org GRAPHS OF PARENT FUNCTIONS Linear Function Absolute Value Function Square Root Function f 共x兲 ⫽ mx ⫹ b f 共x兲 ⫽ ⱍxⱍ ⫽ x, x ⱖ 0 冦⫺x, x < 0 f 共x兲 ⫽ 冪x y y y 2 4 3 f(x) = x (0, b) 1 f(x) = ⏐x⏐ 2 x x (− mb , 0( (− mb , 0( −2 −1 (0, 0) 2 1 −1 x f(x) = mx + b, f(x) = mx + b, −1 (0, 0) 2 3 4 m>0 m<0 −2 −1 Domain: 共⫺ ⬁, ⬁兲 Domain: 共⫺ ⬁, ⬁兲 Domain: 关0, ⬁兲 Range: 共⫺ ⬁, ⬁兲 Range: 关0, ⬁兲 Range: 关0, ⬁兲 x-intercept: 共⫺b兾m, 0兲 Intercept: 共0, 0兲 Intercept: 共0, 0兲 y-intercept: 共0, b兲 Decreasing on 共⫺ ⬁, 0兲 Increasing on 共0, ⬁兲 Increasing when m > 0 Increasing on 共0, ⬁兲 Decreasing when m < 0 Even function y-axis symmetry Greatest Integer Function Quadratic (Squaring) Function Cubic Function f 共x兲 ⫽ 冀x冁 f 共x兲 ⫽ ax2 f 共x兲 ⫽ x3 y y y f(x) = [[x]] 3 3 3 2 2 2 1 1 f(x) = ax 2 , a > 0 (0, 0) x x x −3 −2 −1 1 2 3 −2 −1 1 2 3 4 −3 −2 1 2 3 −1 f(x) = ax 2 , a < 0 −1 f(x) = x 3 −2 −2 −3 −3 −3 Domain: 共⫺ ⬁, ⬁兲 Domain: 共⫺ ⬁, ⬁兲 Domain: 共⫺ ⬁, ⬁兲 Range: the set of integers Range 共a > 0兲: 关0, ⬁兲 Range: 共⫺ ⬁, ⬁兲 x-intercepts: in the interval 关0, 1兲 Range 共a < 0兲 : 共⫺ ⬁, 0兴 Intercept: 共0, 0兲 y-intercept: 共0, 0兲 Intercept: 共0, 0兲 Increasing on 共⫺ ⬁, ⬁兲 Constant between each pair of Decreasing on 共⫺ ⬁, 0兲 for a > 0 Odd function consecutive integers Increasing on 共0, ⬁兲 for a > 0 Origin symmetry Jumps vertically one unit at Increasing on 共⫺ ⬁, 0兲 for a < 0 each integer value Decreasing on 共0, ⬁兲 for a < 0 Even function y-axis symmetry Relative minimum 共a > 0兲, relative maximum 共a < 0兲, or vertex: 共0, 0兲 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 Exponential Function Logarithmic Function 1 f 共x兲 ⫽ f 共x兲 ⫽ ax, a > 1 f 共x兲 ⫽ loga x, a > 1 x y y y 3 1 f(x) = f(x) = loga x x 1 2 1 f(x) = a x f(x) = a −x (0, 1) (1, 0) x x −1 1 2 3 1 2 x −1 Domain: 共⫺ ⬁, 0兲 傼 共0, ⬁) Domain: 共⫺ ⬁, ⬁兲 Domain: 共0, ⬁兲 Range: 共⫺ ⬁, 0兲 傼 共0, ⬁) Range: 共0, ⬁兲 Range: 共⫺ ⬁, ⬁兲 No intercepts Intercept: 共0, 1兲 Intercept: 共1, 0兲 Decreasing on 共⫺ ⬁, 0兲 and 共0, ⬁兲 Increasing on 共⫺ ⬁, ⬁兲 Increasing on 共0, ⬁兲 Odd function for f 共x兲 ⫽ ax Vertical asymptote: y-axis Origin symmetry Decreasing on 共⫺ ⬁, ⬁兲 Continuous Vertical asymptote: y-axis for f 共x兲 ⫽ a⫺x Reflection of graph of f 共x兲 ⫽ ax Horizontal asymptote: x-axis Horizontal asymptote: x-axis in the line y ⫽ x Continuous Sine Function Cosine Function Tangent Function f 共x兲 ⫽ sin x f 共x兲 ⫽ cos x f 共x兲 ⫽ tan x y y y f(x) = tan x 3 3 3 2 f(x) = sin x 2 f(x) = cos x 2 1 1 x x x −π π π 2π −π π π π 2π − π π π 3π 2 2 2 − 2 2 2 −2 −2 −3 −3 Domain: 共⫺ ⬁, ⬁兲 Domain: 共⫺ ⬁, ⬁兲 Domain: all x ⫽ ⫹ n 2 Range: 关⫺1, 1兴 Range: 关⫺1, 1兴 Range: 共⫺ ⬁, ⬁兲 Period: 2 Period: 2 Period: x-intercepts: 共n, 0兲 y-intercept: 共0, 0兲 x-intercepts: 2 ⫹ n , 0冢 冣 x-intercepts: 共n, 0兲 y-intercept: 共0, 0兲 Odd function y-intercept: 共0, 1兲 Vertical asymptotes: Origin symmetry Even function y-axis symmetry x ⫽ ⫹ n 2 Odd function Origin symmetry Copyright 2012 Cengage Learning. All Rights Reserved.
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Cosecant Function Secant Function Cotangent Function f 共x兲 ⫽ csc x f 共x兲 ⫽ sec x f 共x兲 ⫽ cot x 1 1 1 y f(x) = csc x = y f(x) = sec x = y f(x) = cot x = sin x cos x tan x 3 3 3 2 2 2 1 1 x x x −π π π 2π −π π π π 3π 2π −π π π π 2π − − 2 2 2 2 2 2 −2 −3 Domain: all x ⫽ n Domain: all x ⫽ ⫹ n Domain: all x ⫽ n Range: 共⫺ ⬁, ⫺1兴 傼 关1, ⬁兲 2 Range: 共⫺ ⬁, ⬁兲 Period: 2 Range: 共⫺ ⬁, ⫺1兴 傼 关1, ⬁兲 Period: No intercepts Period: 2 Vertical asymptotes: x ⫽ n y-intercept: 共0, 1兲 Vertical asymptotes: x-intercepts: 2 ⫹ n , 0 冢 冣 Odd function Vertical asymptotes: x ⫽ n Origin symmetry x ⫽ ⫹ n Odd function 2 Even function Origin symmetry y-axis symmetry Inverse Sine Function Inverse Cosine Function Inverse Tangent Function f 共x兲 ⫽ arcsin x f 共x兲 ⫽ arccos x f 共x兲 ⫽ arctan x y y y π π π 2 2 f(x) = arccos x x x −1 1 −2 −1 1 2 f(x) = arcsin x f(x) = arctan x −π x −π 2 2 −1 1 Domain: 关⫺1, 1兴 Domain: 关⫺1, 1兴 Domain: 共⫺ ⬁, ⬁兲 Range: 关0, 兴 Range: ⫺ , 冤 2 2 冥 Range: ⫺ , 2 2 冢 冣 Intercept: 共0, 0兲 y-intercept: 0, 2 冢 冣 Intercept: 共0, 0兲 Odd function Horizontal asymptotes: Origin symmetry y⫽± 2 Odd function Origin symmetry www.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.
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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents 1 Functions and Their Graphs 1 1.2 Graphs of Equations 11 1.3 Linear Equations in Two Variables 22 1.5 Analyzing Graphs of Functions 49 1.6 A Library of Parent Functions 60 1.7 Transformations of Functions 67 1.8 Combinations of Functions: Composite Functions 76 1.10 Mathematical Modeling and Variation 93 Chapter Summary 104 Review Exercises 106 Chapter Test 109 Proofs in Mathematics 110 P. Problem Solving 111 2 Polynomial and Rational Functions 113 2.1 Quadratic Functions and Models 114 2.2 Polynomial Functions of Higher Degree 124 2.3 Polynomial and Synthetic Division 138 2.5 Zeros of Polynomial Functions 154 2.7 Nonlinear Inequalities 180 Chapter Summary 190 Review Exercises 192 Chapter Test 194 Proofs in Mathematics 195 P. Problem Solving 197 3 Exponential and Logarithmic Functions 199 3.1 Exponential Functions and Their Graphs 200 3.2 Logarithmic Functions and Their Graphs 211 3.3 Properties of Logarithms 221 3.4 Exponential and Logarithmic Equations 228 3.5 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.
Problem Solving 259 iii Copyright 2012 Cengage Learning. All Rights Reserved.