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This page intentionally left blank This page intentionally left blank cob19545_fm_i-xxxviii.indd Page i 22/12/10 4:21 PM user-f467 /Volume/204/MHDQ268/wea25324_disk1of1/0073525324/wea25324_pagefiles Graphs and Models John W Coburn St Louis Community College at Florissant Valley J.D Herdlick St Louis Community College at Meramec-Kirkwood cob19545_fm_i-xxxviii.indd Page ii 22/12/10 4:21 PM user-f467 /Volume/204/MHDQ268/wea25324_disk1of1/0073525324/wea25324_pagefiles TM COLLEGE ALGEBRA: GRAPHS AND MODELS Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2012 by The McGraw-Hill Companies, Inc All rights reserved No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper DOW/DOW ISBN 978–0–07–351954–8 MHID 0–07–351954–5 ISBN 978–0–07–723057–9 (Annotated Instructor’s Edition) MHID 0–07–723057–4 Vice President, Editor-in-Chief: Marty Lange Vice President, EDP: Kimberly Meriwether David Senior Director of Development: Kristine Tibbetts Editorial Director: Stewart K Mattson Sponsoring Editor: John R Osgood Developmental Editor: Eve L Lipton Marketing Manager: Kevin M Ernzen Senior Project Manager: Vicki Krug Buyer II: Sherry L Kane Senior Media Project Manager: Sandra M Schnee Senior Designer: Laurie B Janssen Cover Image: © Georgette Douwma and Sami Sarkis / Gettyimages Senior Photo Research Coordinator: John C Leland Compositor: Aptara, Inc Typeface: 10.5/12 Times Roman Printer: R R Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Coburn, John W College algebra : graphs and models / John W Coburn, J.D Herdlick p cm Includes index ISBN 978–0–07–351954–8 — ISBN 0–07–351954–5 (hard copy : alk paper) Algebra— Textbooks Algebra—Graphic methods—Textbooks I Herdlick, John D II Title QA154.3.C5953 2012 512.9—dc22 2010035347 www.mhhe.com cob19545_fm_i-xxxviii.indd Page iii 22/12/10 4:21 PM user-f467 /Volume/204/MHDQ268/wea25324_disk1of1/0073525324/wea25324_pagefiles Brief Contents Preface vi Index of Applications CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER CHAPTER R xxxii A Review of Basic Concepts and Skills Relations, Functions, and Graphs 85 More on Functions 187 Quadratic Functions and Operations on Functions 281 Polynomial and Rational Functions 381 Exponential and Logarithmic Functions 479 Systems of Equations and Inequalities 575 Matrices and Matrix Applications 637 Analytic Geometry and the Conic Sections 707 Additional Topics in Algebra 761 Appendix I The Language, Notation, and Numbers of Mathematics Appendix II Geometry Review with Unit Conversions Appendix III More on Synthetic Division Appendix IV More on Matrices A-30 Appendix V Deriving the Equation of a Conic Appendix VI Proof Positive—A Selection of Proofs from College Algebra A-14 A-28 A-32 Student Answer Appendix (SE only) A-34 SA-1 Instructor Answer Appendix (AIE only) Index A-1 IA-1 I-1 iii cob19545_fm_i-xxxviii.indd Page iv 22/12/10 4:21 PM user-f467 /Volume/204/MHDQ268/wea25324_disk1of1/0073525324/wea25324_pagefiles About the Authors John Coburn John Coburn grew up in the Hawaiian Islands, the seventh of sixteen children He received his Associate of Arts degree in 1977 from Windward Community College, where he graduated with honors In 1979 he earned a Bachelor’s Degree in Education from the University of Hawaii After working in the business world for a number of years, he returned to teaching, accepting a position in high school mathematics where he was recognized as Teacher of the Year (1987) Soon afterward, the decision was made to seek a Master's Degree, which he received two years later from the University of Oklahoma John is now a full professor at the Florissant Valley campus of St Louis Community College During his tenure there he has received numerous nominations as an outstanding teacher by the local chapter of Phi Theta Kappa, two nominations to Who’s Who Among America’s Teachers, and was recognized as Post Secondary Teacher of the Year in 2004 by the Mathematics Educators of Greater St Louis (MEGSL) He has made numerous presentations and local, state, and national conferences on a wide variety of topics and maintains memberships in several mathematics organizations Some of John’s other interests include body surfing, snorkeling, and beach combing whenever he gets the chance He is also an avid gamer, enjoying numerous board, card, and party games His other loves include his family, music, athletics, composition, and the wild outdoors J.D Herdlick J.D Herdlick was born and raised in St Louis, Missouri, very near the Mississippi river In 1992, he received his bachelor’s degree in mathematics from Santa Clara University (Santa Clara, California) After completing his master’s in mathematics at Washington University (St Louis, Missouri) in 1994, he felt called to serve as both a campus minister and an aid worker for a number of years in the United States and Honduras He later returned to education and spent one year teaching high school mathematics, followed by an appointment at Washington University as visiting lecturer, a position he held until 2006 Simultaneously teaching as an adjunct professor at the Meramec campus of St Louis Community College, he eventually joined the department full time in 2001 While at Santa Clara University, he became a member of the honorary societies Phi Beta Kappa, Pi Mu Epsilon, and Sigma Xi under the tutelage of David Logothetti, Gerald Alexanderson, and Paul Halmos In addition to the Dean’s Award for Teaching Excellence at Washington University, J.D has received numerous awards and accolades for his teaching at St Louis Community College Outside of the office and classroom, he is likely to be found in the water, on the water, and sometimes above the water, as a passionate wakeboarder and kiteboarder It is here, in the water and wind, that he finds his inspiration for writing J.D and his family currently split their time between the United States and Argentina Dedication With boundless gratitude, we dedicate this work to the special people in our lives To our children, whom we hope were joyfully oblivious to the time, sacrifice, and perseverance required; and to our wives, who were well acquainted with every minute of it iv cob19545_fm_i-xxxviii.indd Page v 1/7/11 6:06 PM user-f468 /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles About the Cover Most coral reefs in the world are 7000–9000 years old, but new reefs can fully develop in as few as 20 years In addition to being home to over 4000 species of tropical or reef fish, coral reefs are immensely beneficial to humans and must be carefully preserved They buffer coastal regions from strong waves and storms, provide millions of people with food and jobs, and prompt advances in modern medicine Similar to the ancient reefs, a course in College Algebra is based on thousands of years of mathematical curiosity, insight, and wisdom In this one short course, we study a wealth of important concepts that have taken centuries to mature Just as the variety of fish in the sea rely on the coral reefs to survive, students in a College Algebra course rely on mastery of this bedrock of concepts to successfully pursue more advanced courses, as well as their career goals From the Authors nges From the ion has seen some enormo us cha cat edu tics ma the ma s, ade dec In the last two to online homework and and the adv ent of the Intern et, ors ulat calc ng phi gra of n ctio elen ting intr odu s ago, the cha nges hav e been unr ade dec ut abo am dre only ld cou visual sup plement s we nce tea ching re a combined 40 yea rs of exp erie sha k dlic Her J.D and urn Cob n of Tog eth er, Joh gies, and hav e dev elop ed a wea lth nolo tech er oth and ors ulat calc ng college alg ebr a wit h gra phi endeav or firs tha nd exp erience related to the ver sat iona l style and Models text, we hav e combined the and phs Gra ck rdli /He urn Cob one of In the , wit h this dep th of exp erience As for wn kno are s text our t tha s the wea lth of application y see functions think visually, to a poin t where the ts den stu help to out set we ls, our primary goa ediately lead to a of gra phs, wit h attr ibutes that imm ily fam a of one as 4x – x = , the nat ure of like f(x) ior, zer oes, solu tions to ineq ualities hav -be end ms, imu and ums discussion of ma xim an equation that es in text — instead of mer ely ibut attr se the of tion lica app the le the roots, and the scr een of a calculat or And whi on ph gra a g etin rpr inte by or off ers much must be solv ed by factor ing nal drudgery, we believe our text atio put com e som eve reli y ma ors gra phing calculat gra phical met hods, wit h ison of algebra ic met hods ver sus par com e -sid -by side ple sim a nua lly n more tha checking answer s to wor k don e ma ply sim n tha role ant ific sign re mo the calculat or playing a sible wit h pap er and investigate far bey ond what’s pos and k wor to d use are ors ulat Gra phing calc age more applications, and e more tru e-to-lif e equations, eng solv to d use gy nolo tech the h wit text is built on pencil, the end we believe you’ll see this In t res inte of ns stio que l ntia explore more substa accent uates the visual and dynamic excursion that a ers off t tha one yet als, ent strong fundam use in all areas of their solv ing acumen that studen ts will blem pro and g nnin pla nal atio aniz l tool for the org Gra phs and Models text as an idea ck rdli /He urn Cob the er off we lives To this end —John Coburn and J.D Her dlick tics tea ching and lear ning of mathema v cob19545_fm_i-xxxviii.indd Page vi 1/7/11 6:02 PM user-f468 /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles Making Connections College Algebra tends to be a challenging course for many students They may not see the connections that College Algebra has to their life or why it is so critical that they succeed in this course Others may enter into this course underprepared or improperly placed and with very little motivation Instructors are faced with several challenges as well They are given the task of improving pass rates and student retention while ensuring the students are adequately prepared for more advanced courses, as a College Algebra course attracts a very diverse audience, with a wide variety of career goals and a large range of prerequisite skills The goal of this textbook series is to provide both students and instructors with tools to address these challenges, so that both can experience greater success in College Algebra For instance, the comprehensive exercise sets have a range of difficulty that provides very strong support for weaker students, while advanced students are challenged to reach even further The rest of this preface further explains the tools that John Coburn, J.D Herdlick, and McGraw-Hill have developed and how they can be used to connect students to College Algebra and connect instructors to their students The Coburn/Herdlick College Algebra Series provides you with strong tools to achieve better outcomes in your College Algebra course as follows: vi ▶ Making Connections Visually, Symbolically, Numerically, and Verbally ▶ Better Student Preparedness Through Superior Course Management ▶ Increased Student Engagement ▶ Solid Skill Development ▶ Strong Mathematical Connections cob19545_fm_i-xxxviii.indd Page vii 1/7/11 5:58 PM user-f468 ▶ /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles Making Connections Visually, Symbolically, Numerically, and Verbally In writing their Graphs and Models series, the Coburn/Herdlick team took great care to help students think visually by relating a basic graph to an algebraic equation at every opportunity This empowers students to see the “Why?” behind many algebraic rules and properties, and offers solid preparation for the connections they’ll need to make in future courses which often depend on these visual skills ▶ Better Student Preparedness Through Superior Course Management McGraw-Hill is proud to offer instructors a choice of course management options to accompany Coburn/ Herdlick If you prefer to assign text-specific problems in a brand new, robust online homework system that contains stepped out and guided solutions for all questions, Connect Math Hosted by ALEKS may be for you Or perhaps you prefer the diagnostic nature and artificial intelligence engine that is the driving force behind our ALEKS 360 Course product, a true online learning environment, which has been expanded to contain hundreds of new College Algebra & Precalculus topics We encourage you to take a closer look at each product on preface pages x through xiii and to consult your McGraw-Hill sales representative to setup a demonstration ▶ Increased Student Engagement There are many texts that claim they “engage” students, but only the Coburn Series has carefully studied and implemented features and options that make it truly possible From the on-line support, to the textbook design and a wealth of quality applications, students will remain engaged throughout their studies ▶ Solid Skill Development The Coburn/Herdlick series intentionally relates the examples to the exercise sets so there is a strong connection between what students are learning while working through the examples in each section and the homework exercises that they complete This development of strong mechanical skills is followed closely by a careful development of problem solving skills, with the use of interesting and engaging applications that have been carefully chosen with regard to difficulty and the skills currently under study There is also an abundance of exercise types to choose from to ensure that homework challenges a wide variety of skills Furthermore, John and J.D reconnect students to earlier chapter material with Mid-Chapter Checks; students have praised these exercises for helping them understand what key concepts require additional practice ▶ Strong Mathematical Connections John Coburn and J.D Herdlick’s experience in the classroom and their strong connections to how students comprehend the material are evident in their writing style This is demonstrated by the way they provide a tight weave from topic to topic and foster an environment that doesn’t just focus on procedures but illustrates the big picture, which is something that so often is sacrificed in this course Moreover, they employ a clear and supportive writing style, providing the students with a tool they can depend on when the teacher is not available, when they miss a day of class, or simply when working on their own vii cob19545_ndx_I1-I16.qxd I-2 12/23/10 9:13 PM Page I-2 Index Boundary lines—Cont linear inequalities and, 615–616 vertical, 122–123, 134 Boundary point, 122 Bounded region, 621 Branches, of hyperbola, 731 Break-even analysis, 584, 610 C Calculators See Graphing calculators Capacity, A–18 – A–19 Cardano, Girolomo, 291 Carrying capacity, 555 Center of circle, 96–98 of hyperbola, 732, 733 Centimeters of mercury, 510 Central circle, 96, 718 Central ellipse, 718 Central hyperbola, 731–732, A–33 Central rectangle, 733 Change in x, 107 Change in y, 107 Change-of-base formula, 522–523 Circles center of, 96–98 central, 96, 718 circumference of, 176, 261, 709, A–15 equation of, 96–97, 715–716, 755 explanation of, 33, 96, 710, 755 on graphing calculator, 99–100, 715 graphs of, 95–99, 715–716 perimeter and area formulas for, 33, A–14 properties of, 714 radius of, 96–98, 103, 491, 710, 714 standard form of, 96 Circumference, 176, 261, 709, A–15 Clark’s rule, A–12 Coefficient matrices, 638–639, 700 Coefficients explanation of, leading, 17, 41 solving linear equations with fractional, 26 Coincident dependence, 598 Collinear points test, 688 Combinations explanation of, 809–810 on graphing calculator, 810 stating probability using, 820 Combined variation, 266 Common difference, for sequence, 773 Common logarithms explanation of, 504 method to find, 505–506 Common ratio, 782 Commutative properties, 5, 81 Complementary events, 818–819 Complements, 818–820 Completing the square applications of, 730–731, 759 explanation of, 104, 297 to graph circles, 715, 716 to graph ellipses, 719–720, 722–723 to graph hyperbolas, 735–736, 738 to graph parabolas, 746 to solve quadratic equations, 297–299 to solve quadratic functions, 313–315 Complex conjugates theorem division and, 288 explanation of, 286, 396 product of, 286 proof of, A–34 – A–35 Complex numbers absolute value of, 291, 408 addition and subtraction of, 284–285 division of, 288 explanation of, 283, 370–371 on graphing calculator, 285, 286 identifying and simplifying, 282–284 multiplication of, 285–287 raised to power, 831 square root of, 408 standard form of, 283–284 Complex polynomial functions, 394 Complex polynomials, 312 Complex zeroes, 475–476 Composite figures explanation of, A–15 – A–16 volume of, A–17 – A–18 Composite functions, 356–357 Composition applications, 362–363 Composition of functions explanation of, 352–354, 374 on graphing calculator, 355, 358–359 method to find, 354–356 notation for, 353 numerical and graphical view of, 357–359 Compounded interest formula, 541 Compound fractions, 58–59 Compound inequalities explanation of, 29 method to solve, 30–31 Compound interest explanation of, 540–542, 569 on graphing calculator, 542 Compressions, 210–211 Conditional equations, 27 Cones explanation of, 710 surface area of, 80 volume of, 35 Conical shells, 52 Conic sections See also Circles; Ellipses; Hyperbolas; Parabolas; specific conic sections applications of foci of, 724–725, 739–740 characteristics of, 710–712 equations of, 736–737, A–32 – A33 explanation of, 708 graphs of, 200 nonlinear systems and, 749–750 Conjugate axis, 732 Conjugates binomial, 19–20 complex, 286, 288, 396 Consecutive integers, 158 Constant of variation, 261 Constant terms, Constraints, 620 Continuous functions explanation of, 137 piecewise and, 247, 251 Continuous graphs, 91 Continuously compounded interest, 541–542, 545, 569 Contradictions, 27 Convenient values method, 685–687 Convex region, 621 Coordinate, A–2 Coordinate grid, 89 Correlation, strong and week, 167–169 Cost-based pricing, 584 Counting techniques combinations, 809–811 distinguishable permutations, 807–808 fundamental principle of, 806–807 listing and tree diagrams, 804–806 nondistinguishable permutations, 809 review of, 840–841 Cramer’s Rule, 680–682, 701, 704 Cube root function, 205 Cube roots, 66, A–8 See also Radical expressions Cubes binomial, 291 explanation of, 35, A–21 perfect, 44–45, 394, A–7 sum of cubes of n natural numbers, 793 sum or difference of two perfect, 44–45 volume of, A–17 Cubic units, A–16 Cubing function, 205 Cylinders, 52, 162, 176, 311, 349, 455 Cylindrical shells, 52 D Data analysis, applications of, 170 Decay rate, 545–547 Decimal notation, 15–16 Decimals, A–2, A–3 Decision variables, 622 Decomposition of composite function, 356–357 for rational expressions, 682, 685–688 of rational terms, Decomposition template, 682–685 Degenerate cases, 104 Degree, of polynomials, 16 Delta (⌬), 107 Denominator, rationalizing the, 71 Dependent system of equations matrices and, 644–645 in three variables, 596–598 in two variables, 581–582 Dependent variable, 88 Depreciation, 498 Descartes’ rule of signs, 402–403 Descriptive variables, Determinants to find area of triangle, 688 of general matrices, A–31 of singular matrices, 668–672 to solve systems, 679–682 ϫ 3, 670–671 ϫ 2, 701 Diagonal entries, of matrices, 638 Difference of two squares, factoring, 43 Difference quotient applications of, 362–364 cob19545_ndx_I1-I16.qxd 1/7/11 9:46 PM Page I-3 Index average rate of change and, 359–361 explanation of, 337, 359, 374 Directrix of ellipse, 711, 712 of hyperbola, 712 of parabola, 710, 711, 746–749 Direct variation applications of, 262–264 explanation of, 261, 277 Discontinuities asymptotic, 431 removable, 446–447 Discontinuous functions, piecewise-defined, 252, 253 Discriminant of cubic equation, 467 explanation of, 301 of quadratic formula, 301–302 use of, 312 Diseconomies of scale, 610 Distance absolute value and, 231 average, 95 perpendicular, 601, 709 Distance formula, 95–96, 708, 755 Distinguishable permutations, 807–808 Distributive property of multiplication over addition, 6–7, 285, 768 Division of complex numbers, 288 factor theorem and, 387–389 on graphing calculator, A–7 long, 382 with nonlinear divisor, 385 of polynomials, 383, 384 of radical expressions, 71–72 of rational expressions, 56 remainder theorem and, 386–387, 390 synthetic, 383–384, 470, A–28 – A–29 with zero, 384–385, A–6 Division algorithm, 383 Divisor, 385 Domain of functions, 123–125, 128 implied, 124–125 of logarithmic functions, 507–508 of piecewise-defined functions, 248–249 of power function, 237–238 of rational functions, 431 of relations, 88, 134–135 solving quadratic inequality to determine, 305–306 Dominant term, of polynomial function, 413 E e, 495–496 Eccentricity, 758–759 Economies of scale, 610 Electrical resistance, 10 Elementary row operations, 640–641 Elements of set, A–1 Elimination explanation of, 579, 580, 634–635 Gaussian, 40, 642, A–30 Gauss-Jordan, 642–645, A–30 to solve system of linear equations in three variables, 593–596 to solve system of linear equations in two variables, 579–581 to solve system of nonlinear equations, 607–608 Ellipses applications using characteristics of, 724–725 area of, 727 central, 718 characteristics of, 711–712, 724–725 definition of, 720, 721 equation of, 716–717, 720–721, 723–724, 755, A–32 focal chord, 723 foci of, 720–725 graphs of, 717–724 horizontal, 717–718 perimeter of, 727 with rational/irrational values, 759 vertical, 718–719 Elongation, 758–759 Empty set, A–1 End-behavior explanation of, 195 of polynomial graphs, 412–415, 419 of rational functions, 233, 234 Endpoint maximum, 196 Endpoints, 122, A–4 Equality additive property of, 25 of matrices, 650–651 multiplicative property of, 25 power property of, 72, 74 square root property of, 295–297 Equations See also Linear equations; System of linear equations; System of linear equations in three variables; System of linear equations in two variables; System of nonlinear equations; specific types of equations absolute value, 220–222, 226, 231 of circle, 96–97, 715–716, 755 conditional, 27 of conic sections, 736–737 of ellipse, 716–717, 720–721, 723–724, 755, A–32 equivalent, 25 explanation of, 25, 150 exponential, 496–499, 503–505, 517–519, 527, 531–533, 569 families of, 25 in function form, 137, 138 of functions, 236, 249, 253–254, 316, 438 of hyperbolas, 731–736, 756, A–32 – A–33 intersection-of-graphs method to solve, 150–153, 183, 519 of line, 138–141 literal, 155–156 logarithmic, 517–519, 527–530, 569 logistic, 533–534, 572 matrix, 663, 667–668 of parabola, 746–749, 756 parametric, 467 of piecewise-defined function, 249, 253–254 polynomial, 46–48, 312 present value, 540 quadratic, 46–48, 292–302, 371 of quadratic function, 316 radical, 72–76 rational, 59–60 of rational function, 236, 438 regression, 171–172, 552, 556–557, 570 relations stated as, 89 roots of, 25 of semi-hyperbola, 742 variation, 261, 263, 265 written information translated into, 31–32 Equivalent equations, 25 Equivalent system of equations, 579, 593–594 Euler, Leonhard, 37 Euler’s polyhedron formula, 37 Even functions, 190–191 Events complementary, 818–819 explanation of, 816 mutually exclusive, 822 nonexclusive, 821–823 probability of, 817, 818 Experiments, 805 Exponential decay, 545–547 Exponential equations explanation of, 569 logarithmic form and, 503–505 method to solve, 517–519, 531–533 uniqueness property to solve, 496–499 Exponential form, 504–505, 519, A–7 Exponential functions See also One-to-one functions applications of, 492, 498–499 base-e, 495–496 evaluation of, 492–493 explanation of, 492, 566–567 on graphing calculator, 496–499 graphs of, 493–495 natural, 496 Exponential growth, 495, 545–546 Exponential notation, 11, A–7 Exponential properties multiplying terms using, 11 simplifying expressions using, 14 summary of, 15 Exponential regression model, 553–554 Exponents explanation of, 11, A–7 power property of, 11–12, 81 product property of, 11, 12, 81 quotient property of, 13, 81 rational, 67–68 zero and negative, 13, 81 Extraneous roots, 60, 527 Extrapolation, 556–557, 570 Extreme values explanation of, 316 quadratic functions and, 316–321 F Factorial formulas, 814 Factorial notation, 765 Factorials, 764–765 I-3 cob19545_ndx_I1-I16.qxd I-4 1/7/11 9:46 PM Page I-4 Index Factoring chart of methods of, 46 difference of two squares, 43 explanation of, 39, 46, 82 by grouping, 40–41 nested, 53 perfect square trinomials, 43–44 polynomial equations, 46–49 quadratic forms, 45–46 quadratic polynomials, 41–43 sum or difference of two perfect cubes, 44–45 trial-and-error method for, 42–43 Factors greatest common, 39–40 Factor theorem explanation of, 387, 471 to find factors of polynomials, 387–389 finding zeroes using, 389 proof of, A–34 Families of curves, 711 of equations, 25 of functions, 204, 236 of polynomials, 16 Feasible region, 621 Finite sequences, 762, 838 Finite series, 765–766 Focal chord of ellipse, 723 explanation of, 723 of hyperbolas, 738, 743 of parabola, 748, 753 Foci applications of, 724–725 of ellipse, 711, 720–725 explanation of, 710 of hyperbola, 712, 737–740 of parabola, 710, 711, 746–749 Foci formula, 723, 738 Focus-directrix form of equation of parabola, 746–749 F-O-I-L method, 19, 41, 285, 399 Folium of Descartes, 467 Formulas area, 33, 37, 455, 491, 627, 660, 727, A–14 – A–16 average rate of change, 361 binomial coefficients, 831–832 binomial cubes, 291 binomial probability, 836 body mass index, 37 change-of-base, 522–523 Clark’s rule, A–12 compounded interest, 541 distance, 95–96, 708, 755 Euler’s polyhedron, 37 explanation of, 155 exponential growth, 495, 545 factorial, 814 foci, 723, 738 inverse of matrices, 704 lift capacity, 37 midpoint, 95, 708, 755 perimeter, 33, 660, 727, A–14 – A–16 perpendicular distance from point to line, 601, 709, 713, 755 Pick’s theorem, 132 pitch diameter, A–12 population density, 381, 441 Pythagorean Theorem, 75–76, 83 radius, 490 required interest rate, 269 right parabolic segment, 752 simple interest, 539 slope, 107, 108, 110, 138 Stirling’s formula, 814 sum of cubes of first n natural numbers, 793 sum of first n natural numbers, 781 sum of squares of first n natural numbers, 781 surface area of cylinder, 162 surface area of rectangular box with square ends, 323 velocity, 79, 80, 236 vertex, 315–316 vertex/intercept, 323 volume, 34, 218, 491, 627 Fractions compound, 58–59 partial, 682–688, 702 Function families, 204 Function form, of linear equations, 137, 138 Function notation, 125–127 Functions absolute value, 204, 225, 257 algebra of, 340–346, 373 applications of, 189, 345–346 base, 378 composite, 356–357 composition of, 352–359, 374 constant, 137 continuous, 247 cube root, 205 cubing, 205 domain and range of, 122–125, 128 evaluation of, 126–127 even, 190–191 explanation of, 119–120, 181–182 exponential, 492–499, 566–567 on graphing calculator, 127, 244–245 graphs of, 120–122, 127–128, 190–197, 204–214, 273, 343–345, 485–487 identification of, 120 identity, 137, 204 increasing or decreasing, 194–195 inverse, 481–487, 527, 566 linear, 138, 169–171 logarithmic, 504–508, 567 maximum and minimum value of, 196, 621–622 nonlinear, 331–332 objective, 620 odd, 191–192 one-on-one, 480–484, 486, 566 piecewise-defined, 247–255, 276 polynomial, 192–193, 394–405, 411–420 positive and negative, 192–193 power, 236–242, 275, 296 products and quotients of, 341–343 quadratic, 293–294, 313–321, 372 range of, 123–124 rational, 232–236, 239–242, 275, 430–439, 473 as relations, 119–122 root, 236, 238–239 sand dune, 257 smooth, 247 square root, 204 squaring, 204 step, 254–255 sums and differences of, 340–341 vertical line test for, 120–122 zeroes of, 192–194 Fundamental principle of counting (FPC), 806–807, 820 Fundamental properties of logarithms explanation of, 517 to solve equations, 516–519 Fundamental property of rational expressions, 53–54 Fundamental theorem of algebra, 394–397, 399 G Galileo Galilei, 236, 363 Gauss, Carl Friedrich, 394, 642 Gaussian elimination, 640, 642, A–30 Gauss-Jordan elimination, 642–645, A–30 General form, of equation of circle, 98 General functions, transformations of, 211–214 General linear equations, 147 Geometric sequences applications of, 789–791 explanation of, 782–783, 839 finding nth partial sum of, 787–788 finding nth term of, 783–787 sum of infinite, 788–789 Geometric series, 782 Geometry analytical, 708–712 perimeter and area formulas, A–14 – A–16 plane, 708–709 unit conversion factors, A–18 – A–21 verifying theorem from basic, 708 volume, A–16 – A–18 Global maximum, 196 Goodness of fit, 167, 376–377 Graphical solutions intersect method for, 150–152 for linear inequalities, 154–155 x-intercept/zeroes method for, 152–154 Graphing calculator features intersect, 150–151 QuadReg command on, 329, 330 repeat graph, 758 split screen viewing, 399 TABLE feature, 92–93, 402, 511–512, 544 window size, 603–604 Graphing calculators absolute value on, 221, 224, 225, A–6 asymptotes on, 240 circles on, 99–100, 715 combinations on, 810 complex numbers on, 285, 286 composition of functions on, 355, 358–359 compound interest on, 542 division on, A–7 elongation and eccentricity on, 758–759 evaluating expressions on, cob19545_ndx_I1-I16.qxd 12/23/10 9:13 PM Page I-5 Index exponential functions on, 496–499 functions on, 127, 344–345 hyperbolas on, 735, 736, 739 imaginary and complex numbers on, 285 intersection-of-graphs method on, 160, 519, 545–546 inverse functions on, 484–487 linear equations on, 114 linear inequalities on, 617, 619 linear programming on, 633–634 logarithms on, 505, 506, 508, 511–512, 520, 528–535 logistic equations on, 533, 534 matrices on, 643–645, 653, 656–658, 665, 667–669, 672–673 maximums and minimums on, 196–197, 317, 319–321 order of operations on, A–9 parallel lines on, 140 partial sums on, 766–767, 769 permutations on, 808 piecewise-defined functions on, 250 polynomial inequalities on, 460 power functions on, 237 probability on, 821, 823, 835 quadratic equations on, 298, 299, 301, 302 rational functions on, 234, 447, 452–454 regression on, 171–173, 178, 242, 376–377, 552, 554–557 relations on, 92–94 repeating decimals on, A–2 sequences on, 763–765, 775–776, 785–786, 789–790 summation on, 767, 769 system of linear equations on, 577, 578, 580–586, 599 system of nonlinear equations on, 605–608 transformations on, 213, 265 translations on, 206–208 variation on, 264, 279 x-intercept on, 153–154, 293–294, 296 zeroes method on, 154 Graphs of circles, 95–99 continuous, 91 of ellipses, 717–724 end-behavior of, 195, 413 of exponential functions, 493–498 of functions, 120–122, 127–128, 190–197, 204–214, 273, 343–345 of hyperbolas, 731–735, 737–740 of linear equations, 105–114, 136–142 of lines, 138–139 of logarithmic functions, 506–507 one-dimensional, 591 of parabolas, 91, 92, 193, 745–749 of piecewise-defined functions, 248–254 of polynomial functions, 411–420, 472 of power functions, 237–239 quadratic, 378 of quadratic functions, 313–321 of rational functions, 232–236, 435–438, 446–450, 472–473, 485–487 of reflections, 208–210 of relations, 89–94 of semicircles, 92 of sequences, 776, 786 to solve inequalities, 193, 194 symmetry and, 190–192 of system of linear equations, 577 of system of nonlinear equations, 604–605 transformations of, 211–214 of translations, 206, 207 two-dimension, 591 of variation, 262–265 vertically stretching/compressing basic, 210–211 Gravity, 236, 244, 269, 271, 363 Greater than symbol, A–4 Greatest common factors (GCF), 39–40 Grid lines, 89 Grouping, factoring by, 40–41 Grouping symbols, 341 Growth rate, 545 H Half planes, 615–616 Horizontal asymptotes, 234, 433–434 Horizontal boundary lines, 123–124, 134 Horizontal change, 107 Horizontal hyperbolas, 732 Horizontal lines, 109–110 Horizontal line test, 480, 486 Horizontal parabolas, 92, 745–747 Horizontal reflections, 209–210 Horizontal translations, 207–208 Hyperbolas applying properties of, 739, 740 branches of, 731 central, 731–732, A–33 directrix of, 712 equation of, 731–736, 756, A–32 – A–33 equation of semi-, 742 explanation of, 731, 737–738, 756 focal chord of, 738, 743 foci of, 712, 737–740 on graphing calculator, 735, 736, 739 graphs of, 731–735, 737–740 horizontal, 732 with rational/irrational values, 759 standard form of equation of, 733–735 vertical, 733, 735–736 I Identities additive, 5–6, 81 explanation of, 27 multiplicative, 5–6, 81 Identity function, 137, 204 Imaginary numbers calculations with, 285, 291 explanation of, 282 on graphing calculator, 285, 286 historical background of, 291 Imaginary unit (i), 282, 287 Implied domain, 124–125 Inclusion, of endpoint, 122 Inconsistent system of equations matrices and, 644–645 in three variables, 596–598 in two variables, 581, 582 Independent variable, 88 Index, A–8 Index of summation, 766, 838 I-5 Induction See Mathematical induction Induction hypothesis, 800 Inequalities See also Linear inequalities absolute value, 223–227, 231, 275 additive property of, 27–28 applications of, 32, 464–465 compound, 29–31 graphs to solve, 193, 194 interval tests to solve, 461–464 joint, 31 linear, 27–29, 154–155 multiplicative property of, 27–28 polynomial, 459–462, 474 push principle to solve, 476–477 quadratic, 303–308, 371 rational, 461–464, 474 writing mathematical models using, A–4 zeroes and, 475–476 Inequality symbols, A–4 Infinite geometric series, 788–789 Infinite sequences, 762, 838 Infinity symbol, 123 Input values, Integers, 81, A–2 Intercept method explanation of, 106, 136, 137 to graph hyperbolas, 713–732 Interest compound, 540–541, 569 continuously compounded, 541–542, 545, 569 simple, 539, 569 Interest rate, 269, 539, 540 Intermediate value theorem (IVT), 397–399 Interpolation, 556–557, 570 Intersection, 29 Intersection-of-graphs method on graphing calculator, 160, 519, 545–546 to solve equations, 150–154, 183, 519, 534 to solve linear inequalities, 154–155 use of, 160, 544 Interval notation, 122, 123 Intervals where function is increasing or decreasing, 194–195 where function is positive or negative, 192–194 Interval test method explanation of, 304, 305 to solve function inequalities, 461–464 to solve quadratic inequalities, 304–305 Inverse additive, 6, 81 of functions, 482, 566 graphs of function and its, 485–487 matrix, 665–667, 696–697, 704–705 multiplicative, 6, 81 Inverse functions algebraic method to find, 482–485 applications of, 487 explanation of, 481–482, 566 on graphing calculator, 484–487 graphs of, 485–487 use of, 527 Inverse variation, 264–265, 277 Irrational numbers, 81, A–3 cob19545_ndx_I1-I16.qxd I-6 1/7/11 10:09 PM Page I-6 Index J Joint inequality, 31 Joint variation explanation of, 266, 277 on graphing calculator, 279 L Latus rectum, 753 Leading coefficients, 17, 41 Least common denominator (LCD), 26, 56–58 Less than symbol, A–4 Lift capacity formula, 37 Like terms, Limited value, 788 Linear association, 166 Linear depreciation, 143 Linear equations See also Equations applications of, 31–32, 113–114, 142–144 explanation of, 105, 181 forms of, 186–187 in function form, 137, 138 general, 147 on graphing calculators, 114 graphs of, 105–106, 136–142 horizontal and vertical lines and, 109–111 intercept/intercept form of, 147 intercept method to graph, 106, 136, 137 methods to solve, 82, 136 in one variable, 25–26, 82 parallel and perpendicular lines and, 111–113 in point-slope form, 141–142 properties of equality to solve, 25–26 in slope-intercept form, 137, 138 slope of line and rates of change and, 106–109 standard form of, 46 in two variables, 105, 106 Linear factorization theorem explanation of, 395, 396 proof of, 8, A–35 Linear function models, 169–171, 184 Linear functions, 138 Linear inequalities See also Inequalities applications of, 615 explanation of, 27 on graphing calculator, 617, 619 intersection-of-graphs method to solve, 154–155 method to solve, 27–29 system of, 618–620 in two variables, 615–616, 618 Linear programming applications of, 622–625 explanation of, 620–621, 631–632 on graphing calculators, 633–634 solutions to problems in, 621–622 Linear regression, 171–173 Linear systems See System of linear equations; System of linear equations in three variables; System of linear equations in two variables Line of best fit, 171–173 Lines auxiliary, A–15 equations of, 138–141 horizontal, 109–110 parallel, 111–112 perpendicular, 112–113 slope-intercept form and graph of, 138–139 slope of, 106–108 vertical, 109–110 Literal equations, 155–156 Local maximum, 196 Logarithmic equations explanation of, 569 forms of, 532 on graphing calculator, 532–533 method to solve, 517–519, 527–530 system of, 608 Logarithmic form, 504–505, 519 Logarithmic functions domain of, 507–508 explanation of, 504, 567 graphs of, 506–507 Logarithmic regression model, 554–555 Logarithmic scales, 508–509 Logarithms applications for, 508–512, 534–535 base-10, 522 base-e, 522 change-of-base formula and, 522–523 common, 504–506 explanation of, 567 fundamental properties of, 516–519 on graphing calculator, 505, 506, 508, 511–512, 520, 528–535 natural, 504–506 product, quotient, and power properties of, 9, 519–521, 527, 568, A–36 uniqueness property of, 528–529 Logistic equations explanation of, 533 on graphing calculator, 533, 534 investigation of, 572 method to solve, 533–534 regression models and, 555–556 Logistic growth, 533–534, 536 Logistic growth model, 555 Long division, 382 Lorentz transformation, 52 Lower bound, 404 M Mapping notation, 88, 89, 354 Market equilibrium, 585–586 Mathematical induction applied to sums, 796–798 explanation of, 796, 840 general principle of, 799–800 to prove statement, 798–799 Mathematical models, 2–3, A–4 Matrices addition and subtraction of, 652–653, 700 applications of, 645–646, 692–697, 701, 702 associated minor, 669–670 augmented, 638–642, 704–705, A–30 coefficient, 638–639 determinants and, 668–672, A–31 equality of, 650–651 explanation of, 638, 700 on graphing calculator, 643–645, 665, 667–669, 672, 673 identity, 663–665 inconsistent and dependent systems and, 644–645 inverse of, 665–667, 696–697, 704–705 multiplication and, 653–658, 663, 700 in reduced row-echelon form, 394, 642 in reduced row-echelon form and, A–30 singular, 668–672 to solve system of equations, 640–643 square, 638, 664 in triangularized form, 640, 678 Matrix equations applications of, 672–673 explanation of, 663, 701 on graphing calculator, 672–673 to solve systems of equations, 667–668, 701 Matrix multiplication explanation of, 654–656 on graphing calculator, 656–658 identity matrices and, 663–665 product matrix and, 663 properties of, 657 Matrix of constraints, 638 Maximum values explanation of, 196 of functions, 196, 621–622 on graphing calculator, 196–197, 317, 319–321 Members of set, A–1 Message encryption, 694–697 Metric units, A–19 – A–20 Midinterval points, 418, 420 Midpoint formula, 95, 708, 755 Midpoint of line segment, 94–95 Minimum values explanation of, 196, 621 of functions, 196 on graphing calculator, 196–197, 317, 319, 320 Minors, 669–670 Mixture problems, 159–160, 582–583 Modeling mathematical, 2–3 step function for, 255 system of linear equations in two variables, 582–586 Monomials, 16, 18 See also Polynomials Multiplication associative property of, 5, 81 commutative property of, 5, 81 of complex numbers, 285–287 matrix, 653–658, 663–665, 700 of polynomials, 18–21 of radical expressions, 70–72 of rational expressions, 55 scalar, 653–654 using exponential properties, 11 Multiplicative identity, 5–6, 81 Multiplicative inverse, 6, 81 Multiplicative property of absolute value, 222 of equality, 25 of inequality, 27–28 cob19545_ndx_I1-I16.qxd 12/23/10 9:13 PM Page I-7 Index Multiplicity vertical asymptotes and, 432–433 zeroes of, 396 Mutually exclusive events, 822 N Nappe, 710 Natural exponential functions, 496 Natural logarithms explanation of, 504 method to find, 505–506 Natural numbers, 81, 781, A–1 Negative association, 165 Negative exponents, 13, 15, 81 Negative numbers, A–2 Negative slope, 108 Nested factoring, 53 Newton’s law of cooling, 498–499 Nondistinguishable permutations, 809 Nonexclusive events, 821–823 Nonlinear association, 166 Nonlinear asymptotes, 448–450 Nonlinear divisor, 385 Nonlinear functions, 331–332 Nonlinear systems applications of, 610–611 conic sections and, 749–750 of equations, 604–608, 631 of inequalities, 609–610, 631 Nonrepeating and nonterminating decimals, A–3 Notation/symbols for composition of functions, 353 delta, 107 explanation of, A–1 exponential, 11, A–7 factorial, 765 function, 125–127 grouping, 341 inequality, A–4 infinity, 123 intersection, 29 interval, 122, 123 mapping, 88, 89, 354 proper subset of, A–1 radical, A–8 rate of change, 495 scientific, 15–16 set, 81, 122, A–1, A–2 subscript, 796 summation or sigma, 766, 767, 838 union, 29 nth term of arithmetic sequence, 774–777 explanation of, 762 of geometric sequence, 783–787 Null set, A–1 Number line, 122, A–2 Numbers See also Integers complex, 282–288, 291, 370–371, 408 imaginary, 282, 285, 286, 291 irrational, 81, A–3 natural, 81, 781, A–1 negative, A–2 positive, A–2 rational, 53, 54, 81, A–2 real, 5–7, 81, A–3, A–4 sets of, 81, A–1 whole, 81, A–1 – A–2 O Objective functions, 620 Objective variables, 155–156, 622 Oblique asymptotes, 448–450 Odd functions, 191–192 One-dimensional graphs, 591 One-to-one functions See also Exponential functions explanation of, 480, 486, 566 identification of, 480–481 inverse of, 482, 566 restricting domain to create, 483–484 Ordered pairs, 88–91 Ordered triples, 591–593 Order of operations, A–8 – A–9 Order property of real numbers, A–4 Origin, symmetry to, 191–192 Output values, P Parabolas applications of analytic, 750 definition of, 747 explanation of, 91, 710–711, 744–745, 756–757 focal chord of, 748, 753 focus-directrix form of equation of, 746–749, 756 graphs of, 91, 92, 193, 745–749 horizontal, 92, 745–747 right parabolic segment of, 752 vertex of, 196, 205, 316 vertical, 91, 745, 747 with vertical axis, 745–746 Parallel lines equations for, 139–141 explanation of, 111 slope of, 111, 112, 139 Parametric equations, 467 Parent functions, transformations of, 212 Parent graph, 205 Pareto’s law, 525 Partial fractions explanation of, 682 rational expressions and, 682–688, 701 Partial sums of arithmetic sequence, 777–778 explanation of, 766 of geometric sequence, 787–788 on graphing calculator, 766–767, 769 of series, 766 Pascal’s triangle, 829–831 Perfect cubes, 44–45, 394, A–7 Perfect squares, 43, A–7 Perfect square trinomials explanation of, 20 factoring, 43–44 Perimeter explanation of, 32, A–14 formulas for, 33, 660, 727, A–14 – A–16 Permutations distinguishable, 807–808 I-7 on graphing calculator, 808 nondistinguishable, 809 Perpendicular distance, 601, 709, 713, 755 Perpendicular lines equations for, 139–141 explanation of, 112 slope of, 112, 113, 139 Pick’s theorem, 132 Piecewise-defined functions applications of, 254–255 continuous, 247, 251 discontinuous, 252, 253 domain of, 248–249 equation of, 249, 253–254 explanation of, 247, 276 on graphing calculator, 250 graphs of, 249–251 Pitch diameter, A–12 Placeholder substitution, 45 Plane, 591, A–14 Plane geometry, relationships from, 708–709 Point of inflection, 124 Point-slope form explanation of, 141 to find function model, 143–144 linear equations in, 141–142 Pointwise defined relations, 88, 89 Poiseuille’s law, 52 Polygons, 37, A–14 Polyhedron formula, 37 Polynomial equations with complex coefficients, 312 explanation of, 46–48 Polynomial form converting between standard form and, 759 of equation of circle, 715 of equation of ellipse, 718 of equation of hyperbola, 735 Polynomial functions applications of, 405 complex, 394 dominant term of, 413 graphs of, 411–420 zeroes of, 192–193, 394–405, 471 Polynomial graphs end-behavior of, 412–415, 419 guidelines for, 419–420 identification of, 411–412 turning points and, 411 zeroes of multiplicity and, 415–418 Polynomial inequalities explanation of, 459, 474 on graphing calculator, 476 interval tests to solve, 461–462 method to solve, 459–461 push principle to solve, 476 Polynomial modeling, 421–422 Polynomials See also Binomials; Monomials; Trinomials addition and subtraction of, 17–18 applications of, 421–422 complex, 312 degree of, 16 division of, 383, 384 evaluation of, 386–387 explanation of, 16, 82 cob19545_ndx_I1-I16.qxd I-8 1/7/11 9:46 PM Page I-8 Index Polynomials—Cont factoring, 39–49 factors of, 387–389, 395 families of, 16 identifying and classifying, 16–17 multiplication of, 18–21 prime, 42, 46 quadratic, 41–43 quartic, 425 rational zeroes of, 400–402 real, 397–399 Population density, 381, 441 Positive association, 165 Positive numbers, A–2 Positive slope, 108 Power functions applications of, 239–242 explanation of, 236, 275, 496 on graphing calculators, 237 graphs of, 236–239 transformations of, 238, 239 variation and, 280 Power property of equality, 72, 74 of exponents, 11, 12, 15 of logarithms, 520–521, 527, 568, A–36 Present value equation, 540 Pressure, 10 Prime polynomials, 42, 46 Principal square roots, 282, 408, A–8 Prisms, A–21 Probability binomial, 834–836 defining events and, 816 elementary, 817 explanation of, 816, 841 on graphing calculator, 821, 823, 835 of mutually exclusive events, 822 of nonexclusive events, 821–823 properties of, 818–820 quick-counting and, 820–821 Problem-solving guide, 157–160 Product property of exponents, 11, 12, 81 of logarithms, 520–521, 527, 568, A–36 of radicals, 68–69, 292 Product to power property, 12, 15, 81 Projectile height, 311, 318–319, 324 Projectile velocity, 333 Proof by induction, 796 Proper subset of whole numbers, A–1 Property of negative exponents, 13 Push principle, 476–477 Pythagorean theorem, 75–76, 83 Q Quadrants, 89 Quadratic equations applications of, 306–307 checking solutions to, 327–328 completing the square to solve, 297–299 explanation of, 46–47, 292, 371 on graphing calculator, 298, 299, 301, 302 quadratic formula to solve, 300, 301 square root property of equality and, 295–297 standard form of, 46 zero product property and, 47–48 Quadratic factors, 396 Quadratic forms explanation of, 45–46 u-substitution to factor expressions in, 45–46, 49, 75 Quadratic formula discriminant of, 301–302 explanation of, 299–300 to solve quadratic equations, 300, 301 Quadratic functions completing the square to graph, 313–315 equation of, 316 explanation of, 293–295, 372 extreme values and, 316–321 finding equation of, 316 vertex formula to graph, 315–316 zeroes of, 293–294 Quadratic graphs, 378 Quadratic inequalities applications of, 30–308 domain and, 305–306 explanation of, 303, 371 interval test method for, 304–305 method to solve, 303–306 Quadratic models, 328–331, 372 Quadratic polynomials, 41–43 Quadratic regression, 331–332, 372 Quadrilaterals, A–14 Quartic polynomials, 425 Quick-counting techniques, 820–821 Quotient property of exponents, 13, 15, 81 of logarithms, 520–521, 527, 568, A–36 of radicals, 69 Quotients difference, 337, 359–364, 374 to power property, 12, 15 R Radical equations explanation of, 72–73 methods to solve, 73–75, 83 Radical expressions See also Cube roots; Square roots addition and subtraction of, 70 evaluation of, 65 explanation of, 65 methods to simplify, 65–66, 68–69 multiplication and division of, 70–72 rational exponents and, 67–68 Radicals explanation of, 65, 83 product property of, 68–69, 292 quotient property of, 69 Radical symbol, A–8 Radicand, 65, 66, A–8 Radioactive elements, 546 Radius of circle, 96–98, 103, 491, 710, 714 of sphere, 490 Range of functions, 123–124 of relations, 88, 134–135 Rate of change applications of, 332 average, 332–334, 359–361, 373 difference quotient and, 359–361 explanation of, 181 nonlinear functions and, 331–332 notation for, 495 rational functions and, 445 slope as, 106–107, 109, 136, 142 Rational equations, 59–60 Rational exponents explanation of, 67 power property of equality and, 74 radical expressions and, 67–68 simplifying expressions with, 68 solving equations with, 74, 83 Rational expressions addition and subtraction of, 56–58 decomposition for, 682, 685–688 explanation of, 53, 82 fundamental property of, 53–54 multiplication and division of, 55–56 partial fractions and, 682–688 in simplest form, 53–55 simplifying compound fractions and, 58–59 Rational functions applications of, 239–241, 439, 451–454 domain of, 431 end-behavior of, 233, 234 equations of, 236, 438 explanation of, 232, 275, 430, 473 on graphing calculator, 234, 452–454 graphs of, 235–236, 435–438, 446–450, 472–473 horizontal asymptotes of, 234, 433–435 with oblique or nonlinear asymptotes, 448–450 reciprocal function and, 232, 233, 235 removable discontinuities and, 446–447 vertical asymptotes of, 235, 430–433 writing equation of, 236 Rational inequalities analysis to solve, 462–463 explanation of, 474 interval test method to solve, 461–464 Rationalizing the denominator, 71 Rational numbers, 53–54, 81, A–2 Rational zeroes theorem, 399–402, 429 Raw data, 165 Real numbers absolute number of, A–5 – A–6 explanation of, 81, A–3 properties of, 5–7, 81, A–4 Real polynomials, 397–399 Reciprocal function, 232, 235 Reciprocal quadratic function, 235 Reciprocal square function, 233 Rectangles central, 733 explanation of, 33, A–14 perimeter and area formulas for, 33, 660, A–14 Rectangular coordinate system, 89 Rectangular solid, 35, 601, A–17 Recursive sequences, 764–765 Reduced row-echelon form, 394, 642, A–30 Reference intensity, 509 Reflections horizontal, 209–210 vertical, 208–209 cob19545_ndx_I1-I16.qxd 12/23/10 9:13 PM Page I-9 Index Regression applications of, 556–557 explanation of, 165 forms of, 552–553 on graphing calculator, 171–173, 178, 242, 376–377, 552, 554–557 quadratic, 331–332, 372 Regression equations applications of, 556–557 explanation of, 171–172, 552, 570 Regression line, 171 Regression models exponential, 553–554 logarithmic, 554–555 logistic equations and, 555–556 selection of, 178, 328, 570 Regular polygon, area of, 37 Relations domain and range of, 88, 134–135 explanation of, 88, 180 functions as, 119–122 on graphing calculator, 92–94 graphs of, 89–94 methods to represent, 88, 89 pointwise defined, 88, 89 Relative maximum, 196 Remainder theorem application of, 390 to evaluate polynomials, 386–387 explanation of, 386, 470, 471 proof of, A–34 Removable discontinuities explanation of, 446 rational functions and, 446–447 Repeated zeros, 475–476 Repeating and nonterminating decimals, A–2 Required interest rate, 269 Residuals, 376–377, 552 Right angles, A–14 Right circular cones, 35, A–17 Right circular cylinders, 35, A–17 Right parabolic segment, 752 Right prisms, A–21 Right pyramids, 35 Right square pyramid, A–17 Right triangles, 75 Root functions explanation of, 236 transformations of, 238–239 Roots cube, 66, A–8 of equation, 25 extraneous, 60, 527 square, 65–66, 282, 408, A–8 Root tests for quartic polynomials, 425 Row-echelon form, 642 Row operations, elementary, 640–641 Run, 107 S Sample outcome, 805 Sample space, 805, 816, 817 Sand dune function, 257 Scalar multiplication, 653–654 Scatterplots explanation of, 165 linear/nonlinear association and, 166–167 positive/negative association and, 165–166 Scientific notation, 15–16, 81 Secant lines, 140, 146 Semicircles, graph of, 92 Semimajor axis, 717 Semiminor axis, 717 Sequences See also Series; specific types of sequences applications of, 768–769, 779, 789–791 arithmetic, 773–779, 839 explanation of, 762, 838 finding terms of, 762–764 finite, 762 geometric, 782–791, 839 on graphing calculator, 763–765, 775–776, 785–786, 789–790 graphs of, 776, 786 graphs of arithmetic, 776 infinite, 762 recursive, 764–765 Series See also Sequences explanation of, 762, 765, 838 finite, 765–766 geometric, 782 properties of, 769 Set notation, 81, 122, A–1, A–2 Sets of numbers, 81, A–1 – A–3 Sigma notation, 766, 767, 838 Similar triangles, A–20 Simple interest, 539, 569 Simplest form radical expressions in, 71 rational expressions in, 53–55 Singular matrices determinants and, 669–672 explanation of, 669 Sinking fund, 544 Slope of line, 106–108 positive and negative, 108 as rate of change, 106–107, 109, 136, 142 Slope formula, 107, 108, 110, 138 Slope-intercept form explanation of, 137 graph of line and, 138–139 linear equations in, 137, 138 Smooth functions, 247 Solution region, 616 Solution sets, 27 Spheres, 35, 218, A–17 Spherical shells, 52 Square matrices, 638, 664 Square root function, 204 Square root property of equality, 295–297 Square roots See also Radical expressions of complex numbers, 408 principal, 282, 408, A–8 simplification of, 65–66 Squares See also Completing the square binomial, 20–21, 43, 759, 830 difference of two, 43 explanation of, 33 factoring difference of two, 43 perfect, 43, A–7 perimeter and area formulas for, 33, A–16 trinomial, 730 I-9 Square systems, 597 Square units, A–14 Squaring function, 204 Standard form of complex numbers, 283–284 converting between polynomial form and, 759 of equation of circle, 96, 716 of equation of ellipse, 717 of equation of hyperbola, 733–734 of linear equations, 46 of polynomial expressions, 17 of quadratic equations, 46 of system of equations, 579 Statistics, 816 Step functions, 254–255 Stevens, Stanley, 571 Stevens’ law, 571 Stirling’s formula, 814 Stretches, vertical, 210–211 Subscripted variables, A–14 Subscript notation, 796 Substitution applying power property after, 12 back, 578 to check complex root, 286, 287 to check solutions, 26 explanation of, 578 placeholder, 45 to solve system of equations, 578–580 Subtraction of complex numbers, 284–285 of matrices, 652–653, 700 of polynomials, 17–18 of radical expressions, 70 of rational expressions, 56–58 Summation applications of, 803–804 explanation of, 766 index of, 766, 838 properties of, 767–768 Summation notation, 766, 767 Surface area See also Area of cone, 80 of cylinder, 52, 311, 349, 455 of rectangular box, 323 Symbols See Notation/symbols Symmetry axis of, 205 to origin, 191–192 polynomial graphs and, 419 y-axis, 190–191 Synthetic division, 383–384, 470, A–28 – A–29 System of linear equations applications of, 645–646 augmented matrix of, 638–640 determinants and Cramer’s Rule to solve, 679–682 equivalent, 579, 593–594 explanation of, 576 on graphing calculators, 643–644 inconsistent and dependent, 581–582, 596–598, 644–645 matrices to solve, 640–643, 663–673, 701, A–30 square, 597 verifying solutions to system of, 576 cob19545_ndx_I1-I16.qxd I-10 12/23/10 9:13 PM Page I-10 Index System of linear equations in three variables applications of, 598–599 coincident dependent, 592 elimination to solve, 593–596, 634–635 explanation of, 591–592, 630–631 inconsistent and dependent, 596–598 linearly dependent, 592 solutions to, 592–593 System of linear equations in two variables applications of, 582–586 elimination to solve, 579–581, 634–635 explanation of, 576, 630 on graphing calculator, 577, 578, 580–586 graphs to solve, 577 inconsistent and dependent, 581–582 modeling and, 582–586 substitution to solve, 578–579 System of linear inequalities applications of, 619–620 explanation of, 615 on graphing calculator, 619 method to solve, 618–620 System of logarithmic equations, 608 System of nonlinear equations conic sections and, 749–750 elimination to solve, 607–608 explanation of, 604, 631 on graphing calculator, 605–608 graphs of, 604–605 possible solutions for, 604–605 substitution to solve, 605–607 System of nonlinear inequalities, 609–610, 631 System of two equations in two variables, 576 T Temperature conversions, 132 Terminating decimals, A–2 Toolbox functions direct variation and, 261–264 explanation of, 204–206, 274 horizontal translations and, 207–208 vertical translations and, 206–207 Transcendental functions, 503 Transformations of general functions, 211–214 to graph exponential functions, 495 on graphing calculator, 213, 265 to graph logarithmic functions, 506 Lorentz, 52 of parent graph, 205 of power functions, 238, 239 of reciprocal functions, 235 review of, 274 of root functions, 238 Translations on graphing calculator, 206–208 horizontal, 207–208 vertical, 206–207 Transverse axis, 732 Trapezoids, 33–34, A–14 Tree diagrams, 804–805 Trial-and-error method, for factoring, 42–43 Trials, 805 Triangles explanation of, 33 Pascal’s, 829–831 perimeter and area formulas for, 33, 455, 688, A–14 right, 75 similar, A–20 Triangularizating augmented matrix, 640–642, 678 Trigonometric graphs, 200 Trinomials See also Polynomials explanation of, 16 factoring, 41–43 perfect square, 20, 43–44 in quadratic form, 45 Trinomial squares, 730 Two-dimension graphs, 591 U Unbounded region, 621 Uniform motion, 158–159, 583–584 Union, 29 Uniqueness property to solve exponential equations, 496–499 to solve logarithms, 528–529 Unique solutions, 592 Unit conversion factors, A–18 – A–21 Upper and lower bounds property, 402, 404 Upper bound, 404 U.S Customary Units, A–19 u-substitution, 45–46, 49, 75 V Variables dependent, 88 descriptive, independent, 88 object, 155–156 subscripted, A–14 Variation constant of, 261 direct, 261–264, 277 on graphing calculators, 264, 279 graphs of, 262–265 inverse, 264–265, 277 joint or combined, 266, 277 power functions and, 280 Variation equations, 261, 263, 265 Velocity, 333, 363, 364 Velocity formula, 79, 80, 236 Vertex of cone, 710 of hyperbola, 732 of parabola, 196, 205, 316 Vertex formula, 315–316 Vertex/intercept formula, 323 Vertical asymptotes, 235, 430–433 Vertical axis, 591 Vertical boundary lines, 122, 134 Vertical change, 106–107 Vertical format, Vertical hyperbolas, 732, 735–736 Vertical lines, 109–110 Vertical line test, 120–122 Vertical parabolas, 91, 745, 747 Vertical reflections, 208–209 Vertical shifts See Vertical translations Vertical stretches, 210–211 Vertical translations, 206–207 Volume of box, 52 of cone, 35, 491 of cylinder, 176 of cylindrical shell, 52 explanation of, 34, A–16, A–18 – A–19 formulas for, 34, 491, A–17 method to compute, 35 of open box, 392 of right prism, A–21 of sphere, 218 of triangular pyramid, 691 W Whole numbers, 81, A–1 – A–2 Written information translated into equations, 31–32 translated into mathematical model, 2–3 X x-intercepts explanation of, 106 on graphing calculators, 153–154, 293–294, 296 of quadratic functions, 293–294 x-intercept/zeroes method, 152–154 xy-plane, 89 Y y-axis, 190–191 y-intercepts, 106 Z Zeroes approximation of real, 429 complex, 475–476 division with, 384–385, A–6 factor theorem to find, 389 of functions, 192–194 intermediate value theorem to find, 398 of multiplicity, 396, 415–416, 475 of polynomial functions, 192–193, 394–405, 415–416, 471 of quadratic functions, 293–294 quotient of, 384–385, A–6 rational, 400–402, 429 repeated, 475–476 Zeroes/x-intercept method, 152–154, 298 Zero exponent property, 13, 15 Zero exponents, 13, 81 Zero product property, 47–48 cob19545_cre_C1-C2.qxd 12/23/10 9:15 PM Page C-1 Photo Credits Chapter R Chapter Chapter p 1: © Royalty-Free/CORBIS; p 8: © Photodisc/Getty Images/RF; p 24: © Royalty-Free/CORBIS; p 79: © Glen Allison/Getty Images/RF Opener/p 381: © Royalty-Free/CORBIS; p 390: © Adalberto Rios Szalay/Sexto 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Pictures/Getty Images/RF C-1 This page intentionally left blank cob19545_es.indd Page Sec1:2 28/01/11 3:41 PM s-60user ▼ /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles Special Constants ▼ Ϸ 3.1416 ▼ e Ϸ 2.7183 12 Ϸ 1.4142 Special Products 1x ϩ a21x ϩ b2 ϭ x2 ϩ 1a ϩ b2x ϩ ab 1a ϩ b21a Ϫ b2 ϭ a2 Ϫ b2 1a ϩ b2 ϭ a2 ϩ 2ab ϩ b2 1a Ϫ b2 ϭ a2 Ϫ 2ab ϩ b2 1a ϩ b2 ϭ a ϩ 3a b ϩ 3ab ϩ b 2 1a Ϫ b2 ϭ a Ϫ 3a b ϩ 3ab Ϫ b 3 2 Special Factorizations x2 ϩ 1a ϩ b2x ϩ ab ϭ 1x ϩ a21x ϩ b2 a2 Ϫ b2 ϭ 1a ϩ b21a Ϫ b2 a2 ϩ 2ab ϩ b2 ϭ 1a ϩ b2 a2 Ϫ 2ab ϩ b2 ϭ 1a Ϫ b2 a Ϫ b ϭ 1a Ϫ b21a ϩ ab ϩ b ▼ Distance between P1 and P2 13 Ϸ 1.7321 a ϩ b ϭ 1a ϩ b21a Ϫ ab ϩ b 2 3 2 ▼ Rectangle Square w P ϭ 2l ϩ 2w P ϭ 4s l A ϭ bh Aϭ b Triangle h 1a ϩ b2 C A b bh a r A ϭ r2 b C ϭ 2r ϭ d Right Parabolic Segment A ϭ ab a b C Ϸ 221a2 ϩ b2 a ▼ H V ϭ LWH S ϭ 21LW ϩ LH ϩ WH2 L W Cube Right Circular Cylinder V ϭ s3 V ϭ r2h S ϭ 6s2 Right Circular Cone Right Square Pyramid Sphere V ϭ r2h V ϭ b2h V ϭ r3 S ϭ r 1r ϩ s2 h S ϭ b2 ϩ b2b2 ϩ 4h2 r ISBN: 0-07-351954-5 Author: John W Coburn Title: College Algebra, 3e Front endsheets Color: Pages: 2, h S ϭ 4r2 b y ϭ mx ϩ b, where b ϭ y1 Ϫ mx1 Parallel Lines Perpendicular Lines Slopes Are Equal: m1 ϭ m2 Slopes Have a Product of Ϫ1: m1m2 ϭ Ϫ1 Intersecting Lines Dependent (Coincident) Lines Slopes and y-Intercepts Are Equal: m1 ϭ m2, b1 ϭ b2 m2 Logarithms and Logarithmic Properties y ϭ logb x b y ϭ x logb b ϭ logb bx ϭ x blogb x ϭ x logb a logb ϭ logc x ϭ M b ϭ logb M Ϫ logb N N logb x logb c logb MP ϭ P # logb M Applications of Exponentials and Logarithms A S amount accumulated P S initial deposit, P S periodic payment n S compounding periods/year r S interest rate per year r R S interest rate per time period a b n t S time in years Interest Compounded n Times per Year Interest Compounded Continuously r nt A ϭ P a1 ϩ b n A ϭ Pert Accumulated Value of an Annuity Payments Required to Accumulate Amount A Pϭ AR 11 ϩ R2 nt Ϫ Sequences and Series: Arithmetic Sequences Geometric Sequences a1, a2 ϭ a1 ϩ d, a3 ϭ a1 ϩ 2d, , an ϭ a1 ϩ 1n Ϫ 12d a1, a2 ϭ a1r, a3 ϭ a1r2, , an ϭ a1r nϪ1 Sn ϭ S ϭ 2r 1r ϩ h2 s y Ϫ y1 ϭ m1x Ϫ x1 a1 S 1st term, an S nth term, Sn S sum of n terms, d S common difference, r S common ratio Formulas from Solid Geometry: S S surface area, V S volume Rectangular Solid Slope-Intercept Form (slope m, y-intercept b) P A ϭ ΄11 ϩ R2 nt Ϫ 1΅ R b ▼ Point-Slope Form b Circle c a2 ϩ b2 ϭ c2 Ellipse Aϭ Pythagorean Theorem A ϩ B ϩ C ϭ 180° ▼ h ¢y y2 Ϫ y1 ϭ x2 Ϫ x ¢x Equation of Line Containing P1 and P2 logb 1MN ϭ logb M ϩ logb N Triangle h Right Triangle B A ϭ ab a Trapezoid h a P ϭ ns a Aϭ P Aϭs A ϭ lw Sum of angles Regular Polygon s s mϭ Equation of Line Containing P1 and P2 Slopes Are Unequal: m1 Formulas from Plane Geometry: P S perimeter, C S circumference, A S area Parallelogram Slope of Line Containing P1 and P2 d ϭ 21x2 Ϫ x1 2 ϩ 1y2 Ϫ y1 2 ▼ Formulas from Analytical Geometry: P1 S (x1, y1), P2 S (x2, y2) h Sn ϭ r r ▼ a1 Ϫ a1rn 1Ϫr a1 Sq ϭ ; ͿrͿ 1Ϫr n 1a1 ϩ an 2 Sn ϭ n ΄2a1 ϩ 1n Ϫ 12d΅ Binomial Theorem n n n 1a ϩ b2 n ϭ a b anb0 ϩ a b anϪ1b1 ϩ a b anϪ2b2 ϩ n! ϭ n1n Ϫ 121n Ϫ 22 # # # 132122112; ### ϩa 0! ϭ n n b a 1bnϪ1 ϩ a b a0bn nϪ1 n n n! a bϭ k k!1n Ϫ k2! cob19545_es.indd Page Sec1:2 28/01/11 3:41 PM s-60user ▼ /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles Special Constants ▼ Ϸ 3.1416 ▼ e Ϸ 2.7183 12 Ϸ 1.4142 Special Products 1x ϩ a21x ϩ b2 ϭ x2 ϩ 1a ϩ b2x ϩ ab 1a ϩ b21a Ϫ b2 ϭ a2 Ϫ b2 1a ϩ b2 ϭ a2 ϩ 2ab ϩ b2 1a Ϫ b2 ϭ a2 Ϫ 2ab ϩ b2 1a ϩ b2 ϭ a ϩ 3a b ϩ 3ab ϩ b 2 1a Ϫ b2 ϭ a Ϫ 3a b ϩ 3ab Ϫ b 3 2 Special Factorizations x2 ϩ 1a ϩ b2x ϩ ab ϭ 1x ϩ a21x ϩ b2 a2 Ϫ b2 ϭ 1a ϩ b21a Ϫ b2 a2 ϩ 2ab ϩ b2 ϭ 1a ϩ b2 a2 Ϫ 2ab ϩ b2 ϭ 1a Ϫ b2 a Ϫ b ϭ 1a Ϫ b21a ϩ ab ϩ b ▼ Distance between P1 and P2 13 Ϸ 1.7321 a ϩ b ϭ 1a ϩ b21a Ϫ ab ϩ b 2 3 2 ▼ Rectangle Square w P ϭ 2l ϩ 2w P ϭ 4s l A ϭ bh Aϭ b Triangle h 1a ϩ b2 C A b bh a r A ϭ r2 b C ϭ 2r ϭ d Right Parabolic Segment A ϭ ab a b C Ϸ 221a2 ϩ b2 a ▼ H V ϭ LWH S ϭ 21LW ϩ LH ϩ WH2 L W Cube Right Circular Cylinder V ϭ s3 V ϭ r2h S ϭ 6s2 Right Circular Cone Right Square Pyramid Sphere V ϭ r2h V ϭ b2h V ϭ r3 S ϭ r 1r ϩ s2 h S ϭ b2 ϩ b2b2 ϩ 4h2 r ISBN: 0-07-351954-5 Author: John W Coburn Title: College Algebra, 3e Front endsheets Color: Pages: 2, h S ϭ 4r2 b y ϭ mx ϩ b, where b ϭ y1 Ϫ mx1 Parallel Lines Perpendicular Lines Slopes Are Equal: m1 ϭ m2 Slopes Have a Product of Ϫ1: m1m2 ϭ Ϫ1 Intersecting Lines Dependent (Coincident) Lines Slopes and y-Intercepts Are Equal: m1 ϭ m2, b1 ϭ b2 m2 Logarithms and Logarithmic Properties y ϭ logb x b y ϭ x logb b ϭ logb bx ϭ x blogb x ϭ x logb a logb ϭ logc x ϭ M b ϭ logb M Ϫ logb N N logb x logb c logb MP ϭ P # logb M Applications of Exponentials and Logarithms A S amount accumulated P S initial deposit, P S periodic payment n S compounding periods/year r S interest rate per year r R S interest rate per time period a b n t S time in years Interest Compounded n Times per Year Interest Compounded Continuously r nt A ϭ P a1 ϩ b n A ϭ Pert Accumulated Value of an Annuity Payments Required to Accumulate Amount A Pϭ AR 11 ϩ R2 nt Ϫ Sequences and Series: Arithmetic Sequences Geometric Sequences a1, a2 ϭ a1 ϩ d, a3 ϭ a1 ϩ 2d, , an ϭ a1 ϩ 1n Ϫ 12d a1, a2 ϭ a1r, a3 ϭ a1r2, , an ϭ a1r nϪ1 Sn ϭ S ϭ 2r 1r ϩ h2 s y Ϫ y1 ϭ m1x Ϫ x1 a1 S 1st term, an S nth term, Sn S sum of n terms, d S common difference, r S common ratio Formulas from Solid Geometry: S S surface area, V S volume Rectangular Solid Slope-Intercept Form (slope m, y-intercept b) P A ϭ ΄11 ϩ R2 nt Ϫ 1΅ R b ▼ Point-Slope Form b Circle c a2 ϩ b2 ϭ c2 Ellipse Aϭ Pythagorean Theorem A ϩ B ϩ C ϭ 180° ▼ h ¢y y2 Ϫ y1 ϭ x2 Ϫ x ¢x Equation of Line Containing P1 and P2 logb 1MN ϭ logb M ϩ logb N Triangle h Right Triangle B A ϭ ab a Trapezoid h a P ϭ ns a Aϭ P Aϭs A ϭ lw Sum of angles Regular Polygon s s mϭ Equation of Line Containing P1 and P2 Slopes Are Unequal: m1 Formulas from Plane Geometry: P S perimeter, C S circumference, A S area Parallelogram Slope of Line Containing P1 and P2 d ϭ 21x2 Ϫ x1 2 ϩ 1y2 Ϫ y1 2 ▼ Formulas from Analytical Geometry: P1 S (x1, y1), P2 S (x2, y2) h Sn ϭ r r ▼ a1 Ϫ a1rn 1Ϫr a1 Sq ϭ ; ͿrͿ 1Ϫr n 1a1 ϩ an 2 Sn ϭ n ΄2a1 ϩ 1n Ϫ 12d΅ Binomial Theorem n n n 1a ϩ b2 n ϭ a b anb0 ϩ a b anϪ1b1 ϩ a b anϪ2b2 ϩ n! ϭ n1n Ϫ 121n Ϫ 22 # # # 132122112; ### ϩa 0! ϭ n n b a 1bnϪ1 ϩ a b a0bn nϪ1 n n n! a bϭ k k!1n Ϫ k2! cob19545_es.indd Page Sec1:3 28/01/11 3:42 PM s-60user ▼ /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles The Toolbox and Other Functions ▼ linear linear y y identity constant y y ϭ mx ϩ b y Fundamental Counting Principle: Given an experiment with two tasks completed in sequence, if the ﬁrst can be completed in m ways and the second in n ways, the experiment can be completed in m ؋ n ways yϭb yϭx Permutations—Order Is a Consideration: (Al, Bo, Ray) and (Ray, Bo, Al) ﬁnish the race in a different order n! The permutations of r objects selected from a set of n (unique) objects is given by nPr ؍ (n ؊ r)! Combinations—Order Is Not a Consideration: (Al, Bo, Ray) and (Ray, Bo, Al) form the same committee n! The combinations of r objects selected from a set of n (unique) objects is given by nCr ؍ r!(n ؊ r)! Basic Probability: Given S is a sample space of equally likely events and E is an event deﬁned relative to S n(E) , where n1E2 and n1S2 represent the number of elements in each The probability of E is P(E) ؍ n(S) For any event E1: Յ P1E1 Յ and P1E1 ϩ P1~E1 ϭ (0, b) (0, b) x x y ϭ mx ϩ b m Ͻ 0, b Ͼ absolute value x m Ͼ 0, b Ͼ m ϭ 1, b ϭ squaring cubing y y x square root y ϭ x3 x y ϭ ͙x x ceiling function Probability of E1 and E2 Probability of E1 or E2 P1E1 ʝ E2 ϭ P1E1 2P1E2 1E1, E2 independent2 P1E1 ´ E2 ϭ P1E1 ϩ P1E2 Ϫ P1E1 ʝ E2 x ﬂoor function y ٘ yϭ x y m ϭ 0, b Ͼ y y ϭ x2 y ϭ ԽxԽ cube root x y Quick-Counting and Probability ▼ reciprocal y y ϭ ٘x٘ Conic Sections y y yϭ y circle with center at (h, k) x ٘ y ϭ ͙x r x x x x k logarithmic y y yϭ x2 logistic y ϭ bx (b Ͼ 0) (Ϫa, 0) x x (0, ϩc a ) y ϭ a f 1x Ϯ h2 Ϯ k vertical reﬂections vertical stretches/compressions ▼ horizontal shift h units, opposite direction of sign vertical shift k units, same direction as sign Average Rate of Change of f(x) f(x2) ؊ f(x1) ⌬y ؍ ⌬x x2 ؊ x1 Front endsheets Color: Pages: 6, k (h, k) ϭ1 ϩ y2 b2 ϭ1 x If a Ͻ b, the ellipse is oriented vertically Ϫ (y Ϫ k)2 b2 h x2 a2 Ϫ y2 b2 c2 ϭ a2 ϩ b2 pϾ0 (0, p) x x If term containing y leads, the hyperbola is oriented vertically ϭ1 y ϭ1 y ϭ Ϫp (a, 0) (c, 0) (Ϫc, 0) x2 ϭ 4py vertical parabola focus (0, p) directrix y ϭ Ϫp y ( p, 0) (x Ϫ h)2 a2 (Ϫa, 0) For linear function models, the average rate of change on the interval 3x1, x2 is constant, and given by the slope formula: ¢y y2 Ϫ y1 ϭ The average rate of change for other function models is nonconstant By writing the slope formula in function form x2 Ϫ x1 ¢x using y1 ϭ f 1x1 and y2 ϭ f 1x2 2, we can compute the average rate of change of other functions on this interval: ISBN: 0-07-351954-5 Author: John W Coburn Title: College Algebra, 3e hyperbola with center at (h, k) central hyperbola S S y ϭ f 1x2 S Transformation of Given Function (y Ϫ k)2 b2 c2 ϭ |a2 Ϫ b2| Transformations of Basic Graphs Given Function ϩ (a, 0) x2 a2 x y ▼ (c, 0) (0, Ϫ b) x (Ϫc, 0) h c ϩ aeϪbx (h ϩ a, k) (0, b) x2 ϩ y2 ϭ r2 yϭc yϭ x h ellipse with center at (h, k), a Ͼ b (x Ϫ h)2 a2 (h, k Ϫ b) central ellipse (x, y) (0, 0) y y y ϭ logb x (b Ͼ 0) (h, k) (x Ϫ h)2 ϩ (y Ϫ k)2 ϭ r2 r exponential (h Ϫ a, k) (h, k) central circle reciprocal square k (h, k ϩ b) pϾ0 x ϭ Ϫp y2 ϭ 4px horizontal parabola focus ( p, 0) directrix x ϭ Ϫp x cob19545_es.indd Page Sec1:3 28/01/11 3:42 PM s-60user ▼ /Volume/204/MHDQ234/cob19545_disk1of1/0073519545/cob19545_pagefiles The Toolbox and Other Functions ▼ linear linear y y identity constant y y ϭ mx ϩ b y Fundamental Counting Principle: Given an experiment with two tasks completed in sequence, if the ﬁrst can be completed in m ways and the second in n ways, the experiment can be completed in m ؋ n ways yϭb yϭx Permutations—Order Is a Consideration: (Al, Bo, Ray) and (Ray, Bo, Al) ﬁnish the race in a different order n! The permutations of r objects selected from a set of n (unique) objects is given by nPr ؍ (n ؊ r)! Combinations—Order Is Not a Consideration: (Al, Bo, Ray) and (Ray, Bo, Al) form the same committee n! The combinations of r objects selected from a set of n (unique) objects is given by nCr ؍ r!(n ؊ r)! Basic Probability: Given S is a sample space of equally likely events and E is an event deﬁned relative to S n(E) , where n1E2 and n1S2 represent the number of elements in each The probability of E is P(E) ؍ n(S) For any event E1: Յ P1E1 Յ and P1E1 ϩ P1~E1 ϭ (0, b) (0, b) x x y ϭ mx ϩ b m Ͻ 0, b Ͼ absolute value x m Ͼ 0, b Ͼ m ϭ 1, b ϭ squaring cubing y y x square root y ϭ x3 x y ϭ ͙x x ceiling function Probability of E1 and E2 Probability of E1 or E2 P1E1 ʝ E2 ϭ P1E1 2P1E2 1E1, E2 independent2 P1E1 ´ E2 ϭ P1E1 ϩ P1E2 Ϫ P1E1 ʝ E2 x ﬂoor function y ٘ yϭ x y m ϭ 0, b Ͼ y y ϭ x2 y ϭ ԽxԽ cube root x y Quick-Counting and Probability ▼ reciprocal y y ϭ ٘x٘ Conic Sections y y yϭ y circle with center at (h, k) x ٘ y ϭ ͙x r x x x x k logarithmic y y yϭ x2 logistic y ϭ bx (b Ͼ 0) (Ϫa, 0) x x (0, ϩc a ) y ϭ a f 1x Ϯ h2 Ϯ k vertical reﬂections vertical stretches/compressions ▼ horizontal shift h units, opposite direction of sign vertical shift k units, same direction as sign Average Rate of Change of f(x) f(x2) ؊ f(x1) ⌬y ؍ ⌬x x2 ؊ x1 Front endsheets Color: Pages: 6, k (h, k) ϭ1 ϩ y2 b2 ϭ1 x If a Ͻ b, the ellipse is oriented vertically Ϫ (y Ϫ k)2 b2 h x2 a2 Ϫ y2 b2 c2 ϭ a2 ϩ b2 pϾ0 (0, p) x x If term containing y leads, the hyperbola is oriented vertically ϭ1 y ϭ1 y ϭ Ϫp (a, 0) (c, 0) (Ϫc, 0) x2 ϭ 4py vertical parabola focus (0, p) directrix y ϭ Ϫp y ( p, 0) (x Ϫ h)2 a2 (Ϫa, 0) For linear function models, the average rate of change on the interval 3x1, x2 is constant, and given by the slope formula: ¢y y2 Ϫ y1 ϭ The average rate of change for other function models is nonconstant By writing the slope formula in function form x2 Ϫ x1 ¢x using y1 ϭ f 1x1 and y2 ϭ f 1x2 2, we can compute the average rate of change of other functions on this interval: ISBN: 0-07-351954-5 Author: John W Coburn Title: College Algebra, 3e hyperbola with center at (h, k) central hyperbola S S y ϭ f 1x2 S Transformation of Given Function (y Ϫ k)2 b2 c2 ϭ |a2 Ϫ b2| Transformations of Basic Graphs Given Function ϩ (a, 0) x2 a2 x y ▼ (c, 0) (0, Ϫ b) x (Ϫc, 0) h c ϩ aeϪbx (h ϩ a, k) (0, b) x2 ϩ y2 ϭ r2 yϭc yϭ x h ellipse with center at (h, k), a Ͼ b (x Ϫ h)2 a2 (h, k Ϫ b) central ellipse (x, y) (0, 0) y y y ϭ logb x (b Ͼ 0) (h, k) (x Ϫ h)2 ϩ (y Ϫ k)2 ϭ r2 r exponential (h Ϫ a, k) (h, k) central circle reciprocal square k (h, k ϩ b) pϾ0 x ϭ Ϫp y2 ϭ 4px horizontal parabola focus ( p, 0) directrix x ϭ Ϫp x ... Coburn, John W College algebra : graphs and models / John W Coburn, J.D Herdlick p cm Includes index ISBN 978–0–07–351954–8 — ISBN 0–07–351954–5 (hard copy : alk paper) Algebra Textbooks Algebra Graphic... students to College Algebra and connect instructors to their students The Coburn/Herdlick College Algebra Series provides you with strong tools to achieve better outcomes in your College Algebra. .. Miami-Dade College Kimberly Graham J.D Herdlick, St Louis Community College, Meramec Jeremy Coffelt Blinn College Nancy Ikeda Fullerton College Vickie Flanders, Baton Rouge Community College Nic
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Xem thêm: College algebra graphs models , College algebra graphs models , B. Translating Written or Verbal Information into a Mathematical Model, D. Properties of Real Numbers, A. The Properties of Exponents, E. The Product of Two Polynomials, A. Solving Linear Equations Using Properties of Equality, F. Solving Applications of Basic Geometry, B. Common Binomial Factors and Factoring by Grouping, D. Factoring Special Forms and Quadratic Forms, E. Polynomial Equations and the Zero Product Property, C. Addition and Subtraction of Rational Expressions, F. Equations and Formulas Involving Radicals, B. The Graph of a Relation, D. The Equation and Graph of a Circle, B. The Slope of a Line and Rates of Change, E. Applications of Linear Equations, 3 Functions, Function Notation, and the Graph of a Function, B. The Domain and Range of a Function, D. Reading and Interpreting Information Given Graphically, B. Slope-Intercept Form and the Graph of a Line, D. Applications of Linear Equations, E. Using a Problem-Solving Guide, E. Linear Regression and the Line of Best Fit, E. Locating Maximum and Minimum Values Using Technology, E. Transformations of a General Function, A. Solving Absolute Value Equations, E. Applications Involving Absolute Value, A. Rational Functions and Asymptotes, C. Graphs of Basic Power Functions, D. Applications of Rational and Power Functions, A. The Domain of a Piecewise-Defined Function, C. Applications of Piecewise-Defined Functions, A. Toolbox Functions and Direct Variation, C. Joint or Combined Variations, C. Multiplying Complex Numbers; Powers of i, D. Division of Complex Numbers, A. Zeroes of Quadratic Functions and x-Intercepts of Quadratic Graphs, D. The Quadratic Formula and the Discriminant, F. Applications of Quadratic Functions and Inequalities, D. Quadratic Functions and Extreme Values, 4 Quadratic Models; More on Rates of Change, C. The Average Rate of Change Formula, D. Applications of the Algebra of Functions, A. The Composition of Functions, C. Average Rates of Change and the Difference Quotient, D. Applications of Composition and the Difference Quotient, A. Long Division and Synthetic Division, A. The Fundamental Theorem of Algebra, C. The Rational Zeroes Theorem, D. Descartes’ Rule of Signs and Upper/Lower Bounds, E. Applications of Polynomial Functions, B. The End-Behavior of a Polynomial Graph, C. Attributes of Polynomial Graphs with Zeroes of Multiplicity, E. Applications of Polynomials and Polynomial Modeling, B. Vertical Asymptotes and Multiplicities, D. The Graph of a Rational Function, E. Applications of Rational Functions, B. Rational Functions with Oblique or Nonlinear Asymptotes, C. Applications of Rational Functions, C. Finding Inverse Functions Using an Algebraic Method, E. Applications of Inverse Functions, D. Solving Exponential Equations Using the Uniqueness Property, D. Finding the Domain of a Logarithmic Function, A. Solving Equations Using the Fundamental Properties of Logarithms, B. The Product, Quotient, and Power Properties of Logarithms, D. Solving Applications of Logarithms, A. Solving Logarithmic and Exponential Equations, B. Applications of Logistic, Exponential, and Logarithmic Functions, A. Simple and Compound Interest, C. Applications Involving Annuities and Amortization, D. Applications Involving Exponential Growth and Decay, C. Logistic Equations and Regression Models, E. Inconsistent and Dependent Systems, C. Solving Systems of Three Equations in Three Variables Using Elimination, D. Inconsistent and Dependent Systems, E. Applications of Nonlinear Systems, A. Linear Inequalities in Two Variables, C. Applications of Systems of Linear Inequalities, C. Solving a System Using Matrices, F. Solving Applications Using Matrices, B. Addition and Subtraction of Matrices, D. Determinants and Singular Matrices, A. Solving Systems Using Determinants and Cramer’s Rule, B. Rational Expressions and Partial Fractions, C. Determinants, Geometry, and the Coordinate Plane, B. Using Matrices to Encrypt Messages, C. Characteristics of the Conic Sections, B. The Equation of an Ellipse, C. The Foci of an Ellipse, A. The Equation of a Hyperbola, C. The Foci of a Hyperbola, B. The Focus-Directrix Form of the Equation of a Parabola, D. Application of the Analytic Parabola, C. Series and Partial Sums, B. Finding the n th Term of an Arithmetic Sequence, C. Finding the n th Partial Sum of an Arithmetic Sequence, B. Find the nth Term of a Geometric Sequence, E. Applications Involving Geometric Sequences and Series, B. Mathematical Induction Applied to Sums, C. The General Principle of Mathematical Induction, B. Fundamental Principle of Counting, E. Probability and Nonexclusive Events, B. Binomial Coefficients and Factorials, D. Finding a Specific Term of the Binomial Expansion