Algebra and Trigonometry: Real Mathematics Real People

1017 Pages·2015·57.97 MB·English

by  Ron Larson

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• Worked-Out Solutions to all odd-numbered exercises at CalcChat.com
• Interactive Activities powered by Desmos
Real Mathematics, Real People 7e
• Worked-Out Solution Videos of all checkpoint exercises in English and Spanish
• Videos explaining the concepts of precalculus
• T ests to gauge progress before and after each chapter
• Projects for each chapter applying the concepts being taught
• D ownloadable Worksheets for each graphing exercise at MathGraphs.com
• Editable Spreadsheets of the data sets in the text
7e
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Library of Parent Functions Summary
Linear Function ( p. 87) Absolute Value Function ( p. 100) Square Root Function ( p. 101)
{x, x ≥ 0
f(x)=x f(x)=∣x∣= f(x)=√x
−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: (−∞, ∞) Domain: (−∞, ∞) Domain: [0, ∞)
Range: (−∞, ∞) Range: [0, ∞) Range: [0, ∞)
Intercept: (0, 0) Intercept: (0, 0) Intercept: (0, 0)
Increasing Decreasing on (−∞, 0) Increasing on (0, ∞)
Increasing on (0, ∞)
Even function
y-axis symmetry
Greatest Integer Function ( p. 115) Quadratic Function ( p. 246) Cubic Function ( p. 255)
f(x)=⟨x⟩ f(x)=x2 f(x)=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 11 22 33
x −1 f(x) = x3
−3 −2 −1 1 2 3
−1 (0, 0) −2
−3 −2 −3
Domain: (−∞, ∞) Domain: (−∞, ∞) Domain: (−∞, ∞)
Range: the set of integers Range: [0, ∞) Range: (−∞, ∞)
x-intercepts: in the interval [0, 1) Intercept: (0, 0) Intercept: (0, 0)
y-intercept: (0, 0) Decreasing on (−∞, 0) Increasing on (−∞, ∞)
Increasing on (0, ∞) Odd function
Constant between each pair of
Even function Origin symmetry
consecutive integers
Axis of symmetry: x=0
Jumps vertically one unit at
Relative minimum or vertex: (0, 0)
each integer value
Rational Function ( p. 298) Exponential Function ( p. 326) Logarithmic Function ( p. 339)
1
f(x)= f(x)=ax, a > 1 f(x)=log x, a > 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: (−∞, 0)∪(0, ∞) Domain: (−∞, ∞) Domain: (0, ∞)
Range: (−∞, 0)∪(0, ∞) Range: (0, ∞) Range: (−∞, ∞)
No intercepts Intercept: (0, 1) Intercept: (1, 0)
Decreasing on (−∞, 0) and (0, ∞) Increasing on (−∞, ∞) Increasing on (0, ∞)
Odd function for f(x)=ax y-axis is a vertical asymptote
Origin symmetry Decreasing on (−∞, ∞) Continuous
Vertical asymptote: y-axis for f(x)=a−x Reflection of graph of f(x)=ax
Horizontal asymptote: x-axis x-axis is a horizontal asymptote in the line y=x
Continuous
Sine Function ( p. 433) Cosine Function ( p. 433) Tangent Function ( p. 444)
f(x)=sin x f(x)=cos x f(x)=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 π π −π π π 2π π π π 3π
−
2 −1 2 2 2 2
−2 −2
−3 −3
Domain: (−∞, ∞) Domain: (−∞, ∞) Domain: x≠ π +nπ
Range: [−1, 1] Range: [−1, 1] 2
Period: 2π Period: 2π Range: (−∞, ∞)
xy--iinntteerrcceepptts: : (0(n, π0), 0) x-intercepts: (π2 +nπ, 0) Px-eirnitoedr:c eπpts: (nπ, 0)
Odd function y-intercept: (0, 1) y-intercept: (0, 0)
Origin symmetry Even function Vertical asymptotes: x= π +nπ
2
y-axis symmetry
Odd function
Origin symmetry
Cosecant Function ( p. 447) Secant Function ( p. 447) Cotangent Function ( p. 446)
f(x)=csc x f(x)=sec x f(x)=cot x
1 1 1
y f(x) = csc x = sin x y f(x) = sec x = cos x y f(x) = cot 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: x≠nπ Domain: x≠ π +nπ Domain: x≠nπ
Range: (−∞, −1]∪[1, ∞) 2 Range: (−∞, ∞)
Period: 2π Range: (−∞, −1]∪[1, ∞) Period: π
No intercepts Period: 2π x-intercepts: (π +nπ, 0)
Vertical asymptotes: x=nπ y-intercept: (0, 1) 2
Odd function Vertical asymptotes:
Vertical asymptotes: x=nπ
Origin symmetry π
x= +nπ Odd function
2
Origin symmetry
Even function
y-axis symmetry
Inverse Sine Function ( p. 459) Inverse Cosine Function ( p. 459) Inverse Tangent Function ( p. 459)
f(x)=arcsin x f(x)=arccos x f(x)=arctan x
y y y
π π
π
2 2
f(x) = arccos x
xx xx
−1 11 −2 −1 11 22
f(x) = arcsin x f(x) = arctan x
ππ π
− x −
2 2
−1 1
Domain: [−1, 1] Domain: [−1, 1] Domain: (−∞, ∞)
Range: [−π, π] Range: [0, π] Range: (−π, π)
2 2 2 2
y-intercept: (0, π)
Intercept: (0, 0) 2 Intercept: (0, 0)
Odd function π
Horizontal asymptotes: y=±
Origin symmetry 2
Odd function
Origin symmetry
Algebra and Trigonometry
Real Mathematics, Real People
Seventh 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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Ron Larson
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Printed in the United States of America
Print Number: 01 Print Year: 2014
Contents
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
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
Chapter Summary 152
Review Exercises 154
Chapter Test 157
Proofs in Mathematics 158
2 Solving Equations and Inequalities 159
2.1 Linear Equations and Problem Solving 160
2.2 Solving Equations Grayphically 170
y 2.3 Complex Numbers 180
2.4 Solving Quadratic Equations Algebraically 186
y 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
8
Chapter Summary 232
L
Review Exercises 234
1
3 Chapter Test 238 L
Cumulative Test: Chapters P–2 239 3
6
Proofs in Mathematics 241
Progressive Summary (Chapters P–2) 242
2
4
x
L
2
2
x
−
2 1 2 3
2x − x
1
f x
iii
4 6 8
x 3 −
2
−
3
iv Contents
3 Polynomial and Rational 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
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
5 Trigonometric Functions 397
5.1 Angles and Their Measure 398 y
y
5.2 Right Triangle Trigonometry 409
5.3 Trigonometric Functions of Any Angle 420
5.4 Graphs of Sine and Cosine Functions 432 y
5.5 Graphs of Other Trigonometric Functions 444
5.6 Inverse Trigonometric Functions 455
8
5.7 Applications and Models 466
L
Chapter Summary 478
1
Review Exercises 480 3 L
Chapter Test 485 3
6
Library of Parent Functions Review 486
Proofs in Mathematics 488
2
4
x
L
2
2
x
−
2 1 2 3
2x − x
1
f x
4 6 8
x 3 −
2
−
3
Contents v
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
Chapter Summary 534
Review Exercises 536
Chapter Test 539
Proofs in Mathematics 540
7 Additional Topics in Trigonometry 543
7.1 Law of Sines 544
7.2 Law of Cosines 553
7.3 Vectors in the Plane 560
7.4 Vectors and Dot Products 574
7.5 Trigonometric Form of a Complex Number 583
Chapter Summary 596
Review Exercises 598
Chapter Test 601
Cumulative Test: Chapters 5–7 602
Proofs in Mathematics 604
Progressive Summary (Chapters P–7) 608
8 Linear Systems and Matrices 609
8.1 Solving Systems of Equations 610
8.2 Systems of Linear Equations in Two Variables 620
8.3 Multivariable Linear Systems 629
8.4 Matrices and Systemsy of Equations 644
y
8.5 Operations with Matrices 658
8.6 The Inverse of a Square Matrix 672
y 8.7 The Determinant of a Square Matrix 681
8.8 Applications of Matrices and Determinants 688
Chapter Summary 698
8
Review Exercises 700
L
Chapter Test 706
1
3 Proofs in Mathematics 707 L
3
6

2
4
x
L
2
2
x
−
2 1 2 3
2x − x
1
f x
4 6 8
x 3 −
2
−
3