College algebra by barnett ziegler 9th edition

bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page i College Algebra bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page iii NINTH EDITION College Algebra Raymond A Barnett Merritt College Michael R Ziegler Marquette University Karl E Byleen Marquette University Dave Sobecki Miami University Hamilton bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page iv COLLEGE ALGEBRA, NINTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2011 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2008, 2001, and 1999 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 978–0–07–351949–4 MHID 0–07–351949–9 ISBN 978–0–07–729713–8 (Annotated Instructor’s Edition) MHID 0–07–729713–X Editorial Director: Stewart K Mattson Sponsoring Editor: John R Osgood Director of Development: Kristine Tibbetts Developmental Editor: Christina A Lane Marketing Manager: Kevin M Ernzen Lead Project Manager: Sheila M Frank Senior Production Supervisor: Kara Kudronowicz Senior Media Project Manager: Sandra M Schnee Designer: Tara McDermott Cover/Interior Designer: Ellen Pettergell (USE) Cover Image: © Bob Pool/Getty Images Senior Photo Research Coordinator: Lori Hancock Supplement Producer: Mary Jane Lampe Compositor: Aptara®, Inc Typeface: 10/12 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 Chapter R Opener: © Corbis RF; p 31: © The McGraw-Hill Companies, Inc./John Thoeming photographer Chapter Opener: © Corbis RF; p 56: © Vol 71/Getty RF; p 92: © Getty RF Chapter Opener: © Vol 88/Getty RF; p 142: © Big Stock Photo; p 147: © Corbis RF; p 151: © Vol 112/Getty RF Chapter Opener: © Getty RF; p 170: © Getty RF; p 187: © Vol 88/Getty RF; p 220: © Corbis RF; p 250: © The McGraw-Hill Companies, Inc./Andrew Resek photographer Chapter Opener: © Corbis RF; p 271: © Corbis RF; p 272: © Vol 4/Getty RF Chapter Opener: © Getty RF; p 333: © Vol 68/Getty RF; p 345: © Corbis RF Chapter Opener: © Brand X/SuperStock RF; p 401: © California Institute of Technology; Chapter Opener: © Corbis RF; p 435: Courtesy of Bill Tapenning, USDA; p 439: © Vol 5/Getty RF; p 456: © Vol 48/Getty RF; p 460: © Getty RF Chapter Opener: © Vol.6/Getty RF; p 531: © ThinkStock/PictureQuest RF; p 543: © Corbis RF Library of Congress Cataloging-in-Publication Data College algebra / Raymond A Barnett [et al.] — 9th ed p cm Rev ed of: College algebra 8th ed / Raymond A Barnett, Michael R Ziegler, Karl E Byleen Includes index ISBN 978-0-07-351949-4 — ISBN 0-07-351949-9 (hard copy : alk paper) Algebra–Textbooks I Barnett, Raymond A QA154.3.B365 2011 512.9–dc22 2009019471 www.mhhe.com bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page v The Barnett, Ziegler, Byleen, and Sobecki Precalculus Series College Algebra, Ninth Edition This book is the same as Precalculus without the three chapters on trigonometry ISBN 0-07-351949-9, ISBN 978-0-07-351-949-4 Precalculus, Seventh Edition This book is the same as College Algebra with three chapters of trigonometry added The trigonometry functions are introduced by a unit circle approach ISBN 0-07-351951-0, ISBN 978-0-07-351-951-7 College Algebra with Trigonometry, Ninth Edition This book differs from Precalculus in that College Algebra with Trigonometry uses right triangle trigonometric to introduce the trigonometric functions ISBN 0-07-735010-3, ISBN 978-0-07-735010-9 College Algebra: Graphs and Models, Third Edition This book is the same as Precalculus: Graphs and Models without the three chapters on trigonometry This text assumes the use of a graphing calculator ISBN 0-07-305195-0, ISBN 978-0-07-305195-6 Precalculus: Graphs and Models, Third Edition This book is the same as College Algebra: Graphs and Models with three additional chapters on trigonometry The trigonometric functions are introduced by a unit circle approach This text assumes the use of a graphing calculator ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3 v This page intentionally left blank bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page vii About the Authors Raymond A Barnett, a native of and educated in California, received his B.A in mathematical statistics from the University of California at Berkeley and his M.A in mathematics from the University of Southern California He has been a member of the Merritt College Mathematics Department and was chairman of the department for four years Associated with four different publishers, Raymond Barnett has authored or co-authored 18 textbooks in mathematics, most of which are still in use In addition to international English editions, a number of the books have been translated into Spanish Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University; Charles Burke, City College of San Francisco; John Fujii, Merritt College; Karl Byleen, Marquette University; and Dave Sobecki, Miami University Hamilton Michael R Ziegler received his B.S from Shippensburg State College and his M.S and Ph.D from the University of Delaware After completing postdoctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he held the rank of Professor in the Department of Mathematics, Statistics, and Computer Science Dr Ziegler published more than a dozen research articles in complex analysis and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Karl Byleen before passing away unexpectedly in 2008 Karl E Byleen received his B.S., M.A., and Ph.D degrees in mathematics from the University of Nebraska He is currently an Associate Professor in the Department of Mathematics, Statistics, and Computer Science of Marquette University He has published a dozen research articles on the algebraic theory of semigroups and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler Dave Sobecki earned a B.A in math education from Bowling Green State University, then went on to earn an M.A and a Ph.D in mathematics from Bowling Green He is an associate professor in the Department of Mathematics at Miami University in Hamilton, Ohio He has written or co-authored five journal articles, eleven books and five interactive CD-ROMs Dave lives in Fairfield, Ohio with his wife (Cat) and dogs (Macleod and Tessa) His passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel, and home improvement projects vii This page intentionally left blank bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page ix Dedicated to the memory of Michael R Ziegler, trusted author, colleague, and friend This page intentionally left blank bar19499_saa_SA1-SA30.qxd 11/20/09 5:17 PM Page SA-29 Student Answer Appendix SA-29 Cumulative Review for Chapters and 17 (A) Domain: x Ϫ2; x intercept: x ϭ Ϫ4; y intercept: y ϭ (C) (4-4) y (B) Vertical asymptote: x ϭ Ϫ2; horizontal asymptote: y ϭ 10 Ϫ10 10 x Ϫ10 (A) Ϫ0.56 (double zero); (simple zero); 3.56 (double zero) (B) Ϫ0.56 can be approximated with a maximum routine; can be approximated with the bisection: 3.56 can be approximated with a minimum routine (4-2) 51 A reflection through the x axis transforms the graph of y ϭ In x into the graph of y ϭ Ϫln x A reflection through the y axis transforms the graph of y ϭ ln x into the graph of y ϭ ln (Ϫx) (5-3) y 55 Vertical asymptote: x ϭ Ϫ2; (4–4) oblique asymptote: y ϭ x ϩ 10 23 Ϫ10 10 x Ϫ10 Cumulative Review for Chapters 6–8 (A) Arithmetic (B) Geometric (C) Neither (D) Geometric (E) Arithmetic (A) 2, 5, 8, 11 (B) a8 ϭ 23 (C) S8 ϭ 100 (8-3) 11 Foci: FЈ ϭ (Ϫ 161, 0), F ϭ ( 161, 0); 13 (6-1) transverse axis length ϭ 12; y conjugate axis length ϭ 10 (6-3) y 10 F FЈ 10 Ϫ10 (8-3) x Ϫ2 F ϭ ΂0, 25 ΃ x Directrix y ϭ Ϫ 25 Ϫ10 17 23 25 31 39 69 93 95 Ϫ3 d d (B) Not defined (C) [3] (D) c (E) [Ϫ1, 8] (F) Not defined (7-4) Ϫ9 Ϫ7 (B) x1 ϭ 2t ϩ 3, x2 ϭ t, t any real number (C) No solution (7-3) (A) x1 ϭ 3, x2 ϭ Ϫ4 Ϫ3 x1 k1 Ϫ5 (A) c (B) AϪ1 ϭ c (C) x1 ϭ 13, x2 ϭ dc d ϭ c d d (D) x1 ϭ Ϫ11, x2 ϭ Ϫ4 (7-5) k2 Ϫ5 x2 Ϫ2 2 Pk: k ϩ k ϩ ϭ 2r for some integer r; Pk ϩ1: (k ϩ 1) ϩ (k ϩ 1) ϩ ϭ 2s for some integer s (8-2) Ϫ1 d (A) c (B) Not defined (7-4) 41 (0, i ), (0, Ϫi), (1, 1), (Ϫ1, Ϫ1) (7-6) (A) Infinite number of solutions (B) No solution (C) Unique solution (7-3) model A truck, model B trucks, and model C trucks; or model A trucks, model B trucks, and model C trucks; or model A trucks and model C trucks (7-3) 82.25 Ann 83 Ann 0.25 0.2 83 Bob 84.8 Bob 0.25 0.2 (A) M ≥ (B) M ≥ ¥ ϭ G 92 W Carol ¥ ϭ G 91.8 W Carol 0.25 0.2 83.75 Dan 85.2 Dan 0.25 0.4 82 Eric 80.8 Eric (C) Class averages Test Test Test Test [0.2 0.2 0.2 0.2 0.2]M ϭ [84.4 81.8 85 87.2] (7-4) (A) c APPENDIX B Exercises B-2 13 Ϫ 3x ϩ 2x Ϫ 15 Ϫ Ϫ x xϪ3 (x Ϫ 3)2 17 3x Ϫ ϩ x x ϩ 2x ϩ 19 2x x2 ϩ ϩ 3x ϩ (x ϩ 2)2 23 2x ϩ ϩ xϪ3 x ϩ 3x ϩ bar19499_saa_SA1-SA30.qxd SA-30 25 11/20/09 5:17 PM Page SA-30 Student Answer Appendix Ϫ ϩ xϪ4 xϩ3 (x ϩ 3)2 29 x ϩ Ϫ xϪ1 ϩ ϩ xϩ2 2x Ϫ 2x Ϫ x ϩ Exercises B-3 y ϭ Ϫ2x Ϫ 2; straight line y ϭ Ϫ2x Ϫ 2, x Յ 0; a ray (part of a straight line) y y Ϫ5 x x y2 ϭ 4x; parabola 11 Ϫ5 Ϫ5 Ϫ5 y y2 ϭ 4x, y Ն 0; parabola (upper half) y 5 Ϫ5 Ϫ5 x Ϫ5 x Ϫ5 29 y Ϫ5 Ϫ5 At ϩ Dt ϩ F , Ϫϱ t ϱ; parabola ϪE (A) The graphs are symmetric about the line y ϭ x (B) y ϭ ex x ϭ ex or y ϭ ln x Function is the inverse of function 21 x ϭ t, y ϭ y ϭ Ϫ23 x; straight line 5 23 y2 Ϫ x2 ϭ 8, x Ն 1, y Ն 3; part of a hyperbola x bar19499_sndx_I1-I10.qxd 11/24/09 3:34 PM Page I-1 SUBJECT INDEX Abscissa, 110 Absolute value definitions for, 65 distance and, 66 method to find, 65–66 to solve radical inequalities, 71–72 Absolute value equations geometric interpretation of, 67–68 method to solve, 66–70, 99–100 verbal statements as, 68–69 Absolute value functions, 188 Absolute value inequalities geometric interpretation of, 67–68 method to solve, 66–70 Absolute value problems method to solve, 69–71 solved geometrically, 67–68 with two cases, 71 Acceptable probability assignment, 547 Actual probability, 552 Addition associative property of, 4, commutative property of, 4, of complex numbers, 76–77 elimination by, 429–434 explanation of, 3–4 of matrices, 457–458 of polynomials, 23 of rational expressions, 34–36 of real numbers, 3–7 Addition properties of equality, 45 of matrices, 477 of real numbers, Additive identity, 4, 77 Additive inverse, 4, Adiabatic process, 530 Algebra, Algebraic equations See also Equations algebraic expressions vs., 49 explanation of, 44 Algebraic expressions algebraic equations vs., 49 containing radicals, 17 explanation of, 21 factor of, 25 Algorithm, division, 267 Analytic geometry basic problems studied in, 122 fundamental theorem of, 110 Approximation by rational numbers, of real zeros, 282–283 Arithmetic sequences explanation of, 520, 564–565 method to find terms in, 522–523 method to recognize, 521 nth term of, 522 Arithmetic series, 523–524 Associative property of addition, 4, of multiplication, Asymptote rectangle, 407 Asymptotes on graphing calculator, 306 horizontal, 303–304 oblique, 308 vertical, 302–304 Augmented matrices explanation of, 443 Gauss-Jordan elimination and, 447 interpretation of, 445 method to write, 443–444 reduced, 450, 451 Axis of cone, 386n conjugate, 407–409 of ellipse, 395 of hyperbola, 405, 407 of parabola, 1111 of symmetry, 205, 387 transverse, 405 Base of exponent, 11 of exponential functions, 329, 331–333 Bell, Alexander Graham, 365 Binomial coefficients, 22–23, 560 Binomial expansion, 558–559 Binomial formula explanation of, 559–560, 565 proof of, 562–563 use of, 560–562 Binomials, 22 See also Polynomials Bisection method, 281–282 Briggsian logarithms See Common logarithms Calculators See Graphing calculators Carbon-14 decay equation, 343–344 Cardano, Girolamo, 108 Cardano’s formula, 108 Cartesian coordinate system, 110, 157–158 Catenary curve, 374, 391 Center of circle, 127, 129 of ellipse, 395 of hyperbola, 405, 408 Change-of-base formula, 361–362 Circles equations of, 126–128 explanation of, 127, 386 formulas for, 599 graphs of, 126–128 Closure property, Coefficient determinant, 492 Coefficient matrix, 443 Coefficients binomial, 22–23, 560 in linear systems, 424 of polynomial functions, 260–261 real, 290–291 Cofactor of element explanation of, 489 method to find, 489–490 Column matrices, 442, 461–462 I-1 bar19499_sndx_I1-I10.qxd I-2 11/24/09 3:34 PM Page I-2 SUBJECT INDEX Combinations, 538–541, 565 Combined properties, of matrices, 477 Combined variation, 319 Common difference, 520 Common factors, 26, 33 Common logarithms, 359, 360 Common ratio, 521 Commutative property, 4, Completing the square, 86–87 Complex numbers addition of, 76–77 division of, 78–79 explanation of, 74–76, 105 historical background of, 74 multiplication of, 77–78 operations with, 76–79 radicals and, 80–81 set of, 75 solving equations involving, 81–82 subtraction of, 76–77 types of, 75 zero of, 77 Composite numbers, 25 Composition of functions, 226–230, 252 inverse functions and, 240 Compound events, 544 Compound fractions, 36, 37 Compound interest applications of, 334, 373 continuous, 335–336 explanation of, 333–334 Conditional equations, 45 Conic sections See also Circles; Ellipses; Hyperbolas; Parabolas explanation of, 386–387 review of, 418–420 Conjecture, 511–512, 517 Conjugate, of a ϩ bi, 75 Conjugate axis, 407–409 Conjugate hyperbolas, 410 Consistent systems, 427 Constant in term of polynomial, 22 of variation, 316–318 Constant functions, 178, 179 Constant matrix, 443 Constant terms, 424 Continuous compound interest, 335–336 Continuous compound interest formula, 336 Continuous graphs, 181 Contraction See Shrinking Coordinate, 3, 110 Coordinate axis, 110 See also x axis; y axis Correspondence, 162, 167 Counterexamples, 511–512 Counting techniques combinations and, 538–541 explanation of, 531–532 factorial notation and, 534–536 multiplication principle and, 532–534 permutations and, 536–538 Cramer, Gabriel, 491 Cramer’s rule explanation of, 491–492, 498 to solve three-variable system, 493–494 to solve two-variable system, 492–494 for three equations in three variables, 493 Cube functions, 189 Cube root functions, 189 Cube roots, 14 Cubic equations, 108 Cubic models, 272 Curve fitting, 151 Curves catenary, 374, 391 explanation of, 151 plane, 592, 593 Data analysis examples of, 271–273 regression and, 346–349, 369 Decibels, 365, 366 Decimal expansions, Decoding matrix, 482 Decreasing functions, 178, 238 Degenerate conic, 387 Degree, of polynomials, 22, 260 Demand, 93, 435 Denominator explanation of, least common, 35 rationalizing the, 18–19 Dependent variables, 164 Descartes, René, 11 Determinants coefficient, 492 explanation of, 487, 498 first-order, 487–488 second-order, 487, 488 to solve systems of equations, 491–494 third-order, 488–491 Diagonal expansion, 495 Difference function, 224–225 Difference of cubes formula, 28, 29 Difference of square formula, 28, 29 Difference quotient, 170 Dimensions, of matrix, 442 Directrix, of parabola, 387 Direct variation, 316 Discriminant, 90–91 Distance absolute value and, 66 in plane, 123–129, 158 between two points, 123–124 Distance formula explanation of, 124 use of, 124, 388, 396, 406–407 Divisibility property, 516 Division of complex numbers, 78 long, 5, 266–267 polynomial, 266–269 of rational expressions, 33–34 of real numbers, synthetic, 268–269 Division algorithm, 267 Division properties, of equality, 45 Divisor, 267 Domain of exponential functions, 355 of functions, 163, 164, 166–167, 169–170, 176–177, 204, 225, 229, 230 implied, 166 of rational functions, 299–300 of variables, 44–45 bar19499_sndx_I1-I10.qxd 11/24/09 3:34 PM Page I-3 SUBJECT INDEX Double inequalities, 61, 67 Double subscript notation, 442 Double zero, 289 Doubling time, 340 Doubling time growth model, 340, 341 Eccentricity, 417 Element of matrix, 442 of set, Elimination, by addition, 429–434 Ellipses applications of, 400–401 equations of, 396–400 explanation of, 386, 395, 418–420 graphs of, 396–400 method to draw, 395–396 Empirical probability application of, 555–556 approximate, 552–554 explanation of, 552, 553 method to find, 553–554 Empty set, Endpoints, 57 Equality explanation of, 76 properties of, 45, 477 symbols for, 57 Equallly likely assumptions, 549–550 Equal polynomials, 585–586 Equations See also Linear equations; Systems of linear equations; specific types of equations of circles, 126–128 conditional, 45 cubic, 108 defining functions by, 164–167 of ellipse, 396–400 equivalent, 45 explanation of, 44, 56 exponential, 372–374, 380 graphs of, 111–112, 118 of hyperbolas, 406–408 as identities, 45 involving complex numbers, 81–82 involving radicals, 97–99 of lines, 132, 133, 137–140, 158 logarithmic, 375–376, 380 matrix, 477–480 of parabola, 209 parametric, 591–595 price-demand, 93 properties of, 69 quadratic, 84–93, 105 of quadratic type, 101–102, 105 solution set of, 44, 111 solutions of, 44, 111 squaring operation on, 98 in two variables, 111 Equilibrium price, 435 Equilibrium quantity, 435 Equivalent equations, 45 Equivalent inequalities, 59–60 Equivalent systems of equations, 429 Euler, Leonhard, 74 Even functions, 196–197 Events compound, 544 explanation of, 546–547 probability of, 547–551 simple, 544, 547 Expected frequency, 553 Experiments, 543–544 Exponential decay, 350 Exponential equations explanation of, 372, 380 method to solve, 372–374 Exponential functions with base e, 331–333 compound interest and, 333–336 domain of, 355 explanation of, 328–329, 379 graphs of, 329–333 inverse of, 329 (See also Logarithmic functions) properties of, 330–331 transformations of, 330 Exponential growth/decay, 349, 350 Exponential models application of, 379 data analysis and regression and, 346–349 exponential growth phenomena and, 349–350 on graphing calculator, 347 mathematical, 340–346 Exponents explanation of, 11, 39 integer, 11–13 rational, 16 Extended principle of mathematical induction, 517 Extraneous solutions, 98, 105 Extrapolation, 153 Face cards, 540 Factorials, 534–536 Factoring explanation of, 25 by grouping, 26–27 of polynomials, 25–29 to solve quadratic equations, 84–86 Factoring formulas, 28, 29 Factors of algebraic expression, 25 common, 26 explanation of, 25 of polynomials with real coefficients, 290–291 Factor theorem, 270 Fermat’s last theorem, 517 Fibonacci, Leonardo, 505 Fibonacci sequences, 505 Finite sequences arithmetic, 523–524 explanation of, 505 Finite series arithmetic, 523–524 explanation of, 507 geometric, 526–527 Finite sets, arithmetic, 523–524 First-degree equations See Linear equations First-degree functions See Linear functions First-order determinants, 487–488 Focal chords, 393, 422 Focus of ellipse, 395–398 of hyperbola, 405 of parabola, 387 Fractional expressions, 32 I-3 bar19499_sndx_I1-I10.qxd I-4 11/24/09 3:34 PM Page I-4 SUBJECT INDEX Fractions compound, 36–37 explanation of, fundamental property of, 32 partial, 585–590 proper, 585 properties of, raised to higher terms, 32 reduced to lowest terms, 32 significant digits in decimal, 583 simple, 36 solving inequalities involving, 61 Frequency, 553 Functions See also specific types of functions absolute value, 188 applications of, 170–171 composition of, 226–230, 240, 252 constant, 178, 179 cube, 189 cube root, 189 decreasing, 178, 238 defined by equations, 164–166 definition of, 163, 235, 250 difference, 224–225 difference quotient and, 170 domains of, 166–167, 169–170, 176–177, 204, 225, 229, 230 even, 196–197 exponential, 328–336, 379 graphs of, 175–184, 188–199, 250–251 greatest integer, 182, 183 identity, 179, 188 increasing, 178, 238 inverse, 235–246, 252 linear, 178–180 logarithmic, 354–362, 379–380 notation for, 167–168 odd, 196–197 one-to-one, 235–238, 358 operations on, 223–226 overview of, 162 piecewise-defined, 180–181 polynomial, 260–266 product, 224–225 quadratic, 203–211, 251–252 quotient, 224–225 range of, 163, 166, 177 rational, 298–310 set form of definition of, 163 square, 188, 203, 204 square root, 189 sum, 224–225 transformations of, 188–197, 251 vertical line test for, 166 Fundamental counting principle See Multiplication principle Fundamental property of fractions, 32 Fundamental sample space, 545 Fundamental theorem of algebra, 288–289 Fundamental theorem of analytic geometry, 110 Fundamental theorem of arithmetic, 25 Gauss, Carl Friedrich, 288, 447 Gauss-Jordan elimination explanation of, 441, 447 on graphing calculators, 449 to solve linear systems, 447–451, 497 use of, 475 General form, of quadratic function, 204 Geometric sequences explanation of, 521, 564–565 method to find terms in, 522–523 method to recognize, 521 nth term of, 522 Geometric series sum formulas for finite, 525 sum formulas for infinite, 526–527 Goldbach’s conjecture, 517 Graphing calculator features INTERSECT, 361 MATRIX-MATH, 488 maximum and minimum, 209 random number generator, 554 rref on, 449 table, 561 TRACE, 134, 280, 588 viewing window, 127 ZERO command, 280 ZSquare, 127 Graphing calculators asymptotes on, 306 circles on, 127 cubic models on, 272 domain of functions on, 225 ellipses on, 399 exponential functions on, 328, 331 exponential models on, 347 graphs of equations on, 112, 118, 143 greatest integer functions on, 183 interest rate on, 335 inverse functions on, 246 linear systems on, 425 logarithms on, 359–360, 370 logistic models on, 349 matrices on, 442, 458, 473, 488 parabolas on, 390 parametric equations on, 592 partial fraction decomposition on, 588 polynomial inequalities on, 284 quadratic regression on, 215, 216 quartic model on, 273 rational inequalities on, 311 reduced echelon form on, 449 regression on, 153 scientific notation on, 13–14 sequences on, 505, 507 sum of series on, 526 Graphs/graphing of circles, 126–128 continuous, 181 of ellipses, 396–400 of equation in two variables, 111 explanation of, 111 of exponential functions, 329–333 of functions, 175–184, 188–199, 250–251 horizontal and vertical shifts in, 189–191 of hyperbolas, 406–412 of inequalities, 58, 59 of intervals, 58, 59 of inverse functions, 244–246 line, 57 of linear functions, 179–180 of lines, 132–133 of logarithmic functions, 354–356, 359–361 multiplicities from, 292 bar19499_sndx_I1-I10.qxd 11/24/09 3:34 PM Page I-5 SUBJECT INDEX of parabolas, 111, 389–390 point-by-point plotting on, 111 of polynomial functions, 260–266, 280 of polynomials, 266, 291–292 of quadratic functions, 204–209 of rational functions, 299–301, 304–310 reflections of, 114, 191–193 stretching and shrinking in, 193–196 symmetry as aid in, 113–117 of systems of linear equations, 424–425 Greatest integer, 182 Greatest integer functions, 182, 183 Half-life, 342 Half-life decay model, 342 Horizontal asymptotes, of rational functions, 303–304 Horizontal axis, 110 See also x axis Horizontal lines, 139, 140 Horizontal line test, 237 Horizontal shifts, 189–191, 195 Horizontal shrinks, 194, 195 Horizontal stretches, 194, 195 Hyperbolas applications of, 412–414 conjugate, 410 equations of, 406–408 explanation of, 387, 405, 420 graphs of, 406–412 method to draw, 406 Hyperbolic paraboloids, 412 Hyperboloids, 412 Identities, 45 Identity functions, 179 Identity matrix, for multiplication, 470–471 Identity property, Imaginary numbers, 75 Imaginary unit, 74–75 Imaginary zeros, of polynomials, 290, 295 Implied domain, 166 Inconsistent systems, in two variables, 427 Increasing functions, 178, 238 Independent systems, 427 Independent variables, 164, 165 Index, 15 Induction See Mathematical induction Inequalities absolute value, 66–70 applications for, 61–62 double, 61, 67 equivalent, 59 explanation of, 57 graphs of, 58, 59 linear, 56–62, 105 polynomial, 283–284, 322 properties of, 60, 69 quadratic, 211–214, 252 radical, 71–72 rational, 310–311, 322–323 solution set for, 59–60 symbols for, 57 Infinite sequences explanation of, 505 geometric, 526–527 I-5 Infinite series explanation of, 507 geometric, 526–527 Infinity, symbol for, 57 Integer exponents explanation of, 11–12 properties of, 12–13 Integers explanation of, 2, greatest, 182, 183 set of, Intercepts See also x intercepts; y intercepts of functions, 176–177 to graph lines, 133 of rational functions, 305 Interest compound, 333–336, 373 explanation of, 333 Interest rate, 333 Interpolation, 153 Intersections, 59 Intervals explanation of, 57 graphs of, 58, 59 notation for, 57–58, 177 Inverse additive, 4, of functions, 238–242, 245–246 method to find, 476 multiplicative, 4, 6, 11, 471–472 to solve linear systems, 478–480, 498 of square matrix, 471–473, 476 Inverse functions explanation of, 235, 252 graphs of, 244–246 method for finding inverse and, 238–242 modeling with, 242–243 one-to-one, 235–238 properties of, 239 Inverse variation, 316–317 Irrational numbers explanation of, 2, historical background of, 74 Joint variation, 318 Lagranges’ four square theorem, 517 Leading term, 264 Learning curves, 344–345 Least common denominator (LCD), 35 Least-squares line, 383 Like terms, of polynomials, 23 Limited growth, 350 Linear and quadratic factors theorem, 290, 586 Linear equations See also Equations; Systems of linear equations explanation of, 104 with more than one variable, 46–47 in one variable, 45–46 Linear factors theorem, 289 Linear functions See also Functions explanation of, 178–179 graphs of, 179–180 Linear inequalities See also Inequalities applications for, 61–62 explanation of, 56, 57, 105 graphs of, 59–62 bar19499_sndx_I1-I10.qxd I-6 11/24/09 3:34 PM Page I-6 SUBJECT INDEX Linear models, construction of, 149–151 Linear regression examples of, 152–154 explanation of, 151 Linear systems See Systems of linear equations Line graph, 57 Lines equations of, 132, 133, 137–140, 158 graphs of, 132–133 horizontal, 139, 140 parallel, 141–142 perpendicular, 141–142 regression, 153 slope-intercept form of, 137–138 slope of, 134–136 vertical, 139, 140, 166 Line segment length of, 66 midpoint of, 124–126 Location theorem, 280–281 Logarithmic equations explanation of, 372, 380 method to solve, 375–376 Logarithmic-exponential conversions, 356–357 Logarithmic-exponential relationships, 360–361 Logarithmic functions change-of-base formula and, 361–362 conversions of, 356–357 explanation of, 329, 354, 379–380 graphs of, 354–356 properties of, 358–359, 380 Logarithmic models applications of, 380 data analysis and regression, 369 logarithmic scales, 365–369 Logarithmic scales, 365–369 Logarithms common, 359, 360 on graphing calculator, 359–361, 370 natural, 359, 360 Logistic growth, 350 Logistic models, 349 Long division explanation of, polynomial, 266–267 Lower triangular matrix, 571 Lowest terms, 32–33 Magnitude, 367 Mathematical induction examples of, 513–517 explanation of, 512–513, 564 extended principle of, 517 principle of, 512 Mathematical models applications of, 230–231, 242–243 explanation of, 147–148 exponential, 340–350 polynomial, 271–273, 285 quadratic, 210–211, 214–215 Matrices addition of, 457–458 applications of, 460–462, 464–465 augmented, 443–445, 447 column, 461–462 decoding, 482 explanation of, 442–443, 497 Gauss-Jordan elimination and, 447–451, 475 on graphing calculators, 442, 458, 473, 488 identity, 470–471 inverse methods to solve linear systems, 498 inverse of square, 471–473 lower triangular, 571 multiplication of, 459–466 negative of, 458 principal diagonal of, 442 properties of, 477 reduced, 444–447 row, 442, 461–462 row-equivalent, 444, 474 singular, 472 size of, 442 square, 442 subtraction of, 458–459 upper triangular, 495, 571 zero, 458 Matrix equations explanation of, 477 method to solve, 477–478 systems of linear equations and, 478–480 Midpoint, of line segment, 124–126 Midpoint formula explanation of, 124 use of, 125–126 Minor of element, in third-order determinant, 489 Mixture problems, 52–53 Models See Mathematical models Monomials, 22 See also Polynomials Multiplication associative property of, 4, commutative property of, 4, of complex numbers, 76–78 identity matrix for, 470–471 of matrices, 459–466 of polynomials, 24 of rational expressions, 33–34 of real numbers, 3–7 Multiplication principle application of, 533–534 explanation of, 532–533, 565 Multiplication properties of equality, 45 of matrices, 477 of real numbers, Multiplicative identity, 4, 78 Multiplicative inverse, 4, 6, 11, 471–472 Multiplicities from graphs, 292 of zero, 289, 291, 292 Multiplier doctrine, 527 Napierian logarithms See Natural logarithms Nappes, of cone, 386n Natural logarithms, 359, 360 Natural numbers, 2, 79 Negative growth, 342 Negative real numbers explanation of, principal square root of, 80 properties of, 7, n factorial, 534–535 Nonrepeating linear factors, 587–588 Nonrigid transformations, 193 Notation/symbols absolute value, 65 composition of function, 226, 228 bar19499_sndx_I1-I10.qxd 11/24/09 3:34 PM Page I-7 SUBJECT INDEX double subscript, 442 empty set, equality and inequality, 57 exponent, 11 factorial, 534–536 function, 167–169, 226, 228 infinity, 57 interval, 57–58, 177 parallel, 141 perpendicular, 141 radical, 15 real number, scientific, 13–14 summation, 507, 508 nth root explanation of, 14–15 principal, 15–16 nth-term formulas, 522–523 Null set, Number line, real, Numbers See also Integers complex, 74–82, 105 composite, 25 imaginary, 75 irrational, 2, 5, 74 natural, 2, 79 pure imaginary, 75 rational, 2, 3–7, 298 real, 2–9, 75 Numerator, Numerical coefficient, 22 See also Coefficients Oblique asymptotes, 308 Odd functions, 196–197 One-to-one functions explanation of, 235–236, 258 identification of, 236–238 Ordered pairs explanation of, 110n, 111 functions as sets of, 163–164 Ordering, 536 Ordinate, 110 Origin explanation of, 3, 110 reflection through, 114, 192, 193 symmetry and, 114, 115 Parabolas See also Quadratic functions applications of, 391–392 coordinate-free definition of, 387 equation of, 209, 388–391 explanation of, 111, 204, 387, 418–419 focal chord of, 393 graphs of, 111, 389–390 method to draw, 387–388 vertex of, 205–208 Paraboloids explanation of, 391, 392 hyperbolic, 412 Parallel lines, 141–142 Parallelograms, 598 Parameter, 432, 592–593 Parametric equations application of, 594–595 explanation of, 591–593 on graphing calculator, 592 Partial fraction decomposition, 586–590 Partial fractions, 585 Particular solutions, 432 PASCAL, 559 Pascal’s triangle, 559 Path of projectile, 594 Perfect square formula, 28, 29 Permutations, 536–538, 565 Piecewise-defined functions, 180–181 Plane, distance in, 123–129, 158 Plane curve, 592, 593 Point, coordinate of, Point-by-point plotting, 111 Point-slope form, 138–140 Polynomial functions explanation of, 260, 321–322 graphs of, 260–266, 280 left and right behavior of, 265 Polynomial inequalities explanation of, 283, 322 on graphing calculators, 284 method to solve, 283–284 Polynomials addition of, 23 bisection method and, 281–282 degree of, 22 division of, 266–269 equal, 585–586 evaluation of, 269–270 explanation of, 21–23, 40 factoring, 25–29 factors of, 270, 290–291 factor theorem and, 270 fundamental theorem of algebra and, 288–289 graphs of, 266, 291–292 location theorem and, 280–281 multiplication of, 24 in one variable, 22 prime, 25, 26 rational zeros of, 292–293, 322 with real coefficients, 290–291 real zeros of, 278–279 reduced, 294 remainder theorem and, 269–279 second-degree, 27–28 subtraction of, 24 Taylor, 365 in two variables, 22 zeros of, 266, 271 Positive real numbers, 3, 81 Predictions, 153 Price-demand equation, 93 Prime numbers, 25 Prime polynomials, 25, 26 Principal, 333, 334 Principal diagonal, 442, 488 Principal nth root, 15–16 Principle square root, of negative real number, 80 Probability actual, 552 empirical, 552–556 of events, 547–551 explanation of, 543 theoretical, 552 Probability function, 547 Problem solving See Word problems Product function, 224–225 Proper fractions, 585 I-7 bar19499_sndx_I1-I10.qxd I-8 11/24/09 3:34 PM Page I-8 SUBJECT INDEX Pure imaginary numbers, 75 Pythagoreans, 74 Pythagorean theorem, 92, 411, 598 Quadrants, 110 Quadratic equations applications for, 91–93 completing the square to solve, 87–89 explanation of, 84, 105 factoring to solve, 84–86 methods to solve, 100–102 quadratic formula to solve, 89–91 square root property to solve, 86–87 Quadratic factors, 589–590 Quadratic formula explanation of, 90 to solve quadratic equations, 89–90 use of, 294–295, 586 Quadratic functions explanation of, 204, 251–252 general form of, 204 graphs of, 204–209 modeling with, 210–211 properties of, 206 Quadratic inequalities explanation of, 211–212, 252 method to solve, 212–214 Quadratic regression, 214–215 Quadratic solving techniques applications using, 102 direct solution, 100 example of, 101 substitution method as, 101 Quantity-rate-time formula, 50 Quartic models, 273 Quotient function, 224–225 Quotients difference, 170 explanation of, 267 of functions, 226 Radical inequalities, 71–72 Radicals equations involving, 97–99 explanation of, 15, 39–40 properties of, 17, 81 in simplified form, 17–19 use of, 16–17 Radius, of circle, 127, 129 Random experiments, 543–546 Range, of functions, 163, 166, 177 Rate of change, 148–149 Rational exponents explanation of, 15–16 use of, 16–17 Rational expressions addition and subtraction of, 34–36 compound fraction, 36–37 explanation of, 32, 40 multiplication and division of, 33–34 reduced to lowest terms, 32–33 Rational functions domain and x intercepts of, 299 explanation of, 298–299, 322–323 graphs of, 299–301, 304–310 oblique asymptotes of, 308 properties of, 300–301 vertical and horizontal asymptotes of, 302–304 Rational inequalities explanation of, 310, 322–323 on graphing calculators, 311 method to solve, 310–311 Rationalizing factor, 18 Rationalizing the denominator, 18–19 Rational numbers addition and multiplication of, 3–7 explanation of, 2, 298 Rational zeros explanation of, 292–293, 322 method for finding, 293–295 Rational zero theorem, 293 Real number line, Real numbers addition of, 3–7 division of, explanation of, 2, 39, 75 multiplication of, 3–7 negative, 3, 7, 8, 80 positive, properties of, roots of, 14–15 set of, 2–3, 6, 8, 164 subtraction of, Real root, 84 Real zeros approximation of, 282–283 explanation of, 278, 322 upper and lower bound for, 278–279 Reciprocals, 78–79 Rectangles, 407, 598 Rectangular coordinate system See Cartesian coordinate system Rectangular solids, 599 Recursion formulas explanation of, 505 use of, 515–516 Recursive formula n factorial, 535 Reduced augmented coefficient, 450, 451 Reduced matrices, 444–447 Reduced polynomials, 294 Reduced system, 447 Reflections explanation of, 114 of graphs of functions, 191–193, 195 Regression on graphing calculators, 153 linear, 151–154 logarithmic, 369 quadratic, 214–215 Regression analysis, 151 Regression line, 153 Regression models, 383 Relative frequency, 553 Relative growth rate, 341 Remainder, 267 Remainder theorem, 269–270 Replacement set, 44 See also Domain Residuals, 383 Revenue, 93 Richter scale, 367 Right circular cones, 386n, 599 Right circular cylinders, 599 Rigid transformations, 193 bar19499_sndx_I1-I10.qxd 11/24/09 3:34 PM Page I-9 SUBJECT INDEX Rise, 134 Rocket equation, 368 Roots See also Square roots cube, 14 of equation, 176 of functions, 261–262 nth, 14–16 real, 84 of real numbers, 14–15 Rounding convention, 583–584 Row-equivalent matrices, 444 Row matrices, 442, 461–462 Row operations, 443 Run, 134 Sample spaces example of, 546 explanation of, 543–544 fundamental, 545 method to choose, 544–545 Scatter plots, 152 Scientific notation, 13–14 Second-degree polynomial functions See Quadratic functions Second-degree polynomials, factoring, 27–28 Second diagonal, 488 Second-order determinants, 487, 488 Sequences arithmetic, 520, 522–523 explanation of, 504–505, 564 Fibonacci, 505–506 finite, 505 general term of, 506–507 geometric, 521–523 on graphing calculators, 505, 507 infinite, 505 terms of, 504 Series explanation of, 507, 564 finite, 507 infinite, 507, 526–527 sum formulas for finite arithmetic, 523–524 sum formulas for geometric, 525–527 in summation notation, 508 terms of, 508 Sets of complex numbers, 75 empty or null, equal, of integers, intersection of, 59 of real numbers, 2–3, 6, 8, 164 replacement, 44 union of, 59 Shrinking, in graphs, 193–196 Significant digits, 582–583 Simple events, 544, 547, 550 Simple fractions, 36 Singular matrix, 472 Slope explanation of, 134 geometric interpretation of, 135 method to find, 134–136 of parallel lines, 141–142 of perpendicular lines, 141–142 as rate of change, 148–149 Slope-intercept form, 137–138, 140 Solutions of equations, 44, 111 extraneous, 98, 105 of linear systems, 424, 426–427 particular, 432 unique, 427 Solution set of equations, 44, 111 of inequalities, 59–60 of linear systems, 424 of quadratic inequalities, 211 Speed, 148 See also Rate of change Spheres, 599 Square functions, 188, 203, 204 Square matrices explanation of, 442 inverse of, 471–473, 476 of order n, 470–471 Square root functions, 189 Square root property, 86–87 Square roots, 14, 80 Squaring operation on equations, 98 Standard deck, 540 Standard form of complex numbers, 80 of equation of circle, 128 of equation of line, 133, 140 of linear equations, 45 of quadratic equations, 84, 100 quadratic inequalities in, 211 Stretching, in graphs, 193–196 Subset, Substitution to solve equations of quadratic type, 101 to solve linear systems, 427–428, 431, 432 Substitution property, of equality, 45 Subtraction of complex numbers, 76–77 of matrices, 458–459 of polynomials, 24 of rational expressions, 34–36 of real numbers, Subtraction properties, of equality, 45 Sum formulas for finite arithmetic series, 523–524 for finite geometric series, 525 for infinite geometric series, 526–527 Sum function, 224–225 Summation formula, proof of, 514–515 Summation notation, 507, 508 Summing index, 507 Sum of cubes formula, 28, 29 Sum of the squares of the residuals (SSR), 383 Supply, 435 Symbols See Notation/symbols Symmetry as aid in graphing, 113–117 axis of, 205 in even and odd functions, 197 tests for, 115–116 Symmetry property, 244–245 Synthetic division, 268–269 Synthetic division table, 279 Systems of linear equations applications of, 434–437, 452–454 basic terms of, 427 Cramer’s rule to solve, 491–494 I-9 bar19499_sndx_I1-I10.qxd I-10 11/24/09 3:34 PM Page I-10 SUBJECT INDEX Systems of linear equations—Cont elimination by addition to solve, 429–434 equivalent, 429 explanation of, 496 Gauss-Jordan elimination to solve, 441, 447–453, 497 graphs of, 424–425 matrices and row operations and, 441–447, 498 matrix equations and, 478–480 modeling with, 501–502 substitution method to solve, 427–428, 431, 432 in two variables, 424 Taylor polynomials, 365 Technology Connections See Graphing calculators Theorems, 511 Theoretical probability explanation of, 552 method to find, 553–554 Third-order determinants, 488–491 Transformations combining graph, 196 even and odd functions and, 196–197 explanation of, 189, 251 of exponential functions, 330 nonrigid, 193 reflections and, 191–193, 195 rigid, 193 stretching and shrinking and, 193–195 vertical and horizontal shifts and, 189–191, 195 Transverse axis, of hyperbola, 405 Trapezoids, 598 Tree diagrams, 532, 544 Triangles formulas for, 598 Pascal’s, 559 similar, 598 Trinomials, 22 See also Polynomials Triple zero, 289 Turning points approximating real zeros at, 282–283 explanation of, 262 Union, of sets, 59 Unique solution, 427 Unlimited growth, 350 Upper and lower bound theorem, 278, 279 Upper triangular matrix, 495, 571 Variables dependent, 164 domains of, 44–45 independent, 164, 165 Variation combined, 319 direct, 316 explanation of, 323 inverse, 316–317 joint, 318 Velocity, 148, 368 See also Rate of change Vertex form, of quadratic functions, 204 Vertical asymptotes, 302–304 Vertical axis, 110 See also y axis Vertical lines, graphs of, 139, 140, 166 Vertical line test, 166 Vertical shifts, 189–191, 195 Vertical shrinks, 194, 195 Vertical stretches, 194, 195 Vertices of cone, 386n of ellipse, 395 of hyperbola, 405 of parabola, 205, 206, 208, 387 Wiles, Andrew, 517 Word problems method to set up, 48, 91 mixture, 52–53 rate, 50–52 strategies to solve, 47, 104 using diagrams in solution of, 48–49 x axis reflection through, 114, 192, 193 symmetry and, 114, 115 x coordinate, 110, 176 x intercepts explanation of, 133 of functions, 176–177 of polynomial functions, 261–262 of rational functions, 299–300 y axis reflection through, 114, 192, 193 symmetry and, 114, 115 y coordinate, 110, 176 y intercepts explanation of, 133 of functions, 176–177 on graphing calculator, 134 Zero factorial, 534–535 Zero matrix, 458 Zero product property, 84–85 Zero property, of real numbers, Zeros complex, 77, 322 double, 289 of functions, 176, 261–262 imaginary, 290, 295 irrational, 294–295 multiplicities of, 289, 291, 292 of polynomials, 266, 271, 278–279 rational, 292–295, 322 real, 278–279, 282–283 triple, 289 bar19499_endsheet.qxd 11/23/09 4:52 PM Page Sets a෈A aԫA л 5x p(x)6 A ( B A´B AʝB Inequalities and Intervals a is an element of set A a is not an element of set A Empty or null set Set of all x such that p(x) is true A is a subset of B 5x x ෈ A or x ෈ B6, union 5x x ෈ A and x ෈ B6, intersection a Ͻ b a is less than b a Յ b a is less than or equal to b a Ͼ b a is greater than b a Ն b a is greater than or equal to b (a, b) Open interval; 5x a x b6 (a, b] Half-open interval; 5x a x Յ b6 [a, b) Half-open interval; 5x a Յ x b6 [a, b] Closed interval; 5x a Յ x Յ b6 Number Systems Absolute Value N Natural numbers Z Integers Q Rational numbers R Real numbers C Complex numbers NʚZʚQʚRʚC x if x Ն Ϫx if x 0 x ϭ x2 2x2 ϭ x 0 x p if and only if Ϫp x p; p 0 x p if and only if x Ϫp or x p; p 0x0 ϭ e Real Number Properties Quadratic Formula For all real numbers a, b, and c: a ϩ b ϭ b ϩ a; ab ϭ ba a ϩ (b ϩ c) ϭ (a ϩ b) ϩ c; a(bc) ϭ (ab)c a(b ϩ c) ϭ ab ϩ ac a ϩ ϭ a; ؒ a ϭ a a ϩ (Ϫa) ϭ 0; a(1րa) ϭ 1, a ab ϭ if and only if a ϭ or b ϭ Exponents and Radicals an ϭ a ؒ a a (n factors of a), n ෈ N a0 ϭ 1, a aϪn ϭ n , a 0, n ෈ R a n bm/n ϭ 2bm (nth root of bm) Special Products (a Ϫ b)(a ϩ b) ϭ a2 Ϫ b2 (a ϩ b)2 ϭ a2 ϩ 2ab ϩ b2 (a Ϫ b)2 ϭ a2 Ϫ 2ab ϩ b2 (a Ϫ b)(a2 ϩ ab ϩ b2) ϭ a3 Ϫ b3 (a ϩ b)(a2 Ϫ ab ϩ b2) ϭ a3 ϩ b3 Commutative properties Associative properties Distributive property Identities Inverses Zero property If ax2 ϩ bx ϩ c ϭ 0, a 0, then: Ϫb Ϯ 2b2 Ϫ 4ac xϭ 2a Rectangular Coordinates (x1, y1) d ϭ 2(x2 Ϫ x1)2 ϩ ( y2 Ϫ y1)2 x1 ϩ x2 y1 ϩ y2 , b 2 y2 Ϫ y1 mϭ , x1 x2 x2 Ϫ x1 a Coordinates of point P1 Distance between P1(x1, y1) and P2(x2, y2) Midpoint of line joining P1 and P2 Slope of line through P1 and P2 bar19499_endsheet.qxd 11/23/09 4:52 PM Page Function Notation Arithmetic Sequence Value of f at x Composite function Value of inverse of f at x f (x) ( f ° g)(x) ϭ f [g(x)] f Ϫ1(x) Linear Equations and Functions y ϭ mx ϩ b ( y Ϫ y1) ϭ m(x Ϫ x1) f (x) ϭ mx ϩ b yϭb xϭa Slope–intercept form Point–slope form Linear function Horizontal line Vertical line Polynomial and Rational Forms f (x) ϭ ax2 ϩ bx ϩ c f (x) ϭ an x n ϩ anϪ1x nϪ1 ϩ ϩ a1x ϩ a0, an 0, n a nonnegative integer p(x) f (x) ϭ , p and q polynomial q(x) functions, q(x) Quadratic function Polynomial function Rational function a1, a2, , an, an Ϫ anϪ1 ϭ d an ϭ a1 ϩ (n Ϫ 1)d Common difference nth-term formula n Sn ϭ a1 ϩ ϩ an ϭ [2a1 ϩ (n Ϫ 1)d ] n Sn ϭ (a1 ϩ an) Sum of n terms Geometric Sequence a1, a2, , an, an ϭr Common ratio anϪ1 nϪ1 an ϭ a1r nth-term formula a1 Ϫ a1r n , r Sn ϭ a1 ϩ ϩ an ϭ 1Ϫr a1 Ϫ ran , r Sn ϭ 1Ϫr a1 , 0r0 Sϱ ϭ a1 ϩ a2 ϩ ϭ 1Ϫr Sum of n terms Sum of infinitely many terms Factorial and Binomial Formulas Exponential and Logarithmic Functions f (x) ϭ b x, b 0, b Exponential function f (x) ϭ logb x, b 0, b Logarithmic function y ϭ logb x if and only if x ϭ b y, b 0, b n factorial n! ϭ n(n Ϫ 1) ؒ 1, n ෈ N 0! ϭ n n! , 0ՅrՅn a bϭ r!(n Ϫ r)! r n n (a ϩ b)n ϭ a a ba nϪk b k, n Ն Binomial formula kϭ0 k Matrices and Determinants a d b e c d f a †d g b e h c f† i c Matrix Determinant Permutations and Combinations For Յ r Յ n: n! Pn,r ϭ (n Ϫ r)! n n! Cn,r ϭ a b ϭ r r!(n Ϫ r)! Permutation Combination (Continued on back endpaper) bar19499_endsheet.qxd 11/23/09 4:52 PM Page Hyperbola Circle (x Ϫ h)2 ϩ ( y Ϫ k)2 ϭ r x2 ϩ y2 ϭ r y2 x2 Ϫ ϭ1 a2 b2 y x2 Ϫ 2ϭ1 a b Center at (h, k); radius r Center at (0, 0); radius r Foci: F¿(Ϫc, 0), F(c, 0); c2 ϭ a2 ϩ b2 Foci: F¿(0, Ϫc), F(0, c); c2 ϭ a2 ϩ b2 Parabola y2 ϭ 4ax, a Ͼ 0, opens right; a Ͻ 0, opens left Focus: (a, 0); Directrix: x ϭ Ϫa a Ͼ 0, opens up; a Ͻ 0, opens down Focus: (0, a); Directrix: y ϭ Ϫa x2 ϭ 4ay, Translation Formulas x ϭ x¿ ϩ h, y ϭ y¿ ϩ k ; New origin (h, k) x¿ ϭ x Ϫ h, y¿ ϭ y Ϫ k Ellipse y2 x2 ϩ ϭ 1, a2 b2 y x2 ϩ ϭ 1, b a a b Foci: F¿(Ϫc, 0), F(c, 0); c2 ϭ a2 Ϫ b2 a b Foci: F¿(0, Ϫc), F(0, c); c2 ϭ a2 Ϫ b2 Metric Units Standard Units of Metric Measure Important Prefixes Meter (m): length (approximately 3.28 ft) kilo (ϫ 1,000) deci (ϫ 101 ) Liter (L): volume (approximately 1.06 ft) hecto (ϫ 100) centi (ϫ 100 ) Gram (g): weight (approximately 0.035 oz) deka (ϫ 10) milli (ϫ 1,000 ) Abbreviations Length m km hm dam meter kilometer hectometer dekameter Volume dm decimeter cm centimeter mm millimeter L kL hL daL liter kiloliter hectoliter dekaliter Weight dL deciliter cL centiliter mL milliliter g kg hg dag gram kilogram hectogram dekagram dg decigram cg centigram mg milligram English-Metric Conversion Length Volume (U.S.) Weight Length Volume (U.S.) in ϭ 2.540 cm ft ϭ 30.48 cm yd ϭ 0.9144 m mi ϭ 1.609 km pt ϭ 0.4732 L qt ϭ 0.9464 L gal ϭ 3.785 L oz ϭ 28.35 g lb ϭ 453.6 kg lb ϭ 0.4536 kg cm ϭ 0.3937 in cm ϭ 0.03281 ft m ϭ 1.0936 yd km ϭ 0.6215 mi L ϭ 2.1133 pt L ϭ 1.0567 qt L ϭ 0.2642 gal Weight g ϭ 0.0353 oz g ϭ 0.002205 lb kg ϭ 2.205 lb ... Page i College Algebra bar19499_fm_i-xxxii.qxd 11/18/09 4:43 PM Page iii NINTH EDITION College Algebra Raymond A Barnett Merritt College Michael R Ziegler Marquette University Karl E Byleen Marquette... Cataloging-in-Publication Data College algebra / Raymond A Barnett [et al.] — 9th ed p cm Rev ed of: College algebra 8th ed / Raymond A Barnett, Michael R Ziegler, Karl E Byleen Includes index ISBN... Page iv COLLEGE ALGEBRA, NINTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2011 by The McGraw-Hill