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www.elsolucionario.org GRAPHS OF PARENT FUNCTIONS Linear Function Absolute Value Function x, x Ն f ͑x͒ ϭ ԽxԽ ϭ ΆϪx, f ͑x͒ ϭ mx ϩ b y Square Root Function f ͑x͒ ϭ Ίx x < y y x (− mb , 0( (− mb , 0( f(x) = mx + b, m>0 (0, b) −2 f(x) = ⏐x⏐ 2 −1 f(x) = mx + b, m0 x −1 −1 Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ x-intercept: ͑Ϫb͞m, 0͒ y-intercept: ͑0, b͒ Increasing when m > Decreasing when m < y x x (0, 0) −1 f(x) = f(x) = ax , a < (0, 0) −3 −2 −1 −2 −2 −3 −3 Domain: ͑Ϫ ϱ, ϱ͒ Range ͑a > 0͒: ͓0, ϱ͒ Range ͑a < 0͒ : ͑Ϫ ϱ, 0͔ Intercept: ͑0, 0͒ Decreasing on ͑Ϫ ϱ, 0͒ for a > Increasing on ͑0, ϱ͒ for a > Increasing on ͑Ϫ ϱ, 0͒ for a < Decreasing on ͑0, ϱ͒ for a < Even function y-axis symmetry Relative minimum ͑a > 0͒, relative maximum ͑a < 0͒, or vertex: ͑0, 0͒ x f(x) = x Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Intercept: ͑0, 0͒ Increasing on ͑Ϫ ϱ, ϱ͒ Odd function Origin symmetry Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Rational (Reciprocal) Function f ͑x͒ ϭ x Exponential Function Logarithmic Function f ͑x͒ ϭ ax, a > f ͑x͒ ϭ loga x, a > y y y f(x) = x f(x) = a −x f(x) = a x 1 (1, 0) (0, 1) x −1 f(x) = loga x x x −1 Domain: ͑Ϫ ϱ, 0͒ ʜ ͑0, ϱ) Range: ͑Ϫ ϱ, 0͒ ʜ ͑0, ϱ) No intercepts Decreasing on ͑Ϫ ϱ, 0͒ and ͑0, ϱ͒ Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑0, ϱ͒ Intercept: ͑0, 1͒ Increasing on ͑Ϫ ϱ, ϱ͒ for f ͑x͒ ϭ ax Decreasing on ͑Ϫ ϱ, ϱ͒ for f ͑x͒ ϭ aϪx Horizontal asymptote: x-axis Continuous Domain: ͑0, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Intercept: ͑1, 0͒ Increasing on ͑0, ϱ͒ Vertical asymptote: y-axis Continuous Reflection of graph of f ͑x͒ ϭ ax in the line y ϭ x Sine Function f ͑x͒ ϭ sin x Cosine Function f ͑x͒ ϭ cos x Tangent Function f ͑x͒ ϭ tan x y y y 3 f(x) = sin x 2 f(x) = cos x 1 x −π f(x) = tan x π π 2π x −π − π π −2 −2 −3 −3 Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓Ϫ1, 1͔ Period: 2 x-intercepts: ͑n, 0͒ y-intercept: ͑0, 0͒ Odd function Origin symmetry π 2π Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓Ϫ1, 1͔ Period: 2 x-intercepts: ϩ n , y-intercept: ͑0, 1͒ Even function y-axis symmetry x − π π π 3π ϩ n Range: ͑Ϫ ϱ, ϱ͒ Period: x-intercepts: ͑n, 0͒ y-intercept: ͑0, 0͒ Vertical asymptotes: x ϭ ϩ n Odd function Origin symmetry Domain: all x Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Cosecant Function f ͑x͒ ϭ csc x y Secant Function f ͑x͒ ϭ sec x sin x f(x) = csc x = y Cotangent Function f ͑x͒ ϭ cot x f(x) = sec x = cos x y 3 2 f(x) = cot x = tan x x x −π π π 2π −π − π π π 3π 2π x −π − π π π 2π −2 −3 ϩ n Range: ͑Ϫ ϱ, Ϫ1͔ ʜ ͓1, ϱ͒ Period: 2 y-intercept: ͑0, 1͒ Vertical asymptotes: x ϭ ϩ n Even function y-axis symmetry Domain: all x n Range: ͑Ϫ ϱ, Ϫ1͔ ʜ ͓1, ϱ͒ Period: 2 No intercepts Vertical asymptotes: x ϭ n Odd function Origin symmetry Domain: all x Inverse Sine Function f ͑x͒ ϭ arcsin x Inverse Cosine Function f ͑x͒ ϭ arccos x y Domain: all x n Range: ͑Ϫ ϱ, ϱ͒ Period: ϩ n , x-intercepts: Vertical asymptotes: x ϭ n Odd function Origin symmetry Inverse Tangent Function f ͑x͒ ϭ arctan x y y π π π f(x) = arccos x x −1 −2 x −1 f(x) = arcsin x −π Domain: ͓Ϫ1, 1͔ Range: Ϫ , 2 Intercept: ͑0, 0͒ Odd function Origin symmetry ΄ ΅ f(x) = arctan x −π x −1 Domain: ͓Ϫ1, 1͔ Range: ͓0, ͔ y-intercept: 0, Domain: ͑Ϫ ϱ, ϱ͒ Range: Ϫ , 2 Intercept: ͑0, 0͒ Horizontal asymptotes: yϭ± Odd function Origin symmetry www.elsolucionario.org Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Precalculus with Limits Third Edition Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C Falvo The Pennsylvania State University The Behrend College Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Precalculus with Limits Third Edition Ron Larson Publisher: Liz Covello Acquisitions Editor: Gary Whalen Senior Development Editor: Stacy Green Assistant Editor: Cynthia Ashton © 2014, 2011, 2007 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher Editorial Assistant: Samantha Lugtu Media Editor: Lynh Pham For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 Senior Content Project Manager: Jessica Rasile For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Art Director: Linda May Rights Acquisition Specialist: Shalice Shah-Caldwell Manufacturing Planner: Doug Bertke Text/Cover Designer: Larson Texts, Inc Compositor: Larson Texts, Inc Library of Congress Control Number: 2012948314 Cover Image: diez artwork/Shutterstock.com Student Edition: ISBN-13: 978-1-133-94720-2 ISBN-10: 1-133-94720-4 Brooks/Cole 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Instructors: Please visit login.cengage.com and log in to access instructor-specific resources Printed in the United States of America 16 15 14 13 12 Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest www.elsolucionario.org Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents Functions and Their Graphs 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 76 Polynomial and Rational Functions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Rectangular Coordinates Graphs of Equations 11 Linear Equations in Two Variables 22 Functions 35 Analyzing Graphs of Functions 49 A Library of Parent Functions 60 Transformations of Functions 67 Combinations of Functions: Composite Functions Inverse Functions 84 Mathematical Modeling and Variation 93 Chapter Summary 104 Review Exercises 106 Chapter Test 109 Proofs in Mathematics 110 P.S Problem Solving 111 Exponential and Logarithmic Functions 3.1 3.2 3.3 3.4 3.5 113 Quadratic Functions and Models 114 Polynomial Functions of Higher Degree 124 Polynomial and Synthetic Division 138 Complex Numbers 147 Zeros of Polynomial Functions 154 Rational Functions 168 Nonlinear Inequalities 180 Chapter Summary 190 Review Exercises 192 Chapter Test 194 Proofs in Mathematics 195 P.S Problem Solving 197 199 Exponential Functions and Their Graphs 200 Logarithmic Functions and Their Graphs 211 Properties of Logarithms 221 Exponential and Logarithmic Equations 228 Exponential and Logarithmic Models 238 Chapter Summary 250 Review Exercises 252 Chapter Test 255 Cumulative Test for Chapters 1–3 256 Proofs in Mathematics 258 P.S Problem Solving 259 iii Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it iv Contents Trigonometry 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 349 Using Fundamental Identities 350 Verifying Trigonometric Identities 357 Solving Trigonometric Equations 364 Sum and Difference Formulas 375 Multiple-Angle and Product-to-Sum Formulas Chapter Summary 391 Review Exercises 393 Chapter Test 395 Proofs in Mathematics 396 P.S Problem Solving 399 382 Additional Topics in Trigonometery 6.1 6.2 6.3 6.4 6.5 Radian and Degree Measure 262 Trigonometric Functions: The Unit Circle 272 Right Triangle Trigonometry 279 Trigonometric Functions of Any Angle 290 Graphs of Sine and Cosine Functions 299 Graphs of Other Trigonometric Functions 310 Inverse Trigonometric Functions 320 Applications and Models 330 Chapter Summary 340 Review Exercises 342 Chapter Test 345 Proofs in Mathematics 346 P.S Problem Solving 347 Analytic Trigonometry 5.1 5.2 5.3 5.4 5.5 261 Law of Sines 402 Law of Cosines 411 Vectors in the Plane 418 Vectors and Dot Products 431 Trigonometric Form of a Complex Number Chapter Summary 450 Review Exercises 452 Chapter Test 456 Cumulative Test for Chapters 4–6 457 Proofs in Mathematics 459 P.S Problem Solving 463 401 440 Systems of Equations and Inequalities 7.1 7.2 7.3 7.4 7.5 7.6 Linear and Nonlinear Systems of Equations Two-Variable Linear Systems 476 Multivariable Linear Systems 488 Partial Fractions 500 Systems of Inequalities 508 Linear Programming 518 Chapter Summary 527 Review Exercises 529 Chapter Test 533 Proofs in Mathematics 534 P.S Problem Solving 535 465 466 Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents Matrices and Determinants 8.1 8.2 8.3 8.4 8.5 10 582 Sequences, Series, and Probability Sequences and Series 606 Arithmetic Sequences and Partial Sums Geometric Sequences and Series 625 Mathematical Induction 634 The Binomial Theorem 644 Counting Principles 652 Probability 662 Chapter Summary 674 Review Exercises 676 Chapter Test 679 Cumulative Test for Chapters 7–9 680 Proofs in Mathematics 682 P.S Problem Solving 685 605 616 Topics in Analytic Geometry 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11 537 Matrices and Systems of Equations 538 Operations with Matrices 551 The Inverse of a Square Matrix 565 The Determinant of a Square Matrix 574 Applications of Matrices and Determinants Chapter Summary 594 Review Exercises 596 Chapter Test 601 Proofs in Mathematics 602 P.S Problem Solving 603 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Lines 688 Introduction to Conics: Parabolas Ellipses 704 Hyperbolas 713 Rotation of Conics 723 Parametric Equations 731 Polar Coordinates 741 Graphs of Polar Equations 747 Polar Equations of Conics 755 Chapter Summary 762 Review Exercises 764 Chapter Test 767 Proofs in Mathematics 768 P.S Problem Solving 771 687 695 Analytic Geometry in Three Dimensions 11.1 11.2 11.3 11.4 v The Three-Dimensional Coordinate System Vectors in Space 781 The Cross Product of Two Vectors 788 Lines and Planes in Space 795 Chapter Summary 804 Review Exercises 806 Chapter Test 808 Proofs in Mathematics 809 P.S Problem Solving 811 773 774 www.elsolucionario.org Copyright 2012 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... www.cengagebrain.com Instructors: Please visit login .cengage. com and log in to access instructor-specific resources Printed in the United States of America 16 15 14 13 12 Copyright 2012 Cengage Learning. .. problem-specific tutorials CengageBrain.com Visit www.cengagebrain.com to access additional course 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Precalculus with Limits Third Edition Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C Falvo The Pennsylvania State University The Behrend College
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