Algebra and Trigonometry: Real Mathematics, Real People

1010 Pages·2011·214.5 MB·English

by  Ron Larson

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LibraryPirate
Library of Parent Functions Summary
Linear Function (p. 87) Absolute Value Function (p. 100) Square Root Function (p. 101)
(cid:9)
f(cid:2)x(cid:3)(cid:2) x f(cid:2)x(cid:3)(cid:2)(cid:8)x(cid:8)(cid:2) x, x ≥ 0 f(cid:2)x(cid:3)(cid:2)(cid:7)x
(cid:4)x, x < 0
y y y
2 4
3 f(x) = x
1
f(x) = x f(x) = ⏐x⏐
2
x x
(0, 0) −2 −1 (0, 0) 2 1
−1
x
−1 (0, 0) 2 3 4
−2 −1
Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:4)0, (cid:3)(cid:3)
Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Range: (cid:4)0, (cid:3)(cid:3) Range: (cid:4)0, (cid:3)(cid:3)
Intercept: (cid:2)0, 0(cid:3) Intercept: (cid:2)0, 0(cid:3) Intercept: (cid:2)0, 0(cid:3)
Increasing Decreasing on (cid:2)(cid:4)(cid:3), 0(cid:3) Increasing on (cid:2)0, (cid:3)(cid:3)
Increasing on (cid:2)0, (cid:3)(cid:3)
Even function
y-axis symmetry
Greatest Integer Function (p. 115) Quadratic Function (p. 246) Cubic Function (p. 255)
f(cid:2)x(cid:3)(cid:2) (cid:5)x(cid:6) f(cid:2)x(cid:3)(cid:2)ax2 f(cid:2)x(cid:3)(cid:2)x3
y y y
f(x) = [ [ x]]
3 4 3
2 3 2
1 2
(0, 0)
x 1 f(x) = x2 x
−3 −2 −1 1 2 3 −3 −2 1 2 3
−3 −2 −1 1 2 3 x −1 f(x) = x3
−1 (0, 0) −2
−3 −2 −3
Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Range: the set of integers Range: (cid:4)0, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
x-intercepts:in the interval (cid:4)0, 1(cid:3) Intercept: (cid:2)0, 0(cid:3) Intercept: (cid:2)0, 0(cid:3)
y-intercept: (cid:2)0, 0(cid:3) Decreasing on (cid:2)(cid:4)(cid:3), 0(cid:3) Increasing on (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Constant between each pair of Increasing on (cid:2)0, (cid:3)(cid:3) Odd function
consecutive integers Even function Origin symmetry
Jumps vertically one unit at Axis of symmetry: x(cid:2)0
each integer value Relative minimum or vertex: (cid:2)0, 0(cid:3)
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Rational Function (p. 29 al Function (p. 326) Logarithmic Function (p. 339)
1
f(cid:2)x(cid:3)(cid:2) f(cid:2)x(cid:3)(cid:2)ax, a > 0, a(cid:6)1 f(cid:2)x(cid:3)(cid:2)log x, a > 0, a(cid:6)1
x a
y y y
3 1
2 f(x) = x 1 f(x) = loga x
1 f(x) = ax f(x) = a−x
(0, 1) (1, 0)
x x
−1 1 2 3 1 2
x
−1
Domain: (cid:2)(cid:4)(cid:3), 0(cid:3)(cid:2)(cid:2)0, (cid:3)) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)0, (cid:3)(cid:3)
Range: (cid:2)(cid:4)(cid:3), 0(cid:3)(cid:2)(cid:2)0, (cid:3)) Range: (cid:2)0, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
No intercepts Intercept: (cid:2)0, 1(cid:3) Intercept: (cid:2)1, 0(cid:3)
Decreasing on (cid:2)(cid:4)(cid:3), 0(cid:3)and (cid:2)0, (cid:3)(cid:3) Increasing on (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Increasing on (cid:2)0, (cid:3)(cid:3)
Odd function for f(cid:2)x(cid:3)(cid:2)ax y-axis is a vertical asymptote
Origin symmetry Decreasing on (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Continuous
Vertical asymptote: y-axis for f(cid:2)x(cid:3)(cid:2)a(cid:4)x Reflection of graph of f(cid:2)x(cid:3)(cid:2)ax
Horizontal asymptote: x-axis x-axis is a horizontal asymptote in the line y (cid:2) x
Continuous
Sine Function (p. 433) Cosine Function (p. 433) Tangent Function (p. 444)
f(cid:2)x(cid:3)(cid:2)sin x f(cid:2)x(cid:3)(cid:2)cos x f(cid:2)x(cid:3)(cid:2)tan x
y y
y f(x) = tan x
3 3 3
f(x) = sin x f(x) = cos x
2 2
2
1
1
x x
x
−π π2 π −π −1 π2 π 2π −π π π 3π
2 2 2
−2 −2
−3 −3
(cid:5)
Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3) Domain:x (cid:6) (cid:7) n(cid:5)
2
Range: (cid:4)(cid:4)1, 1(cid:10) Range: (cid:4)(cid:4)1, 1(cid:10) Range:(cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Period: 2(cid:5) Period: 2(cid:5)
x-intercepts: (cid:2)n(cid:5), 0(cid:3) (cid:11)(cid:5) (cid:12) Period:(cid:5)
y-intercept: (cid:2)0, 0(cid:3) x-intercepts: 2 (cid:7)n(cid:5), 0 x-intercepts:(cid:2)n(cid:5), 0(cid:3)
y-intercept:(cid:2)0, 0(cid:3)
Odd function y-intercept: (cid:2)0, 1(cid:3)
(cid:5)
Origin symmetry Even function Vertical asymptotes: x(cid:2) (cid:7)n(cid:5)
2
y-axis symmetry
Odd function
Origin symmetry
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Cosecant Function (p. 447) p. 447) Cotangent Function (p. 446)
f(cid:2)x(cid:3)(cid:2) csc x f(cid:2)x(cid:3)(cid:2)sec x f(cid:2)x(cid:3)(cid:2)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
(cid:5)
Domain: x(cid:6) n(cid:5) Domain: x(cid:6) (cid:7)n(cid:5) Domain: x(cid:6)n(cid:5)
2
Range: (cid:2)(cid:4)(cid:3), (cid:4)1(cid:10)(cid:2)(cid:4)1, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:4)1(cid:10)(cid:2)(cid:4)1, (cid:3)(cid:3) Range: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Period: 2(cid:5) Period: 2(cid:5) Period: (cid:5)
No intercepts (cid:11)(cid:5) (cid:12)
Vertical asymptotes: x(cid:2)n(cid:5) y-intercept: (cid:2)0, 1(cid:3) x-intercepts: 2 (cid:7) n(cid:5), 0
Vertical asymptotes:
Odd function (cid:5) Vertical asymptotes: x (cid:2) n(cid:5)
Origin symmetry x(cid:2) (cid:7)n(cid:5) Odd function
2
Even function Origin symmetry
y-axis symmetry
Inverse Sine Function (p. 459) Inverse Cosine Function (p. 459) Inverse Tangent Function
(p. 459)
f(cid:2)x(cid:3)(cid:2) arcsin x f(cid:2)x(cid:3)(cid:2)arccos x f(cid:2)x(cid:3)(cid:2)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 π
− x −
2 −1 1 2
Domain: (cid:4)(cid:4)1, 1(cid:10) Domain: (cid:4)(cid:4)1, 1(cid:10) Domain: (cid:2)(cid:4)(cid:3), (cid:3)(cid:3)
Range: (cid:13)(cid:4)(cid:5), (cid:5)(cid:14) Range: (cid:4)0, (cid:5)(cid:10) Range: (cid:11)(cid:4)(cid:5), (cid:5)(cid:12)
2 2 (cid:11) (cid:5)(cid:12)
2 2
Intercept: (cid:2)0, 0(cid:3) y-intercept: 0, 2 Intercept: (cid:2)0, 0(cid:3)
Odd function (cid:5)
Horizontal asymptotes: y (cid:2)±
Origin symmetry 2
Odd function
Origin symmetry

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Algebra and Trigonometry
Real Mathematics, Real People
Sixth 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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− x 2 4
x 2
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
4 6 8 − x 2 1
3
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Algebra and Trigonometry: Real Mathematics, Real People © 2012, 2008 Brooks/Cole, Cengage Learning
Sixth Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright
Ron Larson
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Algebra and Trigonometry
Real M cs, Real People
Sixth Edition
Contents
Chapter P Prerequisites
1
P.1 Real Numbers 2
P.2 Exponents and Radicals 12
P.3 Polynomials and Factoring 23
P.4 Rational Expressions 35
P.5 The Cartesian Plane 46
P.6 Representing Data Graphically 57
Chapter Summary 66 Review Exercises 68
Chapter Test 71 Proofs in Mathematics 72
Chapter 1 Functions and Their Graphs
73
Introduction to Library of Parent Functions 74
1.1 Graphs of Equations 75
1.2 Lines in the Plane 84
1.3 Functions 97
1.4 Graphs of Functions 110
1.5 Shifting, Reflecting, and Stretching Graphs 122
1.6 Combinations of Functions 131
1.7 Inverse Functions 141
y
Chapter Summary 152 Review Exercises 154
Chapter Test 157 Proofs in Mathematics 158
Chapter 2 Solving Equations and Inequalities
159
2.1 Linear EquaLtions and Problem Solving 160
2.2 Solving Equation1s Graphically 170
L
2.3 Complex Numbers 180 3
2.4 Solving Quadratic Equations Algebraically 186
2.5 Solving Other Types of Equations Algebraically 200
2.6 Solving Inequalities Algebraically and Graphically 210
2.7 Linear Models and Scatter Plots 223
x
Chapter Summary 232 Review Exercises 234
Chapter Test 238 Cumulative Test: Chapters P–2 239
L
Proofs in Mathematics 241
2
x
Progressive Summary (Chapters P–2) 242
iii
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iv Contents
Chapter 3 Polyno nal Functions
243
3.1 Quadratic Functions 244
3.2 Polynomial Functions of Higher Degree 254
3.3 Real Zeros of Polynomial Functions 266
3.4 The Fundamental Theorem of Algebra 281
3.5 Rational Functions and Asymptotes 288
3.6 Graphs of Rational Functions 297
3.7 Quadratic Models 307
Chapter Summary 314 Review Exercises 316
Chapter Test 320 Proofs in Mathematics 321
Chapter 4 Exponential and Logarithmic Functions
323
4.1 Exponential Functions and Their Graphs 324
4.2 Logarithmic Functions and Their Graphs 336
4.3 Properties of Logarithms 347
4.4 Solving Exponential and Logarithmic Equations 354
4.5 Exponential and Logarithmic Models 365
4.6 Nonlinear Models 377
Chapter Summary 386 Review Exercises 388
Chapter Test 392 Cumulative Test: Chapters 3–4 393
Proofs in Mathematics 395
Progressive Summary (Chapters P–4) 396
Chapter 5 Trigonometric Functions
397
5.1 Angles and Their Measure 398
5.2 Right Triangle Trigonometry 409
5.3 Trigonometric Functions of Any Angle 420
5.4 Graphs of Sine and Cosine Functions 432
5.5 Graphs of Other Trigonometric Functions 444 y
5.6 Inverse Trigonometric Functions 455
5.7 Applications and Models 466
Chapter Summary 478 Review Exercises 480
Chapter Test 485 Library of Parent Functions Review 486
Proofs in Mathematics 488

Chapter 6 Analytic Trigonometry
489
6.1 Using Fundamental Identities 490
6.2 Verifying Trigonometric Identities 497
6.3 Solving Trigonometric Equations 505
6.4 Sum and Difference Formulas 517
6.5 Multiple-Angle and Product-to-Sum Formulas 524 x
Chapter Summary 534 Review Exercises 536
Chapter Test 539 Proofs in Mathematics 540
2x