www.elsolucionario.net Innovative Technology to Help You Succeed MyMathLab can improve any learning environment—whether you are taking a lab-based, hybrid, fully online, or a traditional lecture-style course INTERACTIVE FIGURES Math comes alive with new Interactive Figures in MyMathLab! Your instructor may choose to assign assessment questions that are written to accompany each figure This interaction will lead you to fully understand key mathematical concepts in a hands-on, engaging way A HISTORY OF SUCCESS Results show that you can improve your grade by using the videos, animations, interactive figures, step-by-step examples, and personalized feedback in MyMathLab To see the growing list of case studies for yourself, visit www.mymathlab.com/success-stories www.mymathlab.com www.elsolucionario.net Prepare for Class “Read the Book” Feature Description Benefit Page Every chapter begins with… Chapter Opening Article & Project Each chapter begins with a current article and ends with a related project The Article describes a real situation The Project lets you apply what you learned to solve a related problem 398, 501 NEW! Internet-based Projects The projects allow for the integration of spreadsheet technology that students will need to be a productive member of the workforce The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 398, 501 Every section begins with… Learning Objectives Each section begins with a list of objectives Objectives also appear in the text where the objective is covered These focus your studying by emphasizing what’s most important and where to find it 419 Most sections contain… Most sections begin with a list of key concepts to review with page numbers Ever forget what you’ve learned? This feature highlights previously learned material to be used in this section Review it, and you’ll always be prepared to move forward 419 Now Work the ‘Are You Prepared?’ Problems Problems that assess whether you have the prerequisite knowledge for the upcoming section Not sure you need the Preparing for This Section review? Work the ‘Are You Prepared?’ problems If you get one wrong, you’ll know exactly what you need to review and where to review it! 419, 430 “Now Work ” These follow most examples and direct you to a related exercise We learn best by doing You’ll solidify your understanding of examples if you try a similar problem right away, to be sure you understand what you’ve just read 428 PROBLEMS WARNING Warnings are provided in the text These point out common mistakes and help you to avoid them 453 Explorations and Seeing the Concept These represent graphing utility activities to foreshadow a concept or solidify a concept just presented You will obtain a deeper and more intuitive understanding of theorems and definitions 252, 425 These provide alternative descriptions of select definitions and theorems Does math ever look foreign to you? This feature translates math into plain English 421 Calculus Icon These appear next to information essential for the study of calculus Pay attention if you spend extra time now, you’ll better later! 365 Showcase EXAMPLES These examples provide “how-to” instruction by offering a guided, stepby-step approach to solving a problem With each step presented on the left and the mathematics displayed on the right, students can immediately see how each step is employed 337–338 Model It! Marked with These are examples and problems that require you to build a mathematical model from either a verbal description or data The homework Model It! problems are marked by purple numbers It is rare for a problem to come in the form, “Solve the following equation” Rather, the equation must be developed based on an explanation of the problem These problems require you to develop models that will allow you to describe the problem mathematically and suggest a solution to the problem 444, 473 PREPARING FOR THIS SECTION In Words Examples and Problems www.elsolucionario.net Practice “Work the Problems” Feature Description Benefit Page “Assess Your Understanding” contains a variety of problems at the end of each section ‘Are You Prepared?’ Problems These assess your retention of the prerequisite material you’ll need Answers are given at the end of the section exercises This feature is related to the Preparing for This Section feature Do you always remember what you’ve learned? Working these problems is the best way to find out If you get one wrong, you’ll know exactly what you need to review and where to review it! 419, 430 Concepts and Vocabulary These short-answer questions, mainly Fill-in-the-Blank and True/False items, assess your understanding of key definitions and concepts in the current section It is difficult to learn math without knowing the language of mathematics These problems test your understanding of the formulas and vocabulary 431 Skill Building Correlated to section examples, these problems provide straightforward practice It’s important to dig in and develop your skills These problems provide you with ample practice to so 431–433 Mixed Practice These problems offer comprehensive assessment of the skills learned in the section by asking problems that relate to more than one concept or objective These problems may also require you to utilize skills learned in previous sections Learning mathematics is a building process Many concepts are interrelated These problems help you see how mathematics builds on itself and also see how the concepts tie together 433 Applications and Extensions These problems allow you to apply your skills to real-world problems These problems also allow you to extend concepts leamed in the section You will see that the material learned within the section has many uses in everyday life 433–435 Explaining Concepts: Discussion and Writing “Discussion and Writing” problems are colored red These support class discussion, verbalization of mathematical ideas, and writing and research projects To verbalize an idea, or to describe it clearly in writing, shows real understanding These problems nurture that understanding Many are challenging but you’ll get out what you put in 436 NEW! Interactive Exercises In selected exercise sets, applets are provided to give a “hands-on” experience “Now Work ” Many examples refer you to a related homework problem These related problems are marked by a pencil and yellow numbers If you get stuck while working problems, look for the closest Now Work problem and refer back to the related example to see if it helps 429 Every chapter concludes with a comprehensive list of exercises to pratice Use the list of objectives to determine the objective and examples that correspond to the problems Work these problems to verify you understand all the skills and concepts of the chapter Think of it as a comprehensive review of the chapter 495–499 PROBLEMS Chapter Review Problems The applets allow students to interact with mathematics in an active learning environment By exploring a variety of scenarios, the student is able to visualize the mathematics and develop a deeper conceptual understanding of the material www.elsolucionario.net 345–346 Review “Study for Quizzes and Tests” Feature Description Benefit Page Chapter Reviews at the end of each chapter contain… “Things to Know” A detailed list of important theorems, formulas, and definitions from the chapter Review these and you’ll know the most important material in the chapter! 494–495 “You should be able to…” Contains a complete list of objectives by section, examples that illustrate the objective, and practice exercises that test your understanding of the objective Do the recommended exercises and you’ll have mastery over the key material If you get something wrong, review the suggested examples and page numbers and try again 495–496 Review Exercises These provide comprehensive review and practice of key skills, matched to the Learning Objectives for each section Practice makes perfect These problems combine exercises from all sections, giving you a comprehensive review in one place 496–499 CHAPTER TEST About 15–20 problems that can be taken as a Chapter Test Be sure to take the Chapter Test under test conditions—no notes! Be prepared Take the sample practice test under test conditions This will get you ready for your instructor’s test If you get a problem wrong, watch the Chapter Test Prep video 499–500 CUMULATIVE REVIEW These problem sets appear at the end of each chapter, beginning with Chapter They combine problems from previous chapters, providing an ongoing cumulative review These are really important They will ensure that you are not forgetting anything as you go These will go a long way toward keeping you constantly primed for the final exam 500 CHAPTER PROJECTS The Chapter Project applies what you’ve learned in the chapter Additional projects are available on the Instructor’s Resource Center (IRC) The Project gives you an opportunity to apply what you’ve learned in the chapter to solve a problem related to the opening article If your instructor allows, these make excellent opportunities to work in a group, which is often the best way of learning math 501 NEW! Internet-based Projects In selected chapters, a web-based project is given The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 501 www.elsolucionario.net This page intentionally left blank www.elsolucionario.net ALGEBRA & TRIGONOMETRY Enhanced with Graphing Utilities Sixth Edition Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.elsolucionario.net Editor in Chief: Anne Kelly Sponsoring Editor: Dawn Murrin Assistant Editor: Joseph Colella Executive Marketing Manager: Roxanne McCarley Marketing Manager: Peggy Sue Lucas Marketing Assistant: Justine Goulart Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Peggy McMahon Procurement Manager/Boston: Evelyn Beaton Procurement Specialist: Debbie Rossi Procurement Media Specialist: Ginny Michaud Senior Author Support/Technology Specialist: Joe Vetere Associate Director of Design, USHE North and West: Andrea Nix Senior Design Specialist: Heather Scott Interior and Cover Design: Tamara Newnam Cover Image (background): iStockphoto/Simfo Image Manager:/Image Management Services: Rachel Youdelman Photo Research: PreMedia Global Permissions Project Manager: Michael Joyce Media Producer: Christina Maestri Software Development: Kristina Evans, Mary Durnwald, and Marty Wright Full-Service Project Management: Cenveo Publisher Services/Nesbitt Graphics, Inc Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this text appear on page xxviii of the book Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and Pearson was aware of a trademark claim, the designations have been printed in initial caps or all caps Microsoft® and Windows® are registered trademarks of the Microsoft Corporation in the U.S.A and other countries Screen shots and icons reprinted with permission from the Microsoft Corporation This book is not sponsored or endorsed by or affiliated with the Microsoft Corporation Library of Congress Cataloging-in-Publication Data Sullivan, Michael, 1942Algebra & trigonometry : enhanced with graphing utilities/Michael Sullivan, Michael Sullivan, III –6th ed p cm Includes bibliographical references and index ISBN 978-0-321-78483-4 (alk paper) Algebra–Textbooks Trigonometry–Textbooks Algebra–Graphic methods Trigonometry–Graphic methods I Sullivan, Michael, 1967 July 2- II Title III Title: Algebra and trigonometry QA154.3.S75 2013 512’.13–dc23 2011024234 Copyright ©2013, 2009, 2006, 2003, 2000 Pearson Education, Inc All rights reserved Manufactured in the United States of America This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290 10—CRK—15 14 13 12 11 ISBN-10: 0-321-78483-9 ISBN-13: 978-0-321-78483-4 www.elsolucionario.net For the Family Katy (Murphy) and Pat Mike and Yola Dan and Sheila Colleen (O’Hara) and Bill Shannon, Patrick, Ryan Michael, Kevin, Marissa Maeve, Sean, Nolan Kaleigh, Billy, Timmy www.elsolucionario.net I2 INDEX Axis/axes (continued) formulas for, 810 identifying conic without, 813–814 of symmetry of parabola, 296, 774 of quadratic function, 298 Babylonians, ancient, 15, 105, 116, 962 Back substitution, 844 Barry, Rick, 226 Base of exponent, 22 Basic trigonometric identities, 631 Bearing (direction), 672 Bernoulli, Jakob, 736, 1002 Bernoulli, Johann, 832 Bessel, Friedrich, 674 Best fit cubic function of, 340–341 line of, 290–291 Beta of a stock, 277 Bezout, Étienne, 910 Binomial(s), 40 cubing, 44–45 squares of (perfect squares), 44 Binomial coefficient, 971, 972 Binomial Theorem, 969–974 to evaluate (n/j), 969–971 expanding a binomial, 971–972 historical feature on, 974 proof of, 973 using, 971–973 Bisection Method, 358–359 Blood alcohol concentration (BAC), 444 Bode, Johann, 949 Bode’s Law, 949 Bonds, zero-coupon, 474 Book value, 282–283 Boole, George, 1002 Bounded graphs, 920 Bounds on zeros, 353–355 Box, volume and surface area of, 32 Brachistochrone, 831–832 Brancazio, Peter, 226 Branches of hyperbola, 795 Break-even point, 286 Brewster’s Law, 630 Briggs, Henry, 456 Bürgi, Joost, 456 Calculator(s), 6–7 See also Graphing utility(ies) approximating roots on, 74 converting between decimals and degrees, minutes, seconds on, 505 converting from polar coordinates to rectangular coordinates, 717 to evaluate powers of 2, 420 functions on, 211–212 inverse sine on, 605–606 kinds of, logarithms on, 454 trigonometric equations solved using, 624 Calculus, 628 approximating e x, 949 area under curve, 614, 615 area under graph, 239 complicated functions in, 215 composite functions in, 403 derivative, 374 difference quotient in, 211, 240, 435, 457, 649 double-angle formulas in, 652 e in, 427 end behavior of graphs, 335 exponential equations in, 462 factoring problems occurring in, 54 functions and exponential, 420, 949 increasing, decreasing, or constant, 231, 482 local maxima and local minima in, 232, 234 graph of polynomial functions in, 328 independent variable in, 560 integral, 654 Intermediate Value Theorem, 355 limits, 335, 365 logarithms and, 452, 462 partial fraction decomposition and, 897 polar equations and, 736 projectile motion, 826–828 quadratic equations in, 115 secant line and, 236, 240 simplifying expressions with rational exponents in, 77–78 Simpson’s Rule, 314 Snell’s Law and, 629 tangent line and, 697 trigonometric functions and equations in, 626, 628, 652, 660 trigonometric identities useful in, 654 turning points and, 333, 334 of variations, 832 Cantor, Georg, 15 Carbon dating, 478 Cardano, Girolamo, 356, 745, 1001 Cardioid, 729–730, 735 Caret key, 25 Carlson, Tor, 489 Carrying capacity, 480–482 Cartesian (rectangular) coordinates, 81, 82–83 converted to polar coordinates, 718–719 polar coordinates converted to, 716–717 polar coordinates vs., 714 polar equations graphed by converting to, 723–724 Cartesian (rectangular) form of complex number, 740–741 Catenary, 310n, 783 Cayley, Arthur, 840, 893 Ceilometer, 533 Cell division, 475, 480 Cellular telephones, 205 Center of circle, 188, 195 of hyperbolas, 795 Central angle, 506 Change-of-Base Formula, 454–455 Chebyshëv, P L., 652n Chebyshëv polynomials, 652n Chu Shih-chieh, 974 Circle(s), 188–195, 773 arc length of, 507 area of, 32 area of sector of, 510–511 center of, 188, 195 central angle of, 506 circumference of, 22, 32 defined, 188 general form of equation of, 191–192 graphing, 188–190, 735 inscribed, 698 intercepts of, 190 polar equation of, 723–724, 726–728 radius of, 188, 195 standard form of equation of, 188–189 unit, 189, 549–553 Circular functions, 551 Circular motion, 511–512 simple harmonic motion and, 699–700 Circumference, 22, 32 Clark, William, 669, 711 Clock, Huygens’s, 832 Closed interval, 146 Coefficient, 40 binomial, 971, 972 damping, 702 leading, 40, 359 Coefficient matrix, 856 www.elsolucionario.net Cofactors, 874 Cofunctions, 522–523 names of, 524 Coincident lines, 843 Column index, 856, 880 Column vector, 884 Combinations, 989–991 defined, 989 listing, 989–990 of n distinct objects taken r at a time, 990 Combinatorics, 981 Combined inequalities, 150 Combined variation, 197–198 Common difference, 950 Common logarithms (log), 441, 454, 456 Common ratio, 956 Commutative property of dot products, 761 of matrix addition, 882, 887 of real numbers, 10 of vector addition, 748 Complementary angles, 522 Complementary Angle Theorem, 522–523 Complement of event, 1000 Complement of set, Complement Rule, 1000–1001 Complete graph, 89 Completing the square, 56–57, 112–113 identifying conics without, 809 Complex number(s), 120–128, 756, 765 argument of, 740 conjugates of, 122, 123, 124, 740 defined, 120 De Moivre’s Theorem and, 742–743 equality, addition, subtraction, and multiplication of, 121–124 geometric interpretation of, 739 magnitude (modulus) of, 740 parts of, 120 in polar form converting from rectangular form to, 740–741 converting to rectangular form, 740–741 products and quotients of, 741–742 powers of i, 124 product of, 741–742 quadratic equations in, 124–126 quotient of, 741–742 in standard form, 120, 122–124 Complex plane, 739–741 defined, 739 imaginary axis of, 739 plotting points in, 739–741 real axis of, 739 Complex polynomial function, 359 Complex rational expressions, 68–70 Complex roots, 743–745 Complex variable, 359 Complex zeros of polynomials, 359–363 Conjugate Pairs Theorem, 360–361 defined, 359 finding, 362–363 polynomial function with specified zeros, 361–362 Components of vectors, 750, 751 Composite functions, 399– 406 calculus application of, 403 components of, 403 defined, 399 domain of, 400– 403 equal, 404– 405 evaluating, 400 finding, 400– 403 forming, 399–400 Compound interest, 466– 471 computing, 466– 467 continuous, 468– 469 defined, 466 doubling or tripling time for money, 471 INDEX effective rates of return, 469 formula, 467 future value of lump sum of money, 465–469 present value of lump sum of money, 470 Compound probabilities, 998 Compressions, 254–256, 258 Comps (home valuation method), 163 Conditional equation, 631 Cone axis of, 773 generators of, 773 right circular, 773 vertex of, 773 Confidence interval, 156 Congruent triangles, 33–36 Conics defined, 816 degenerate, 773 directrix of, 816 eccentricity of, 816 ellipse, 773, 783–794 with center at (h, k), 788–790 with center at the origin, 784–788 with center not at origin, 789–790 center of, 784 defined, 784, 816 eccentricity of, 794, 817–818 foci of, 784 graphing of, 786–788 length of major axis, 784 major axis of, 784, 816 minor axis of, 784 solving applied problems involving, 791 vertices of, 784 focus of, 816 general form of, 808–809 hyperbolas, 772, 773, 794–807 asymptotes of, 800–801 branches of, 795 with center at (h, k), 801–803 with center at the origin, 795–799 with center not at the origin, 802 center of, 795 conjugate, 807 conjugate axis of, 795 defined, 795, 816 eccentricity of, 807, 818 equilateral, 807 foci of, 795 graphing equation of, 796–799 solving applied problems involving, 803–804 transverse axis of, 795, 816 vertices of, 795 identifying, 808–809 without a rotation of axes, 813–814 names of, 773 parabola, 296–298, 773, 774–783 axis of symmetry of, 296, 774 defined, 774, 816 directrix of, 774 focus of, 774 graphing equation of, 775 solving applied problems involving, 779–780 with vertex at (h, k), 778–779 with vertex at the origin, 774–778 vertex of, 296, 774 paraboloids of revolution, 772, 779–780 parametric equations, 822–835 applications to mechanics, 831–832 for curves defined by rectangular equations, 829–832 cycloid, 831 defined, 822 describing, 825–826 graphing using graphing utility, 823–824 rectangular equation for curve defined parametrically, 824–826 time as parameter in, 826–829 polar equations of, 816–822 analyzing and graphing, 816–820 converting to rectangular equation, 820 focus at pole; eccentricity e, 817–818 rotation of axes to transform equations of, 809–814 analyzing equation using, 811–813 formulas for, 810 Conjugate(s), 122, 123–124 of real number, 123 Conjugate axis, 795 Conjugate golden ratio, 949 Conjugate hyperbola, 807 Conjugate of complex numbers, 740 Conjugate Pairs Theorem, 360–361 Connected mode, 246 Consistent systems of equations, 842, 843, 847 Constant(s), 20, 40 of proportionality, 196 Constant functions, 231–232, 233, 243 Constant linear functions, 281–282 Constant rate job problems, 142 Constraints, 923 Consumer Price Index (CPI), 474 Continued fractions, 73 Continuous compounding, 468–469 Continuous function, 245, 355 Continuous graph, 328 Convergent geometric series, 959–962 Cooling, Newton’s Law of, 479–480 Coordinates, 82, 84 See also Rectangular (Cartesian) coordinates of point on number line, 18 Copernicus, 512 Corner points, 920 Correspondence between two sets, 206 Cosecant, 539–540, 553 defined, 517 graph of, 579–580 periodic properties of, 556 Cosecant function, 551 domain of, 554, 555 inverse, 617 approximate value of, 618 calculator to evaluate, 617–618 definition of, 617 exact value of, 617 range of, 555 Cosine(s), 539–540, 553 defined, 517 exact value of, 639 Law of, 686–692 in applied problems, 688–689 defined, 686 historical feature on, 689 proof of, 687 Pythagorean Theorem as special case of, 687 SAS triangles solved using, 687–688 SSS triangles solved using, 688 periodic properties of, 556 Sum and Difference Formula for, 638–639 trigonometric equations linear in, 645–646 Cosine function, 550 domain of, 554, 555, 563 graphs of, 560–575 amplitude and period, 564–566 equation for, 569–570 key points for, 566–569 hyperbolic, 435 inverse, 607–609 defined, 607 exact value of, 608–609 exact value of expressions involving, 616 implicit form of, 607 properties of, 563 range of, 554, 555, 563 www.elsolucionario.net Cost(s) fixed, 185 marginal, 305 variable, 185 Cotangent, 539–540, 553 defined, 517 periodic properties of, 556 Cotangent function, 551 domain of, 554, 555 graph of, 578–579 inverse, 617 approximating the value of, 618 calculator to evaluate, 617–618 definition of, 617 range of, 555 Coterminal angles, 541–543 Counting, 981–986 addition principle of, 982–983 combinations, 989–991 defined, 989 listing, 989–990 of n distinct objects taken r at a time, 990 formula, 982 multiplication principle of, 983–984 number of possible meals, 983–984 permutations, 986–989 computing, 989 defined, 986 distinct objects without repetition, 987–989 distinct objects with repetition, 987 involving n nondistinct objects, 991–992 Counting numbers (natural numbers), 4, 5, 967 Cramer, Gabriel, 840 Cramer’s Rule, 840, 870 inconsistent or dependent systems, 876 for three equations containing three variables, 875–876 for two equations containing two variables, 871–873 Cross (vector) product, 756 Cube(s) of binomials (perfect cubes), 44 difference of two, 44–45, 50–51 sum of two, 44–45, 50–51 Cube function, 210, 244 Cube root, 73, 241–242, 244 complex, 743, 744–745 Cubic equations, theory of, 105 Cubic function of best fit, 340–341 Cubic models from data, 340–341 Curve(s) defined by rectangular equations, 829–832 defined parametrically, 823–829 of quickest descent, 831–832 sawtooth, 706 Curve fitting, 851 sinusoidal, 586–590 hours of daylight, 589–590 sine function of best fit, 590 temperature data, 586–588 Curvilinear motion, 826 Cycle of sinusoidal graph, 560, 565 Cycloid, 831 Damped motion, 699n, 702–703 Damping factor (damping coefficient), 702 Dantzig, George, 923n Data arrangement in matrix, 880 cubic models from, 340–341 fitting exponential functions to, 487–488 linear models from, 288–294 quadratic models from, 311–312 sinusoidal model from, 586–590 Data (Euclid), 116 Day length, 326, 502 “Deal or No Deal” (TV show), 980 I3 I4 INDEX Decay, Law of, 477– 479 See also Exponential growth and decay Decimals, approximate, approximating, converting between degrees, minutes, seconds and, 505–506 converting between scientific notation, 25 Decimal system, 15 Declination of the Sun, 613–614 Decomposition, 763–764 Decreasing functions, 231–232, 233, 234–235 Decreasing linear functions, 281–282 Dedekind, Richard, 15 Deflection, force of, 713 Degenerate conics, 773 Degree of monomial, 40, 47 Degree of polynomial, 40, 47, 327–330 odd, 353, 361 Degree of power function, 328 Degrees, 504–506 converting between decimals and, 505–506 converting between radians and, 507–510 historical note on, 504 Demand equation, 307 De Moivre, Abraham, 742 De Moivre’s Theorem, 742–743 Denominator, 4, 63 rationalizing the, 75–76 Dependent systems of equations, 843 containing three variables, 850–851 containing two variables, 846–847 Cramer’s Rule with, 876 matrices to solve, 862–864 Dependent variable, 210 Depreciation, 398 Depressed equation, 351 Depression, angle of, 533 Derivative, 374 Descartes, René, 81, 82n, 205, 629 Descartes’s Law, 629n Determinants, 840, 870–879 cofactors, 874 Cramer’s Rule to solve a system of three equations containing three variables, 875–876 Cramer’s Rule to solve a system of two equations containing two variables, 871–873 expanding across a row or column, 874 minors of, 873–874 properties of, 876–877 by 3, 873–875 by 2, 870, 876–877 Diagonal entries, 887 Difference(s), 7, 8, 12 See also Subtraction common, 950 of complex numbers, 121 first, 582 of logarithms, 452 of two cubes, 44–45, 50–51 of two functions, 214–216 of two matrices, 881–882 of two squares, 43, 50–51 of vectors, 748 Difference quotient, 211, 240, 435, 649 Diophantus, 962 Directed line segment, 747 Direction angle, 753 Direction (bearing), 672 Direction of vectors, 747, 752–754 Directrix, 816 of parabola, 774 Direct variation, 196, 197, 198 Dirichlet, Lejeune, 205 Discontinuity, 245–246 Discontinuous function, 245–246 Discriminant, 113 negative, 124 Disjoint sets, Distance mean, 794, 839 on real number line, 19–20 Distance formula, 84–86 proof of, 85 Distributive Property of dot products, 761 of matrix multiplication, 887 of real numbers, 11 Divergent geometric series, 959–962 Dividend, 45, 347 Division, 7, See also Quotient(s) of complex numbers, 121–124 in standard form, 123 in order of operations, of polynomials, 45–47, 347–349 algorithm for, 347 synthetic, 59–62 properties of, 12–13 of rational expressions, 63–64 of two integers, 45 Divisor, 45, 347 Domain, 207, 212–214 of absolute value function, 245 of composite function, 400–403 of constant function, 243 of cosecant function, 554, 555 of cosine function, 554, 555, 563 of cotangent function, 554, 555 of cube function, 244 of cube root function, 244 defined by an equation, 213 of difference function, 214 of greatest integer function, 245 of identity function, 243 of inverse function, 410 of logarithmic function, 438–439 of logistic models, 481 of one-to-one function, 407 of product function, 214 of quotient function, 215 of rational function, 364–366 of reciprocal function, 245 of secant function, 554, 555 of sine function, 554, 555, 561 of square function, 244 of square root function, 244 of sum function, 214 of tangent function, 554, 555, 577 of trigonometric functions, 554–555 unspecified, 216 of variable, 21–22 Domain-restricted function, 414–415 Doppler, Christian, 383 Doppler effect, 383 Dot mode, 246 Dot product, 756, 760–767 angle between vectors using, 761–762 to compute work, 764–765 defined, 760 finding, 760–761 historical feature on, 765 orthogonal vectors and, 762–764 parallel vectors and, 762 properties of, 761 of two vectors, 760–761 Double-angle Formulas, 650–654 to establish identities, 651–654 to find exact values, 650–651 Double root (root of multiplicity 2), 110 Drag, 771 Dry adiabatic lapse rate, 955 e, 427– 428, 435 defined, 427 Earthquakes, magnitude of, 448– 449 www.elsolucionario.net Eccentricity, 816 of ellipse, 794, 817–818 of hyperbola, 807, 818 Eddin, Nasir, 512, 689 Effective rates of return, 469 Egyptians, ancient, 116, 962 Elements (Euclid), 689, 962 Elements of sets, 2, 981–983 Elevation, angle of, 533 Elimination, Gauss-Jordan, 862 Elimination method, 840, 844–847 systems of nonlinear equations solved using, 905–907 Ellipse, 773, 783–794 with center at (h, k), 788–790 with center at the origin, 784–788 major axis along x-axis, 785 major axis along y-axis, 787 with center not at origin, 789–790 center of, 784 defined, 784, 816 eccentricity of, 794, 817–818 foci of, 784 graphing of, 786–788 major axis of, 784, 816 length of, 784 minor axis of, 784 solving applied problems involving, 791 vertices of, 784 Ellipsis, Elliptical orbits, 772 Elongation angle, 685 Empty (null) sets, 2, 981 End behavior, 335–336 Endpoints of interval, 146 Entries of matrix, 856, 880 diagonal, 887 Equality of complex numbers, 121 properties of, 10 of sets, 2, 981 of vectors, 747, 751 Equally likely outcomes, 997–998 Equal sign, Equation(s) algebraically solving, 100–101 approximate solutions of, 99 conditional, 631 demand, 307 depressed, 351 domain of a function defined by, 213 equivalent, 90, 100–101 even and odd functions identified from, 230–231 exponential, 428–430, 443, 460–462 quadratic in form, 462 in the form y = {expression in x}, 90 as function, 209 graphing utility to solve, 99–100, 101 historical feature on, 105 intercepts from, 164–165 inverse function defined by, 412–415 linear See Linear equation(s) polar See Polar equations quadratic in form, 130–132 solving, 130–132 satisfying the, 87, 98 sides of, 87, 98 solution set of, 98 solving, 98 systems of See Systems of equations in two variables, graphs of, 87–93 intercepts from, 92–93 by plotting points, 87–89 symmetry test using, 165–167 x = y 2, 168 y = , x, 168–169 y = x 3, 167 INDEX Equilateral hyperbola, 807 Equilateral triangle, 96 Equilibrium, static, 755–756 Equilibrium price, 283–284 Equilibrium quantity, 283–284 Equilibrium (rest) position, 699 Equivalent equations, 90, 100–101 Equivalent inequalities, 147, 149 Equivalent systems of equations, 844–845 Error triangle, 97 Euclid, 116, 689, 962, 974 Euler, Leonhard, 205, 512, 1002 Even functions, 563, 565 determining from graph, 229–230 identifying from equation, 230–231 Evenness ratio, 447–448 Even-Odd identity, 631 Even-odd properties, 557 Events, 997 complement of, 1000–1001 mutually exclusive, 999–1000 probabilities of union of two, 999–1000 Exact numbers, 6–7 Explicit form of function, 212 Exponent(s), 22 Laws of, 22–24, 420, 429 logarithms related to, 436–437 negative, 23 Exponential equations, 428–430 defined, 428 logarithm properties to solve, 453, 460 solving, 428–430, 443, 460–462 equations quadratic in form, 462 using graphing utility, 459–460, 461–463 Exponential expressions, changing between logarithmic expressions and, 437 Exponential functions, 419–436 defined, 421 e, 427–428, 435 evaluating, 419–423 fitting to data, 487–488 graph of, 423–427 using transformations, 426–427, 428 identifying, 421–423 power function vs., 421 properties of, 424–425, 426, 430 ratio of consecutive outputs of, 421–423 Exponential growth and decay, 420–430, 475–486 law of decay, 477–479 logistic models, 480–483 defined, 480 domain and range of, 481 graph of, 480–482 properties of, 481 uninhibited growth, 475–477 Exponential law, 475 Extended Principle of Mathematical Induction, 968 Extraneous solutions, 103, 129 Extreme values of functions, 233 Extreme Value Theorem, 234 Factored completely, 50 Factorial symbol, 939–940 Factoring defined, 49 of expression containing rational exponents, 78 over the integers, 49–50 polynomials, 49–58 Ax + Bx + C, 54–55 difference of two squares and the sum and the difference of two cubes, 50–51 by grouping, 53–54 perfect squares, 51–52 x + Bx + C, 52–53 quadratic equations, 109–111, 133 Factors, 8, 49 linear, 897–900 nonrepeated, 897–898 repeated, 899–900 quadratic, 352–353, 900–902 synthetic division to verify, 61 Factor Theorem, 347–349 Family of lines, 186 of parabolas, 262 Feasible point, 924, 925 Fermat, Pierre de, 81, 435, 1001 Ferrari, Lodovico, 356 Ferris, George W., 194, 628 Fertility rate, 936 Fibonacci (Leonardo Pisano Bigollo), 962 Fibonacci numbers, 941 Fibonacci sequences, 941, 948–949 Financial models, 465–475 compound interest, 466–471 doubling time for investment, 471 effective rates of return, 469 future value of a lump sum of money, 465–469 present value of a lump sum of money, 467, 470 tripling time for investment, 471 Finck, Thomas, 512, 524 Finite sets, 981 First-degree equation See Linear equation(s) First differences, 582 Fixed costs, 185 Focus/foci, 816 of ellipse, 784 of hyperbola, 795 of parabola, 774 FOIL method, 43 Foot-pounds, 764 Force(s), 699 aerodynamic, 771 of deflection, 713 resultant, 754 Force vector, 753 Formulas, geometry, 32–33 Foucault, Jean-Bernard-Leon, 136 Fractions continued, 73 least common multiple to add, 14–15 partial, 897 Frequency, 574 in simple harmonic motion, 700 Frobenius, Georg, 893 Function(s), 205–276 See also Composite functions; Exponential functions; Inverse functions; Linear functions; Polynomial functions; Trigonometric functions absolute value, 242–243, 245 argument of, 210 average cost, 223 average rate of change of, 235–237 finding, 235–237 secant line and, 236–237 building and analyzing, 264–266 on calculators, 211–212 circular, 551 constant, 231–232, 233, 243 continuous, 245, 355 cube, 210, 244 cube root, 241–242, 244 decreasing, 231–232, 233, 234–235 defined, 207 difference of two, 214–216 difference quotient of, 211 discontinuous, 245–246 domain of, 207, 212–214 unspecified, 216 domain-restricted, 414–415 equation as, 209 even and odd, 563, 565 determining from graph, 229–230 identifying from equation, 230–231 www.elsolucionario.net I5 explicit form of, 212 graph of, 219–228, 252–264 combining procedures, 254, 258–260 determining odd and even functions from, 229–230 determining properties from, 231–232 identifying, 220–221 information from or about, 221–223 using compressions and stretches, 254–256, 258 using reflections about the x-axis or y-axis, 256–257 using vertical and horizontal shifts, 252–254, 258 greatest integer, 245–246 identically equal, 631 identity, 243–244 implicit form of, 212 important facts about, 212 increasing, 231–232, 233, 234–235 library of, 241–246 local maxima and local minima of, 232–233, 234–235 nonlinear, 278 objective, 923–927 one-to-one, 406–409 periodic, 555–556 piecewise-defined, 246–248 power, 328–330 graph of, 329–330 of odd degree, 329–330 properties of, 329–330 product of two, 214–216 quotient of two, 214–216 range of, 207 reciprocal, 244–245, 579 See also Cosecant function; Secant function relation as, 206–209 square, 244 square root, 241, 244 step, 246 sum of two, 214–216 graph of, 703–704 value (image) of, 207, 209–212 zeros of, Bisection Method for approximating, 358–359 Function keys, Function notation, 216 Fundamental identities of trigonometric functions, 518–520 quotient, 518 reciprocal, 518 Fundamental period, 556 Fundamental Theorem of Algebra, 359–360 Conjugate Pairs Theorem and, 360–361 proof of, 359n, 360 Future value, 465–469 Galois, Evariste, 356 Gauss, Karl Friedrich, 359, 745, 840 Gauss-Jordan method, 862 General addition principle of counting, 983 General form of conics, 808–809 of equation of circle, 191–192 linear equation in, 179–180 General term, 938 Generators of cone, 773 Geometric mean, 156 Geometric progression See Geometric sequences Geometric sequences, 956–959 common ratio of, 956 defined, 956 determining, 956–957 formula for, 957–958 nth term of, 957–958 sum of, 958–959 Geometric series, 959–962 infinite, 959–960 I6 INDEX Geometric vectors, 747–749 Geometry essentials, 30–39 formulas, 32–33 Pythagorean Theorem and its converse, 30–32, 35–36 Geometry problems, algebra to solve, 86 Gibbs, Josiah, 756 Golden ratio, 949 conjugate, 949 Grade, 187 Graph(s)/graphing bounded, 920 of circles, 188–190, 735 complete, 89 of cosecant function, 579–580 using transformations, 580 of cosine function, 562–564 of cotangent function, 578–579 of De Moivre’s Theorem, 743 of ellipse, 786–788 of equations in two variables, 87–93 intercepts from, 92–93 by plotting points, 87–89 symmetry test using, 165–167 x = y 2, 168 y = , x, 168–169 y = x 3, 167 of exponential functions, 423–427 using transformations, 426–427, 428 of function, 219–228, 252–264 combining procedures, 254, 258–260 determining odd and even functions from, 229–230 determining properties from, 231–232 identifying, 220–221 information from or about, 221–223 in library of functions, 241–246 using compressions and stretches, 254–256, 258 using reflections about the x-axis or y-axis, 256–257 using vertical and horizontal shifts, 252–254, 258 of inequalities, 18–19, 914–920 linear inequalities, 915–916 steps for, 915 of inverse functions, 411–412 of inverse of matrix, 891–892 of linear function, 278 of lines given a point and the slope, 175 using intercepts, 179–180 to locate absolute maximum and absolute minimum of function, 233–234 of logarithmic functions, 439–442 base not 10 or e, 455 inverse, 440–442 of logistic models, 480–482 of matrix multiplication, 885–886 of parabola, 775, 779 of parametric equations, 823–824 of piecewise-defined functions, 246–248 of polar equations, 723–739 cardioid, 729–730, 735 circles, 735 of conics, 817–820 by converting to rectangular coordinates, 723–724 defined, 723 lemniscate, 733734, 735 limaỗon with inner loop, 731732, 735 limaỗon without inner loop, 730731, 735 by plotting points, 729–734 polar grids for, 723 rose, 732–733, 735 sketching, 735–736 spiral, 734 using graphing utility, 725–728 of polynomial functions, 329–334 analyzing, 337–340 end behavior of, 335–336 smooth and continuous, 328 turning points of, 333–334 using bounds on zeros, 354 using transformations, 330–331 of polynomial inequalities, 385–386 of quadratic functions properties of, 298 steps for, 303 using its vertex, axis, and intercepts, 298–302 using transformations, 296–298 of rational functions, 375–384 analyzing, 375–381 constructing rational function from, 380–381 using transformations, 366 of rational inequalities, 386–388 of secant function, 579–580 using transformations, 580 of sequences, 937, 938 of sine and cosine functions, 560–575, 586, 704 amplitude and period, 564–566 equation for, 569–570 key points for, 566–569 to solve quadratic equation, 110, 114 to solve systems of equations, 843 of system of linear equations in row echelon form, 860–862 using determinants, 872–873 of systems of nonlinear inequalities, 919 of by determinant, 870 of vectors, 749 of y = 1/x 2, 365–366 Graphing calculator(s), caret key on, 25 composite functions on, 400 exponents evaluated on, 25 Graphing utility(ies), 83–84 absolute value equations using, 133 absolute value inequalities using, 151–152 to approximate intercepts, 93 circles graphed on, 191–192 connected mode, 246 coordinates of a point on, 84 dot mode, 246 equations graphed on, 90–91 equations solved using, 99–100 equation using a rotation of axes on, 812 eVALUEate feature, 348 factoring on, 133 to find sum of arithmetic sequence, 952 to fit exponential function to data, 487– 488 to fit logarithmic function to data, 488– 489 to fit logistic function to data, 489– 490 functions on, 234–235 geometric sequences using, 958, 959 identity established with, 633 inequalities solved using, 150–151 involving quadratic function, 317–319 INTERSECT feature of, 99–100 line of best fit from, 290–291 local maxima and local minima approximated with, 234–235 logarithmic and exponential equations solved using, 459–460, 461–463 matrix addition and subtraction on, 881–882 matrix operations on, 881–882 MAXIMUM and MINIMUM features, 235 PARametric mode, 828 polar equations using, 725–728 polynomial function analyzed with, 339–340 radical equations solved using, 129–130 REF command, 861 REGression options, 487 RREF command, 862 sequences on, 940–941, 942, 943–944 www.elsolucionario.net sine function of best fit on, 590 square screens on, 174–175 system of nonlinear equations solved using, 904–909 TABLE feature, 938 TRACE feature, 938 trigonometric equations solved using, 626 turning points in, 333–334 ZERO (or ROOT) feature, 99, 310 Zoom-Fit feature, 91 Zoom-Standard feature, 91 Grassmann, Hermann, 756, 765 Greatest integer function, 245–246 Greek letters, to denote angles, 503 Greeks, ancient, 15, 512 Grouping, factoring by, 53–54 Growth, uninhibited, 475–477 Growth factor, 421 Hale-Bopp Comet, orbit of, 772, 839 Half-angle Formulas, 655–656 to find exact values, 655–656 for tangent, 656 Half-life, 478 Half-line (ray), 503 Half-open/half-closed intervals, 146 Half-planes, 915 Hamilton, William Rowan, 756 Harmonic mean, 156 Harriot, Thomas, 116 Heron of Alexandria, 694, 695, 962 Heron’s Formula, 694–695 historical feature on, 695 proof of, 694–695 Hindus, ancient, 116 Home, valuing a, 163, 204 Horizontal component of vector, 751 Horizontal compression or stretches, 255 Horizontal lines, 176–177, 724–725, 734 Horizontal-line test, 408–409 Horizontal (oblique) asymptote, 367, 369–372 Horizontal shifts, 252–254, 258 Huygens, Christiaan, 832, 1001 Huygens’s clock, 832 Hyperbolas, 772, 773, 794–807 asymptotes of, 800–801 branches of, 795 with center at (h, k), 801–803 with center at the origin, 795–799 transverse axis along x-axis, 796, 802 transverse axis along y-axis, 798–799, 802 with center not at the origin, 802 center of, 795 conjugate, 807 conjugate axis of, 795 defined, 795, 816 eccentricity of, 807, 818 equilateral, 807 foci of, 795 graphing equation of, 796–799 solving applied problems involving, 803–804 transverse axis of, 795, 816 vertices of, 795 Hyperbolic cosine function, 435 Hyperbolic sine function, 435 Hyperboloid, 807 Hypocycloid, 835 Hypotenuse, 30–31, 516 i, 120 powers of, 124 Ibn MÛsâ al-Khowârizmỵ, Mohammed, 27 Identically equal functions, 631 Identity(ies), 98 definition of, 631 multiplicative, 11 polarization, 767 INDEX Pythagorean, 519, 631 trigonometric, 630–637 basic, 631 establishing, 632–635, 640–643, 651–654 Even-Odd, 631 Pythagorean, 631 Quotient, 631 Reciprocal, 518, 631 trigonometric equations solved using, 625–626 Identity function, 243–244 Identity matrix, 887–888 Identity Properties, 888 of real numbers, 11 Image (value) of function, 207, 209–212 Imaginary axis of complex plane, 739 Imaginary part of complex number, 120 Imaginary unit (i), 120 Implicit form of function, 212 Improper rational expression, 897 Improper rational function, 369–371 Incidence, angle of, 629 Inclination, 559 Inconsistent systems of equations, 842, 843, 847, 849 containing three variables, 849 containing two variables, 846 Cramer’s Rule with, 876 matrices to solve, 864 Increasing functions, 231–232, 233, 234–235 Increasing linear functions, 281–282 Independent systems of equations, 843 Independent variable, 210 in calculus, 560 Index/indices of radical, 73, 78 of refraction, 629 row and column, 856, 880 of sum, 941 Induction, mathematical, 965–968 Extended Principle of, 968 principle of, 965–966, 968 proving statements using, 965–967 Inequality(ies), 145–157 absolute value, 151–153 combined, 150–151 equivalent, 147, 149 graphing, 18–19, 914–920 linear inequalities, 915–916 steps for, 915 interval notation for, 146 –147 involving quadratic functions, 316 –320 multiplication of, 148 –149 nonstrict, 19 in one variable, 149 polynomial, 385–386 algebraically and graphically solving, 385–386 steps for solving, 388 procedures for manipulating symbol, 149 properties of, 147–149 rational algebraically and graphically solving, 386–388 steps for solving, 388 satisfying, 914 sides of, 19 solutions of, 149–150 solving, 149–150 strict, 19 systems of, 914–922 graphing, 917–920 in two variables, 914 Inequality symbols, 18 Inertia moment of, 663 product of, 659 Infinite geometric series, 959–960 Infinite limit, 335–336 Infinite sets, 981 Infinity approaches, 365 limits at, 336 Inflation, 473 Inflection point, 481 Initial point of directed line segment, 747 Initial side of angle, 503 Initial value of exponential function, 421 Input to relation, 206 Inscribed circle, 698 Integers, 4, dividing, 45 factoring over the, 49–50 Integrals, 654 Intercept(s) of circle, 190 from an equation, 164–165 from a graph, 92–93 graphing an equation in general form using, 179–180 graphing utility to approximate, 93 graph of lines using, 179–180 of quadratic function, 298–301 Interest compound, 466–471 computing, 466–467 continuous, 468–469 defined, 466 doubling or tripling time for money, 471 effective rates of return, 469 formula, 467 future value of lump sum of money, 465–469 present value of lump sum of money, 470 problems involving, 138 –139 rate of, 138, 465–466 effective, 469 on loans, 162 simple, 138, 466 Intermediate Value Theorem, 355–356 Intersection of sets, 2–3 Interval notation, 146–147 Intervals confidence, 156 writing, using inequality notation, 147 Invariance, 815 Inverse additive, 11, 65 of inverse of matrix finding, 888–891 of matrix, 888–891 multiplying matrix by, 888–890 solving system of linear equations using, 891–892 multiplicative, 11–12 Inverse functions, 409–415, 602–620 See also Logarithmic functions cosine, 607–609 defined, 607 exact value of, 608–609 exact value of expressions involving, 616 implicit form of, 607 defined by a map or an ordered pair, 409– 411 domain of, 410 of domain-restricted function, 414– 415 finding, 409– 411, 611– 612 defined by an equation, 412– 415 graph of, 411– 412 range of, 410 secant, cosecant, and cotangent, 617 approximating the value of, 618 calculator to evaluate, 617– 618 definition of, 617 sine, 602– 606 approximate value of, 604 –605 defined, 603 exact value of, 603– 604 exact value of expressions involving, 616, 644 implicit form of, 603 www.elsolucionario.net I7 properties of, 605–606 solving equations involving, 612 Sum and Difference Formulas involving, 644–645 tangent, 609–611 defined, 610 exact value of, 610–611 exact value of expressions involving, 616 implicit form of, 610 verifying, 411 written algebraically, 618–619 Inverse trigonometric equations, 612 Inverse variation, 196–197, 198 Irrational numbers, 4, 5, 15, 120 decimal representation of, Irreducible quadratic factor, 352–353, 900–902 Isosceles triangle, 96 J i ba, 524 J i va, 524 Joint variation, 197–198 Jordan, Camille, 840 Joules (newton-meters), 764 Karmarkar, Narendra, 923n Kepler, Johannes, 198 Kepler’s Third Law of Planetary Motion, 202 Khayyám, Omar, 974 Kirchhoff’s Rules, 854, 869 Kôwa, Takakazu Seki, 840 Latitude, 326 Latus rectum, 775, 776 Law of Cosines, 686–692 in applied problems, 688–689 defined, 686 historical feature on, 689 proof of, 687 Pythagorean Theorem as special case of, 687 SAS triangles solved using, 687–688 SSS triangles solved using, 688 Law of Decay, 477–479 See also Exponential growth and decay Law of Sines in applied problems, 680–682 defined, 676 historical feature on, 689 proof of, 681–682 SAA or ASA triangles solved using, 676–677 SSA triangles solved using, 677–680 Law of Tangents, 686, 689 Laws of Exponents, 22–24, 420, 429 Leading coefficient, 40, 359 Leading term, 40 Least common multiple (LCM) to add rational expressions, 66–68 to add two quotients, 14–15 Left endpoint of interval, 146 Left stochastic transition matrix, 896 Legs of triangle, 30, 516 Leibniz, Gottfried Wilhelm, 205, 840 Lemniscate, 733–734, 735 Length of arc of a circle, 507 Lensmaker’s equation, 72 Lewis, Meriwether, 669, 711 Lift, 713, 771 Light detector, 533 Light projector, 533 Like radicals, 75 Like terms, 40 Limaỗon with inner loop, 731–732, 735 without inner loop, 730–731, 735 Limits, 335–336, 365 infinite, 335–336 at infinity, 336 Line(s), 172–188 of best fit, 290–291 I8 INDEX Line(s) (continued) coincident, 843 equations of See also Linear equation(s); Systems of linear equations secant, 236–237 family of, 186 graphing given a point and the slope, 175 using intercepts, 179–180 horizontal, 176–177, 724–725, 734 number line, 18 point-slope form of, 176 polar equation of, 724, 734 slope of, 172–175, 178–179 containing two points, 173 from linear equation, 178–179 tangent, 194 vertical, 172, 173, 725–726, 734 equation of, 176 y-intercept of, 178–179 Linear algebra, 880 Linear equation(s), 101–102 See also Line(s); Systems of linear equations applied problems involving, 104–105 defined, 180 in general form, 179–180 given two points, 177 historical feature on, 105 for horizontal line, 176–177 in one variable, 98, 101–102 for parallel line, 180–181 for perpendicular line, 181–183 slope from, 178–179 in slope-intercept form, 177–178 solving equations that lead to, 102 steps for solving, 104 for vertical line, 175–176 Linear factors, 897–900 nonrepeated, 897–898 repeated, 899–900 Linear functions, 278–287 average rate of change of, 278–281 building from data, 288–294 defined, 278 graphing utility to find the line of best fit, 290–291 graph of, 278 identifying, 421–423 increasing, decreasing, or constant, 281–282 nonlinear relations vs., 289–290 scatter diagrams, 288–289 Linear models from data, 288–294 from verbal descriptions, 282–284 Linear programming problems, 840, 923–927 defined, 924 maximum, 926–927 minimum, 925–926 setting up, 923–924 solution to, 925 location of, 925 solving, 924–927 in two variables, 924, 925 Linear speed, 511 Linear trigonometric equation, 622–623 Line segment, 747 length of, 85–86 midpoint of, 87 Local maxima and local minima of functions, 232–233, 234–235 Logarithmic equations, 458–465 defined, 443 logarithm properties to solve, 453, 460 solving, 443–444, 458–460 Logarithmic functions, 436–449 changing between logarithmic expressions and exponential expressions, 437 defined, 436 domain of, 438–439 evaluating, 437–438 exact value of, 437–438 fitting to data, 488–489 graph of, 439–442 base not 10 or e, 455 properties of, 439, 445 range of, 438 Logarithmic spiral, 734 Logarithms, 449– 457 on calculators, 454 common (log), 441, 454, 456 evaluating, with bases other than 10 or e, 453– 455 historical feature on, 456 logarithmic expression as single, 452– 453 logarithmic expression as sum or difference of, 452 natural (ln), 440, 454, 456 properties of, 449– 457 establishing, 450 proofs of, 450– 451 summary of, 455 using, with even exponents, 460 relating to exponents, 436– 437 Logistic functions, fitting to data, 489–490 Logistic models, 480– 483 defined, 480 domain and range of, 481 graph of, 480– 482 properties of, 481 Loudness, 448 Lowest terms rational function in, 365, 368 reducing rational expressions to, 62–63 Magnitude of earthquake, 448– 449 of vectors, 747, 749, 751, 752, 753–754 Magnitude (modulus), 740, 741 Major axis, 816 Mandel, Howie, 980 Mandelbrot sets, 746 Mapping, 206 Marginal cost, 305 Marginal propensity to consume, 964 Markov chains, 934–935 Mathematical induction, 965–968 Extended Principle of, 968 principle of, 965–966, 968 proving statements using, 965–967 Mathematical modeling, 136 See also Model(s) process of, 136–137 Matrix/matrices, 840, 855–869, 879–896 arranging data in, 880 augmented, 856 coefficient, 856 defined, 855, 880 entries of, 856, 880, 887 equal, 881 examples of, 881 graphing utilities for, 881–882 historical feature on, 893 identity, 887–888 inverse of, 888–891 finding, 888–891 multiplying matrix by, 888–890 solving system of linear equations using, 891–892 left stochastic transition, 896 m by n, 880 nonsingular, 888, 890 product of two, 883–888 in reduced row echelon form, 862, 864, 865 row and column indices of, 856, 880 in row echelon form, 858–866 row operations on, 857–858 scalar multiples of, 882–883 singular, 888 www.elsolucionario.net to solve system of linear equations, 858–866 square, 880 sum and difference of two, 881–882 transition, 934–935 zero, 882 Maxima of functions absolute, 233–234 local, 232–233, 234–235 Maximum value of a quadratic function, 302 Mean arithmetic, 156 geometric, 156 harmonic, 156 Mean distance, 794, 839 Mechanics, parametric equations applied to, 831–832 Medians of triangle, 96 Menelaus of Alexandria, 512 Metrica (Heron), 695 Midpoint formula, 87 Mind, mapping of, 601 Mindomo (software), 668 Minima of functions absolute, 233–234 local, 232–233, 234–235 Minimum value of a quadratic function, 302 Minors, 873–874 Minutes, 505–506 Mixed numbers, Mixture problems, 139–140 Model(s), 104, 136 linear from data, 288–294 from verbal descriptions, 282–284 sinusoidal, 586–590 best-fit, 590 daylight hours, 589–590 temperature data, 586–588 using direct variation, 196, 197, 198 using inverse variation, 196–197, 198 using joint variation or combined variation, 197–198 Modeling process, 136–137 Modulus (magnitude), 740, 741 Mollweide, Karl, 685 Mollweide’s Formula, 685 Moment of inertia, 663 Monomial(s), 40 common factors, 50 degree of, 40, 47 examples of, 40 recognizing, 40 in two variables, 47 Monter, 187 Motion circular, 511–512, 699–700 curvilinear, 826 damped, 699n, 702–703 Newton’s second law of, 749 projectile, 826–828 simple harmonic, 699–703 uniform, 140–141 Multiplication, 7, See also Product(s) of complex numbers, 121–122 horizontal, 42–43 of inequalities, 148–149 in order of operation, 8, of polynomials, 42– 43 of quotients, 13–14 of rational expressions, 63–64 scalar, 882–883 of vectors, by numbers See also Dot product of vectors, by numbers geometrically, 748–749 vertical, 43–44 by zero, 12 Multiplication principle of counting, 983–984 Multiplication properties, 18 for inequalities, 149 INDEX Multiplicative identity, 11 Multiplicative inverse, 11–12 Multiplicity of polynomial function, 331–333 Multiplier, 964 Mutually exclusive events, 999–1000 Napier, John, 456 Nappes, 773 Natural logarithms (ln), 440, 454, 456 Natural numbers (counting numbers), 4, 5, 967 Nautical miles, 515 Negative angle, 503 Negative numbers real, 18 square roots of, 125 Newton-meters (joules), 764 Newton’s Law of Cooling, 479–480, 484 Newton’s Law of Heating, 484 Newton’s Law of universal gravitation, 391 Newton’s Method, 374 Newton’s Second Law of Motion, 699, 749 Niccolo of Brescia (Tartaglia), 356, 745 Nonlinear equations, systems of, 904–913 elimination method for solving, 905–906 historical feature on, 910 substitution method for solving, 904–905 Nonlinear functions, 278 Nonlinear inequalities, systems of, 919 Nonlinear relations, 289–290 Nonnegative property of inequalities, 147 Nonsingular matrix, 888, 890 Nonstrict inequalities, 19 nth roots, 73–74 complex, 743–745 historical feature, 78 rationalizing the denominator, 75–76 simplifying, 74 simplifying radicals, 74–75 Null (empty) sets, 2, 981 Number(s) classification of, 4–5 Fibonacci, 941 irrational, 4, 5, 6, 15 mixed, natural (counting), 4, 5, 967 negative, 18 rational, 4, of significant digits, 6–7 triangular, 949 whole, Number lines, 18, 19–20 Numerator, 4, 63 Numerical expressions, 8–10 Objective function, 923–927 Oblique (horizontal) asymptote, 367, 369–372 Oblique triangle, 675–676 Odd functions, 565 determining from graph, 229–230 identifying from equation, 230–231 One-to-one functions, 406–409 defined, 407 horizontal-line test for, 408–409 Open interval, 146 Opens down, 296 Opens up, 296 Optical (scanning) angle, 659 Optimization, quadratic functions and, 307 Orbits elliptical, 772 planetary, 794 Ordered pair(s), 82 inverse function defined by, 409–411 as relations, 206 Order of operations, 8, Ordinary annuity, 944 Ordinary (statute) miles, 515 Ordinate (y-coordinate), 82 Orientation, 823 Origin, 82 distance from point to, 265 of real number line, 18 symmetry with respect to, 165–166, 728–729 Orthogonal vectors, 762–764 Outcome of probability, 995 equally likely, 997–998 Output of relation, 206 Parabola, 296–298, 773, 774–783 axis of symmetry of, 296, 774 defined, 774, 816 directrix of, 774 family of, 262 focus of, 774 graphing equation of, 775, 779 solving applied problems involving, 779–780 with vertex at (h, k), 778–779 with vertex at the origin, 774–778 finding equation of, 777–778 focus at (a, 0), a > 0, 775 vertex of, 296, 774 Paraboloids of revolution, 772, 779–780 Parallax, 674 Parallel lines, 180–181 Parallel vectors, 762 Parameter, 822 time as, 826–829 Parametric equations, 822–835 for curves defined by rectangular equations, 829–832 applications to mechanics, 831–832 cycloid, 831 defined, 822 describing, 825–826 graphing, 823–824 rectangular equation for curve defined parametrically, 824–826 time as parameter in, 826–829 Parentheses, order of operations and, 9, 23 Partial fraction decomposition, 840, 896–903 defined, 897 where denominator has nonrepeated irreducible quadratic factor, 901–902 where denominator has only nonrepeated linear factors, 897–898 where denominator has repeated irreducible quadratic factors, 902 where denominator has repeated linear factors, 899–900 Partial fractions, 897 Participation rate, 219 Pascal, Blaise, 832, 971, 1001 Pascal triangle, 971, 974 Payment period, 466 Peano, Giuseppe, 1002 Pendulum Foucault’s, 136 period of, 80, 199 simple, 199 Perfect cubes, 44 Perfect roots, 74 Perfect squares, 44, 51–52 Perfect triangle, 697 Perihelion, 794, 822, 839 Perimeter, formulas for, 32 Period fundamental, 556 of pendulum, 80, 199 of simple harmonic motion, 699 of sinusoidal functions, 564–566, 583–586 Periodic functions, 555–556 Periodic properties, 555–556 Permutations, 986–989 computing, 989 www.elsolucionario.net I9 defined, 986 distinct objects without repetition, 987–989 distinct objects with repetition, 987 involving n nondistinct objects, 991–992 Phase angle, 526 Phase shift, 582–586 to graph y = A sin 1vx @ f2 + B, 582–586 Phones, cellular, 205 Physics, vectors in, 747 Piecewise-defined functions, 246–248 Pitch, 187 Pixels, 83 Plane(s) complex, 739–741 defined, 739 imaginary axis of, 739 plotting points in, 739–741 real axis of, 739 Plane curve, 822 Planetary motion, Kepler’s Third Law of, 202 Planets, orbit of, 794 Plotting points, 82, 714–716 graph equations by, 87–89 Point(s) coordinate of, 84 on number line, 18 corner, 920 distance between two, 84 distance from the origin to, 265 feasible, 924, 925 inflection, 481 initial, 747 plotting, 82, 714–716 graph equations by, 87–89 polar coordinates of, 715–716 of tangency, 194 terminal, 747 turning, 333–334 Point-slope form of equation of line, 176 Polar axis, 714 Polar coordinates, 714–722 conversion from rectangular coordinates, 718–719 conversion to rectangular coordinates, 716–717 defined, 714 plotting points using, 714–716 of a point, 715–716 polar axis of, 714 pole of, 714 rectangular coordinates vs., 714 Polar equations calculus and, 736 of circle, 723–724, 726–728 classification of, 734–735 of conics, 816–822 analyzing and graphing, 817–820 converting to rectangular equation, 820 focus at pole; eccentricity e, 817–818 defined, 723 graph of, 723–739 cardioid, 729–730, 735 circles, 735 by converting to rectangular coordinates, 723724 defined, 723 lemniscate, 733734, 735 limaỗon with inner loop, 731732, 735 limaỗon without inner loop, 730731, 735 by plotting points, 729–734 polar grids for, 723 rose, 732–733, 735 sketching, 735–736 spiral, 734 using graphing utility, 725–728 historical feature on, 736 of horizontal line, 724–725 identifying, 723–724 testing for symmetry, 728–729 transforming rectangular form to, 720–721 I10 INDEX Polar equations (continued) transforming to rectangular form, 720 of vertical line, 725–726 Polar form of complex number, 740–741 Polar grids, 723 Polarization identity, 767 Pole, 714 symmetry with respect to, 728–729 Polynomial(s), 40–58 adding, 41–42 Chebyshëv, 652n degree of, 40, 47, 327–330 odd, 353, 361 second-degree, 52–53 dividing, 45–47, 347–349 synthetic division, 59–62 examples of, 41 factoring, 49–58 Ax + Bx + C, 54–55 difference of two squares and the sum and the difference of two cubes, 50–51 by grouping, 53–54 perfect squares, 51–52 x + Bx + C, 52–53 multiplying, 42–43 prime, 50, 53 recognizing, 40–41 solving, 352–353 special products formulas, 43–45 in standard form, 41 subtracting, 41–42 terms of, 40–41 in two variables, 47 zero, 41 Polynomial functions, 326–364 complex, 359 complex zeros of, 359–363 Conjugate Pairs Theorem, 360–361 defined, 359 finding, 362–363 polynomial function with specified zeros, 361–362 cubic models from data, 340–341 defined, 327 end behavior of, 335–336 graph of, 329–334 analyzing, 337–340 end behavior of, 335–336 smooth and continuous, 328 turning points of, 333–334 using bounds on zeros, 354 using transformations, 330–331 historical feature on, 356 identifying, 327–330 multiplicity of, 331–333 behavior near zero and, 333 real zeros (roots) of, 331–333, 346–359 finding, 350–352 Intermediate Value Theorem, 355–356 number of, 349 Rational Zeros Theorem, 349–350, 362 Remainder Theorem and Factor Theorem, 347–349 repeated, 332 theorem for bounds on, 353–355 solving, 350–352 unbounded in the negative direction, 335 Polynomial inequalities, 385–386 algebraically and graphically solving, 385–386 steps for solving, 388 Population, world, 936 Position vector, 750 Positive angle, 503 Positive real numbers, 18 Power(s), 22, 574 See also Exponent(s) of i, 124 log of, 451 Power functions, 328–330 exponential function vs., 421 graph of, 329–330 of odd degree, 329–330 properties of, 329–330 Present value, 467, 470 Price, equilibrium, 283–284 Prime polynomials, 50, 53 Principal, 138, 465 Principal nth root of real number, 73 Principal square root, 24, 125 Probability(ies), 934–935, 995–1005 Complement Rule to find, 1000–1001 compound, 998 constructing models, 995–997 defined, 995 of equally likely outcomes, 997–998 of event, 997 mutually exclusive, 999–1000 historical feature on, 1001–1002 outcome of, 995 sample space, 995 of union of two events, 999–1000 Product(s), 7, See also Dot product; Multiplication of complex numbers, 121–122 in polar form, 741–742 of inertia, 659 log of, 451 special, 43–44, 47 of two functions, 214–216 of two matrices, 883–888 vector (cross), 756 Product function, 214–216 Product-to-Sum Formulas, 660–662 Projectile, range of, 535 Projectile motion, 826–828 Projection, vector, 763–764 Projection of P on the x-axis, 699–700 Projection of P on the y-axis, 700 Prolate spheroid, 793 Proper rational expressions, 897 Proper rational function, 369–370 Proper subsets, 981 Proportionality, constant of, 196 Ptolemy, 630, 689 Pure imaginary number, 120 Pythagorean Brotherhood, 15 Pythagorean Identities, 519, 631 Pythagorean Theorem, 30–32, 516 applying, 31–32 converse of, 31 proof of, 35–36 as special case of Law of Cosines, 687 Pythagorean triples, 39 Quadrant, angle lying in, 504 Quadrantal angles, 504, 540–541 Quadrants, 83 Quadratic equations, 109–119 applied problems involving, 115–116 character of solutions of, 126 completing the square to solve, 112–113 in complex number system, 124–126 defined, 109 discriminant of, 113 negative, 124 factoring, 109–111, 133 historical feature on, 116 procedure for solving, 115 quadratic formula for, 112–114, 125 Square Root Method for solving, 111 in standard form, 109 Quadratic factors, irreducible, 352–353, 900–902 Quadratic formula, 112–114, 125 Quadratic functions, 295–306 defined, 295 graph of www.elsolucionario.net properties of, 298, 299–300 steps for, 303 using its vertex, axis, and intercepts, 298–302 using transformations, 296–298 inequalities involving, 316–320 maximum or minimum value of, 302, 307 optimizations and, 307 vertex and axis of symmetry of, 298 Quadratic models, 307–316 from data, 311–312 from verbal descriptions, 307–311 Quantity, equilibrium, 283–284 Quantity demanded, 283–284 Quantity supplied, 283–284 Quaternions, 756 Quotient(s), 7, 8, 12, 45, 347 See also Division arithmetic of, 13–14 of complex numbers in polar form, 741–742 in standard form, 123 difference, 211, 240, 435, 649 log of, 451 subtraction of, 13–14 synthetic division to find, 60–61 of two functions, 214–216 Quotient identity(ies), 631 of trigonometric functions, 518 Radians, 506 converting between degrees and, 507–510 Radical equations, 128–130 defined, 128 solving, 128–130 Radicals, 73 fractional exponents as, 76–77 index of, 73, 78 like, 75 properties of, 74 rational exponents defined using, 76–77 simplifying, 74–75 Radical sign, 24, 78 Radicand, 73 Radioactive decay, 477–479 Radius, 188, 195 Range, 207 of absolute value function, 245 of constant function, 243 of cosecant function, 555 of cosine function, 554, 555, 563 of cotangent function, 555 of cube function, 244 of cube root function, 244 of greatest integer function, 245 of identity function, 243 of inverse function, 410 of logarithmic function, 438 of logistic models, 481 of one-to-one function, 407 of projectile, 535, 653–654 of reciprocal function, 245 of secant function, 555 of sine function, 554, 555, 561 of square function, 244 of square root function, 244 of tangent function, 555, 577 of trigonometric functions, 554–555 Rate of change, average, 173, 234–236, 278–281 of linear and exponential functions, 422– 423 Rate of interest, 138, 465–466 Rates of return, effective, 469 Ratio common, 956 golden, 949 Rational equations, 102–104 algebraic solution to, 103–104 defined, 102 with no solution, 103–104 INDEX Rational exponents, 76–78 Rational expressions, 62–73 adding and subtracting, 64–66 least common multiple (LCM) method for, 66–68 application of, 70 complex, 68–70 decomposing See Partial fraction decomposition defined, 62 improper, 897 multiplying and dividing, 63–64 proper, 897 reducing to lowest terms, 62–63 Rational functions, 364–374 applied problems involving, 381 asymptotes of, 366–372 horizontal or oblique, 367 vertical, 367–369 defined, 364 domain of, 364–366 examples of, 364 graph of, 375–384 analyzing, 375–381 constructing rational function from, 380–381 using transformations, 366 with a hole, 379–380 improper, 369–371 in lowest terms, 365, 368 proper, 369–370 unbounded in positive direction, 365 Rational inequalities, solving algebraically and graphically, 386–388 steps for, 388 Rationalizing the denominator, 75–76 Rational numbers, 4, 5, 120, 364 Rational Zeros Theorem, 349–350, 362 Rays (half-lines), 503 of central angle, 506 vertex of, 503 Real axis of complex plane, 739 Real number(s), 2–17, 120 approximating decimals, conjugate of, 123 defined, historical feature on, 15 number line representation of, 18 numerical expressions, 8–10 positive and negative, 18 principal nth root of, 73 properties of, 10–15 square of, 120 Real number line, 18 distance on, 19–20 Real part of complex number, 120 Real zeros (roots) of polynomial functions, 346–359 finding, 350–352 Intermediate Value Theorem, 355–356 number of, 349 Rational Zeros Theorem, 349–350, 362 Remainder Theorem and Factor Theorem, 347–349 repeated, 332 theorem for bounds on, 353–355 Reciprocal, 12 of complex number in standard form, 122–123 Reciprocal function, 244–245, 579 See also Cosecant function; Secant function Reciprocal identities, 518, 631 Rectangle, area and perimeter of, 32 Rectangular (Cartesian) coordinates, 81, 82–83 converted to polar coordinates, 718–719 polar coordinates converted to, 716–717 polar coordinates vs., 714 polar equations graphed by converting to, 723–724 Rectangular (Cartesian) form of complex number, 740–741 Rectangular equations for curve defined parametrically, 824–826 polar equations converted to, 720, 820 transforming to polar equation, 720–721 Rectangular grid, 723 Recursive formula, 940–941 for arithmetic sequences, 952 terms of sequences defined by, 940–941 Reduced row echelon form, 862, 864, 865 Reduction properties, 13 Reference angles, 544–545 exact value of trigonometric function using, 545–546 Reflections about x-axis or y-axis, 256–257 Reflexive property, 10 Refraction, 629 Regiomontanus, 512, 689 Relation(s) See also Function(s) defined, 206 as function, 206–209 input to, 206 nonlinear, 289–290 ordered pairs as, 206 output of, 206 Relative maxima and minima of functions, 232–233 Remainder, 45, 347 synthetic division to find, 60–61 Remainder Theorem, 347–349 Repeated zeros (solutions), 110, 332 Repose, angle of, 620 Rest (equilibrium) position, 699 Resultant force, 754 Review, 1–80 of algebra, 17–30 distance on the real number line, 19–20 domain of variable, 21–22 evaluating algebraic expressions, 20–21 graphing calculator to evaluate exponents, 25 graphing inequalities, 18–19 historical feature, 27 Laws of Exponents, 22–24 multiplication properties of positive and negative numbers, 18 real number line, 18 scientific notation, 25–26 square roots, 24 of geometry, 30–39 formulas, 32–33 Pythagorean Theorem and its converse, 30–32, 35–36 of nth roots, 73–74 historical feature, 78 rationalizing the denominator, 75–76 simplifying, 74 simplifying radicals, 74–75 of polynomials, 40–58 adding, 41– 42 dividing, 45– 47 factoring, 49–58 monomials, 40 multiplying, 42– 43 recognizing, 40– 41 special products formulas, 43– 45 subtracting, 41– 42 synthetic division of, 59– 62 in two variables, 47 of rational exponents, 76 –78 of rational expressions, 62 –73 adding and subtracting, 64– 66 application of, 70 complex, 68–70 multiplying and dividing, 63– 64 reducing to lowest terms, 62– 63 of real numbers, 2–17 approximating decimals, defined, historical feature on, 15 number line representation of, 18 www.elsolucionario.net I11 numerical expressions, –10 properties of, 10 –15 significant digits, –7 Revolutions per unit of time, 512 Rhaeticus, 512 Rhind papyrus, 962 Richter scale, 448– 449 Right angle, 30, 504 Right circular cone, 773 Right circular cylinder, volume and surface area of, 32 Right endpoint of interval, 146 Right triangles, 30, 516, 670– 672 applications of, 671– 672 solving, 670– 671 Right triangle trigonometry, 516–527, 670–675 Complementary Angle Theorem, 522–523 finding values of trigonometric functions when one is known, 520–522 fundamental identities, 518–520 modeling and solving applied problems involving, 531–534 values of trigonometric functions of acute angles, 516–518 Rise, 172 Root(s), 98 See also Solution(s); Zeros complex, 743–745 of multiplicity (double root), 110 perfect, 74 Rose, 732–733, 735 Roster method, Rotation of axes, 809–814 analyzing equation using, 811–813 formulas for, 810 identifying conic without, 813–814 Rounding, Round-off errors, 671 Row echelon form, 858–866 reduced, 862, 864, 865 Row index, 856, 880 Row operations, 857–858 Row vector, 884 Rudolff, Christoff, 78 Ruffini, P., 356 Rules of Signs, 13 Rumsey, David, 669 Run, 172 Rutherford, Ernest, 807 SAA triangles, 676–677 Saffir-Simpson Hurricane Scale, 591 Sample space, 995 SAS triangles, 676, 687–688, 693–694 Satisfying equations, 87, 98 Satisfying inequalities, 914 Sawtooth curve, 706 Scalar, 748–749, 751, 882–883 Scalar multiples of matrix, 882–883 Scalar product See Dot product Scale of number line, 18 Scanning (optical) angle, 659 Scatter diagrams, 288–289, 586–588 Schroeder, E., 1002 Scientific calculators, Scientific notation, 25–26 Secant, 539–540, 553 defined, 517 graph of, 579–580 periodic properties of, 556 Secant function, 551 domain of, 554, 555 inverse, 617 approximating the value of, 618 calculator to evaluate, 617–618 definition of, 617 range of, 555 Secant line, 236–237 I12 INDEX Second-degree equation See Quadratic equations Second-degree polynomials, 52–53 Seconds, 505–506 Seed, 746 Sequences, 937–965 amortization, 945–946 annuity problems, 944–945 arithmetic, 950–954 common difference in, 950 defined, 950 determining, 950–951 formula for, 951 nth term of, 951 recursive formula for, 952 sum of, 952–954 defined, 937 factorial symbol, 939–940 Fibonacci, 941, 948–949 geometric, 956–959 common ratio of, 956 defined, 956 determining, 956–957 formula for, 957–958 nth term of, 957–958 sum of, 958–959 graph of, 937, 938 historical feature on, 962 from a pattern, 939 properties of, 942 summation notation, 941 sum of, 942–944 terms of, 937–941 alternating, 939 defined by a recursive formula, 940–941 general, 938 Set(s), 2–3, 15 complement of, correspondence between two, 206 defined, 981 disjoint, elements of, 2, 981 empty (null), 2, 981 equal, 2, 981 finite, 981 infinite, 981 intersection of, 2–3 Mandelbrot, 746 of numbers, 4–5 subsets of, 981 proper, 981 union of, 2–3 universal, 3, 982 Set-builder notation, Set theory, 15 Setting the viewing rectangle, 83 Shannon’s diversity index, 447 Shifts, graphing functions using vertical and horizontal, 252–254, 258 Side-angle-side case of congruent triangle, 33 Side-angle-side case of similar triangle, 34–35 Sides of equation, 87, 98 of inequality, 19 Side-side-side case of congruent triangle, 33 Side-side-side case of similar triangle, 34 –35 Significant digits, 6–7 Signs, Rules of, 13 Similar triangles, 33–36 Simple harmonic motion, 699–703 amplitude of, 699, 700 analyzing, 701 circular motion and, 699–700 defined, 699 equilibrium (rest) position, 699 frequency of object in, 700 model for, 699–701 period of, 699, 700 Simple interest, 138, 466 Simple pendulum, 199 Simplex method, 923n Simplifying complex rational expressions, 68–70 expressions with rational exponents, 76–78 nth roots, 74 radicals, 74–75 rational expression, 62–63 Simpson’s rule, 314 Sine, 539–540, 553 defined, 517 historical feature on, 524 Law of in applied problems, 680–682 defined, 676 historical feature on, 689 proof of, 681–682 SAA or ASA triangles solved using, 676–677 SSA triangles solved using, 677–680 periodic properties of, 555, 556 Sum and Difference Formula for, 640–641 trigonometric equations linear in, 645–646 Sine function, 550 of best fit, 590 domain of, 554, 555, 561 graphs of, 560–575 amplitude and period, 564–566 equation for, 569–570 key points for, 566–569 hyperbolic, 435 inverse, 602–606 approximate value of, 604–605 defined, 603 exact value of, 603–604 exact value of expressions involving, 616 implicit form of, 603 properties of, 605 properties of, 561 range of, 554, 555, 561 Singular matrix, 888 Sinusoidal graphs, 560–575, 704 amplitude and period, 564–566 equation for, 569–570 key points for, 566–569 steps for, 586 Sinusoidal models, 586–590 best-fit, 590 daylight hours, 589–590 temperature data, 586–588 Six trigonometric functions of u, 539–541, 552–553 See also Trigonometric functions of t, 550–551 Slope, 172–175, 178–179 containing two points, 173 graphing lines given, 175 from linear equation, 178–179 of secant line, 236 Slope-intercept form of equation of line, 177–178 Smooth graph, 328 Snell, Willebrord, 629 Snell’s Law of Refraction, 629 Solution(s), 98 See also Zeros extraneous, 103, 129 of inequalities, 149–150 of linear programming problems, 925 location of, 925 repeated, 110, 332 of systems of equations, 842, 847–848 of trigonometric equations, 621 Solution set of equation, 98 Special products, 43–45, 47 Speed angular, 511–512 linear, 511–512 Sphere, volume and surface area of, 32 www.elsolucionario.net Spherical trigonometry, 711 Spheroid, prolate, 793 Spiral, 734 Square(s) of binomials (perfect squares), 44, 50 completing the, 56–57 difference of two, 43, 50–51 perfect, 44, 51–52 Square function, 244 Square matrix, 880 Square root(s), 24, 73 complex, 743 of negative numbers, 125 principal, 24, 125 Square root function, 241, 244 Square Root Method, 111 SSA triangles, 677–680 SSS triangles, 676, 688, 694 Standard deviation, 156 Standard form complex numbers in, 120, 122–124 of equation of circle, 188–189 polynomials in, 41 quadratic equations on, 109 Standard position, angle in, 503–504 Statements, writing using symbols, Static equilibrium, 755–756 Statute (ordinary) miles, 515 Step function, 246 Stevin, Simon, 15 Stirling’s formula, 975 Stock valuation, 277 Straight angle, 504 Stretches, graphing functions using, 254–256, 258 Strict inequalities, 19 Subscripted letters, 938 Subsets, 2, 981 proper, 981 Substitution, principle of, 10 Substitution method, 840, 843–844 systems of nonlinear equations solved using, 904–905 Subtraction, 7, See also Difference(s) of complex numbers, 121 horizontal, 42 in order of operations, of polynomials, 41–42 of quotients, 13–14 of rational expressions, 64–66 least common multiple (LCM) method for, 66–68 of vectors, 751–752 vertical, 42 Sum, 7, See also Addition of arithmetic sequences, 952–954 of complex numbers, 121 of geometric sequences, 958–959 index of, 941 of infinite geometric series, 960 of logarithms, 452 of sequences, 942–944 of two cubes, 44–45, 50 of two functions, 214–216 graph of, 703–704 of two matrices, 881–882 Sum and Difference Formulas, 638–650 for cosines, 638–639 defined, 638 to establish identities, 640–643 to find exact values, 639, 641–642 involving inverse trigonometric function, 644–645 for sines, 640–641 for tangents, 643 Sum function, 214–216 Summation notation, 941 Sum-to-Product Formulas, 661–662 Sun, declination of, 613–614 INDEX Surface area, formulas for, 32 Sylvester, James J., 893 Symbols, writing statements using, Symmetric property, 10 Symmetry, 165–167 axis of of parabola, 296 of quadratic function, 298 axis of, of parabola, 774 of polar equations, 728–729 with respect to origin, 165–166 with respect to the line u = p/2 (y-axis), 728–729 with respect to the polar axis (x-axis), 728–729 with respect to the pole (origin), 728–729 with respect to the x-axis, 165–166 with respect to the y-axis, 165–166 Synthetic division, 59–62 Systems of equations consistent, 842, 847 dependent, 843 containing three variables, 850–851 containing two variables, 846–847 Cramer’s Rule with, 876 equivalent, 844–845 graphing, 843 inconsistent, 842, 847, 849 containing three variables, 849 containing two variables, 846 Cramer’s Rule with, 876 independent, 843 solutions of, 842, 847–848 Systems of inequalities, 914–922 graphing, 917–920 bounded and unbounded graphs, 920 vertices or corner points, 920 Systems of linear equations, 841–879 consistent, 843, 847 defined, 842–843 dependent, 843 containing three variables, 850–851 containing two variables, 846–847 matrices to solve, 862–864 determinants, 870–879 cofactors, 874 Cramer’s Rule to solve a system of three equations containing three variables, 875–876 Cramer’s Rule to solve a system of two equations containing two variables, 871–873 minors of, 873–874 properties of, 876–877 by 3, 873–875 by 2, 870, 876–877 elimination method of solving, 844–847 equivalent, 844–845 examples of, 841–842 graphing, 843 inconsistent, 843, 847, 849 containing three variables, 849 containing two variables, 846 matrices to solve, 864 independent, 843 matrices See Matrix/matrices partial fraction decomposition, 896–903 defined, 897 where denominator has a nonrepeated irreducible quadratic factor, 901–902 where denominator has only nonrepeated linear factors, 897–898 where denominator has repeated irreducible quadratic factors, 902 where denominator has repeated linear factors, 899–900 solution of, 842, 847–848 solving, 842 substitution method of, 843–844 three equations containing three variables, 847–849 Systems of nonlinear equations, 904–913 elimination method for solving, 905–907 historical feature on, 910 substitution method for solving, 904–905 Systems of nonlinear inequalities, graphing, 919 Tables, graphing utility to create, 92 Tangency, point of, 194 Tangent(s), 539–540, 553 defined, 517 graph of, 575–578 Half-angle Formulas for, 656 historical feature on, 524 Law of, 686, 689 periodic properties of, 556 Sum and Difference Formulas for, 643 Tangent function, 551 domain of, 554, 555, 577 inverse, 609–611 defined, 610 exact value of, 610–611 exact value of expressions involving, 616 implicit form of, 610 properties of, 577 range of, 555, 577 Tangent line, 194 Greek method for finding, 194 Tartaglia (Niccolo of Brescia), 356, 745 Tautochrone, 831, 832 Terminal point of directed line segment, 747 Terminal side of angle, 503 Terms like, 40 of polynomial, 40–41 of sequences, 937–941 alternating, 939 defined by a recursive formula, 940–941 general, 938 by determinants, 873–875 Thrust, 771 TI-84 Plus, 90 Time, as parameter, 826–829 Transformations, 252–264, 778, 789, 802 combining, 254, 258–260 compressions and stretches, 254–256, 258 cosecant and secant graphs using, 580 of cosine function, 563–564 defined, 252 graphs using of exponential functions, 426–427, 428 of polynomial functions, 330–331 of quadratic functions, 296–298 of rational functions, 366 reflections about the x-axis or y-axis, 256–257 of sine function, 561–563 vertical and horizontal shifts, 252–254, 258 Transition matrix, 934–935 Transitive property, 10 Transverse axis, 795, 816 Tree diagram, 984 Triangle(s) See also Law of Sines area of, 32, 692–698 ASA, 676–677 congruent, 36 equilateral, 96 error, 97 isosceles, 96 legs of, 30, 516 medians of, 96 oblique, 675–676 Pascal, 971, 974 perfect, 697 right, 30, 516, 670–672 applied problems involving, 671–672 solving, 670–671 SAA, 676–677 SAS, 676, 687–688, 693–694 www.elsolucionario.net I13 similar, 33–36 SSA, 677–680 SSS, 676, 688, 694 Triangular addition, 973 Triangular numbers, 949 Trigonometric equations, 621–630 calculator for solving, 624 graphing utility to solve, 626 identities to solve, 625–626 involving single trigonometric function, 621–624 linear, 622–623 linear in sine and cosine, 645–646 quadratic in form, 624–625 solutions of, defined, 621 Trigonometric expressions, written algebraically, 618–619, 644 Trigonometric functions of acute angles, 516–518, 527–539, 670 calculator to approximate values of, 530–531 exact values of p/4 = 45°, 528–529 exact values of p/6 = 30° and p/3 = 60°, 529–530 of any angle, 539–549 coterminal angles to find exact values of, 541–543 exact values of, 539–541 reference angle of, 544–545 signs of, in a given quadrant, 543–544 applications of, 669–712 damped motion, 702–703 graphing sum of two functions, 703–704 involving right triangles, 671–672 Law of Cosines, 688–689 Law of Sines, 687 Law of Tangents, 686, 689 simple harmonic motion, 699–703 cosecant and secant graphs, 579–580 fundamental identities of, 518–520 quotient, 518 reciprocal, 518 historical feature on, 512 phase shift, 582–586 to graphy = A sin 1vx - f2 + B, 582–586 properties of, 554–557 domain and range, 554–555 even-odd, 557 periodic, 555–556 right triangle trigonometry, 516–527, 670–675 Complementary Angle Theorem, 522–523 finding values of trigonometric functions when one is known, 520–522 modeling and solving applied problems involving, 531–534 sine and cosine graphs, 560–575 amplitude and period, 564–566, 583–586 equation for, 569–570 key points for, 566–569 sinusoidal curve fitting, 586–590 of t, 550–551 tangent and cotangent graphs, 575–579 of u, 539–541, 552–553 unit circle approach to finding exact values of, 550–553 Trigonometric identities, 630–637 basic, 631 establishing, 632–635 Double-angle Formulas for, 651–654 Sum and Difference Formulas for, 640–643 Even-Odd, 631 Pythagorean, 631 Quotient, 631 Reciprocal, 631 Trinomials, 40 factoring, 52–53, 55 Truncation, Turning points, 333–334 I14 INDEX by determinants, 870 proof for, 876–877 Umbra versa, 524 Unbounded graphs, 920 Unbounded in positive direction, 365 Unbounded in the negative direction, polynomial functions, 335 Uniform motion, 140–141 Uninhibited growth, 475–477 Union of sets, 2–3 of two events, probabilities of, 999–1000 Unit circle, 189, 549–553 Unit vector, 749, 752–753 Universal sets, 3, 982 Value (image) of function, 207, 209–212 Variable(s), 20, 40 complex, 359 dependent, 210 domain of, 21–22 independent, 210 in calculus, 560 Variable costs, 185 Variation, 195–200 combined, 197–198 direct, 196, 197, 198 inverse, 196–197, 198 joint, 197–198 Vector(s), 747–760 adding, 748, 751–752 algebraic, 750 angle between, 761–762 column, 884 components of, 750, 751 horizontal and vertical, 751 decomposing, 763–764 defined, 747 difference of, 748 direction of, 747, 752–754 dot product of two, 760–761 equality of, 747, 751 finding, 753–754 force, 753 geometric, 747–749 graphing, 749 historical feature on, 756 magnitudes of, 747, 749, 752, 753–754 modeling with, 754–756 multiplying by numbers geometrically, 748–749 objects in static equilibrium, 755–756 orthogonal, 762–764 parallel, 762 in physics, 747 position, 750 row, 884 scalar multiples of, 748–749, 751, 760 subtracting, 751–752 unit, 749, 752–753 velocity, 753–755 writing, 754 zero, 747 Vector product See Cross (vector) product Vector projection, 763–764 Velocity vector, 753–755 Venn diagrams, Verbal descriptions linear models from, 282–284 quadratic models from, 307–311 Vertex/vertices, 920 of cone, 773 of ellipse, 784 of hyperbola, 795 of parabola, 296, 774 of quadratic function, 298–302 of ray, 503 Vertical asymptote, 367–369 Vertical component of vector, 751 Vertical line, 172, 173, 725–726, 734 equation of, 176 Vertical-line test, 220 Vertically compressed or stretched graphs, 254256 Vertical shifts, 252254, 258 Viốte, Franỗois, 116, 689 Viewing angle, 614 Viewing rectangle, 83 Vinculum, 78 Volume, formulas for, 32 Wallis, John, 745 Waves See also Simple harmonic motion Waves, traveling speeds of, 629 Weight, 771 Whispering galleries, 791 Whole numbers, 4, Wings, airplane, 713, 771 www.elsolucionario.net Work, dot product to compute, 764–765 World population, 936 x-axis, 82 projection of P on the, 699–700 reflections about, 256–257 symmetry with respect to, 165–166, 728–729 x-coordinate, 82 x-intercept, 92 of quadratic function, 299 xy-plane, 82 Yang Hui, 974 y-axis, 82 projection of P on the, 700 reflections about, 256–257 symmetry with respect to, 165–166, 728–729 y-coordinate (ordinate), 82 y-intercept, 92, 178–179 from linear equation, 178–179 Zero, 18 multiplication by, 12 significant digits and, Zero-coupon bonds, 474 Zero-level earthquake, 449 Zero matrix, 882 Zero polynomial, 41 Zero-Product Property, 13 Zeros bounds on, 353–355 complex, of polynomials, 359–363 Conjugate Pairs Theorem, 360–361 defined, 359 finding, 362–363 polynomial function with specified zeros, 361–362 real, of polynomials, 331–333, 346–359 finding, 350–352 Intermediate Value Theorem, 355–356 number of, 349 Rational Zeros Theorem, 349–350, 362 Remainder Theorem and Factor Theorem, 347–349 repeated, 332 theorem for bounds on, 353–355 Zero vector, 747 CONICS Parabola D: x = –a y V D: x = a y y F = (–a, 0) V F = (a, 0) x y F = (0, a) V V x x x D: y = –a y2 ϭ 4ax Ellipse y2 ϭ Ϫ4ax x2 ϭ Ϫ4ay y V = (0, a) F = (0, c) (0, b) V2 = (a, 0) x F2 = (c, 0) (0, –b) F1 = (–c, 0) (b, 0) x (– b, 0) F = (0, – c) V = (0, – a) y2 x2 + = 1, a b, c = a - b2 a b2 y2 x2 + = 1, a b, c = a - b2 b2 a y Hyperbola y V = (– a, 0) F = (– c, 0) F = (0, –a) x2 ϭ 4ay y V1 = (–a, 0) D: y = a F = (0, c ) V = (a, 0) F = (c, 0) x V = (0, a) x V = (0, –a ) F = (0, –c ) y2 x2 = 1, c = a + b2 a b2 b b Asymptotes: y = x, y = - x a a y2 x2 = 1, c = a + b2 a b2 a a Asymptotes: y = x, y = - x b b - PROPERTIES OF LOGARITHMS BINOMIAL THEOREM log a 1MN2 = log a M + log a N n n 1a + b2 n = a n + a b ba n - + a b b2a n - 2 log a a M b = log a M - log a N N + g+ a r log a M = r log a M log a M = log M ln M = log a ln a a x = e x ln a PERMUTATIONS/COMBINATIONS n b b n - 1a + b n n -1 ARITHMETIC SEQUENCE a + 1a + d2 + 1a + 2d2 + g+ a + 1n - 12d n n = 2a + 1n - 12d = a + a n 2 0! = 1! = n! = n 1n - 12 # p # 132 122 112 n! P1n, r2 = 1n - r2! GEOMETRIC SEQUENCE n n! C 1n, r2 = a b = 1n - r2! r! r If r 1, a + a 1r + a 1r + g = a a 1r k - a + a 1r + a 1r + g + a 1r n - = a # GEOMETRIC SERIES - rn 1-r ϱ k=1 = www.elsolucionario.net a1 1-r LIBRARY OF FUNCTIONS Identity Function f1x2 = x Cube Function f 1x2 = x3 Square Function f1x2 = x2 y y y (1, 1) (1, 1) (0, 0) –3 (–1, –1) (2, 4) ( – 2, 4) (– 1, 1) 3x (1, 1) 4x (0, 0) –4 (0, 0) x –4 (– 1, – 1) –4 Square Root Function Reciprocal Function f1x2 = x f1x2 = 1x Cube Root Function f 1x2 = x y y y (1, 1) 1– (2, ) (4, 2) x (0, 0) –1 –2 (Ϫ2,Ϫ ) Ϫ3 –2 Exponential Function f1x2 = e x y (2, e (2, 2) (1, 1) y 2) (e, 1) 3x (0, 0) –3 (–1, 1–e ) x (0, 0) (Ϫ1, Ϫ1) Natural Logarithm Function f 1x2 = ln x y (–2, 2) (–1, 1) ( 1–8 , 1–2) Ϫ3 2x (– 1, – 1) Absolute Value Function f1x2 = x (1, 1) (Ϫ 1–8,Ϫ 1–2) (1, 1) (2, ) (1, 0) x ( 1–e , Ϫ1) (1, e) (0, 1) x Sine Function f1x2 = sin x Cosine Function f1x2 = cos x y Ϫ – Ϫ y 3–– – Ϫ1 x 2 5–– Cosecant Function f1x2 = csc x Ϫ – 3 Ϫ––– y Ϫ1 Ϫ Ϫ –Ϫ1 2 – 3–– 2 5–– y x Secant Function f1x2 = sec x Ϫ 3–– Ϫ –– Ϫ 5–– 2Ϫ1 2 x 3 Ϫ ––– Ϫ –– 2Ϫ1 – 3–– 2 5 –– Cotangent Function f 1x2 = cot x y 3 ––– – Tangent Function f1x2 = tan x y – 3 –– x www.elsolucionario.net 3 Ϫ ––– Ϫ – Ϫ1 – 2 3 ––– 5 ––– x x ... Congress Cataloging-in-Publication Data Sullivan, Michael, 194 2Algebra & trigonometry : enhanced with graphing utilities/ Michael Sullivan, Michael Sullivan, III ? ?6th ed p cm Includes bibliographical... 978-0-321-78483-4 (alk paper) Algebra? ??Textbooks Trigonometry? ??Textbooks Algebra? ??Graphic methods Trigonometry? ??Graphic methods I Sullivan, Michael, 1967 July 2- II Title III Title: Algebra and trigonometry QA154.3.S75... representative NEW! Resources for Sullivan/ Sullivan, Algebra & Trigonometry Enhanced with Graphing Utilities, 6e • Author Solves It videos feature author Michael Sullivan, III, solving MathXL® exercises