiplying each of its y-coordinates by c Vertically shrink f, multiplying each of its y-coordinates by c y = f(cx); c y = f(cx); c Horizontally shrink f, dividing each of its x-coordinates by c Horizontally stretch f, dividing each of its x-coordinates by c e Reflections e Vertical Stretching or Shrinking e Horizontal Stretching or Shrinking e Distance and Midpoint Formulas The distance from (x1, y1) to (x2, y2) is 2 3(x2 - x1) + (y2 - y1) Logarithmic Function: f(x) = logb x, b 0, b ≠ y = logb x is equivalent to x = by Graph: y y=x The midpoint of the line segment with endpoints (x1, y1) and (x2, y2) is x1 + x2 y1 + y2 a , b 2 Quadratic Formula f(x) = bx The solutions of ax + bx + c = with a ≠ are x = (0, 1) - b { 2b - 4ac 2a x (1, 0) Functions f -1(x) = logb x Linear Function: f(x) = mx + b Graph is a line with slope m and y-intercept b Quadratic Function: f(x) = ax + bx + c, a ≠ b Graph is a parabola with vertex at x = - 2a Quadratic Function: f(x) = a(x - h)2 + k In this form, the parabola’s vertex is (h, k) nth-Degree Polynomial Function: f(x) = anx n + an - 1x n - + an - 2x n - + g + a1x + a0, an ≠ For n odd and an 0, graph falls to the left and rises to the right For n odd and an 0, graph rises to the left and falls to the right For n even and an 0, graph rises to the left and rises to the right For n even and an 0, graph falls to the left and falls to the right p(x) Rational Function: f(x) = , p(x) and q(x) are polynomials, q(x) q(x) ≠ x Exponential Function: f(x) = b , b 0, b ≠ Graphs: y f(x) = bx b>1 f(x) = bx 0