Precalculus with CalcChat and CalcView

Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

GRAPHS OF PARENT FUNCTIONS
Linear Function Absolute Value Function Square Root Function
f(x)=mx+b f (x)=∣
x∣
={
x,
−x,
x ≥0
x
<0

f(x)=√x
x
y
(0, b)
b
m( (− , 0
b
m( (− , 0
f(x) = mx + b,
m > 0
f(x) = mx + b,
m < 0
x
y
−1−22
−1
−2
1
2
(0, 0)
f(x) = ⎮x⎮
x
y
−1 234
−1
1
2
3
4 (0, 0)
f(x) = x
Domain: (−∞, ∞) Domain: (−∞, ∞) Domain: [0, ∞)
Range (m≠0): (−∞, ∞) Range: [0, ∞) Range: [0, ∞)
x-intercept:(−bcm, 0) Intercept: (0, 0) Intercept: (0, 0)
y-intercept:(0, b) Decreasing on (−∞, 0) Increasing on (0, ∞)
Increasing when m >0 Increasing on (0, ∞)
Decreasing when m <0 Even function
y-axis symmetry
Greatest Integer Function Quadratic (Squaring) Function Cubic Function
f(x)=⟨x⟩

f(x)=ax
2
f(x)=x
3
x
y
1−1−2−3 23
−3
1
2
3
f(x) = x[[ ]]
x
y
−1−2 1234
1
−1
−2
−3
2
3
f(x) =ax,a>0
2
f(x) =ax,a<0
2
x
y
(0, 0)
f(x) = x
3
−2−3 123
−2
−1
−3
2
3
Domain: (−∞, ∞) Domain: (−∞, ∞) Domain: (−∞, ∞)
Range: the set of integers Range (a>0):[0, ∞) Range: (−∞, ∞)
x-intercepts: in the interval [0, 1) Range (a<0): (−∞, 0] Intercept: (0, 0)
y-intercept:(0, 0) Intercept: (0, 0) Increasing on (−∞, ∞)
Constant between each pair of Decreasing on (−∞, 0) for a >0 Odd function
consecutive integers Increasing on (0, ∞) for a >0 Origin symmetry
Jumps vertically one unit at Increasing on (−∞, 0) for a <0
each integer value Decreasing on (0, ∞) for a <0
Even function
y-axis symmetry
Relative minimum (a>0),
relative maximum (a<0),
or vertex: (0, 0)Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

Rational (Reciprocal) Function Exponential Function Logarithmic Function
f(x)=
1
x
f(x)=a
x
, a>1 f (x)=log
a
x, a >1
x
y
f(x) =
1
x
−1 123
1
2
3
x
y
(0, 1)
f(x) = a
−x
f(x) = a
x
x
y
f(x) = log
a
x
12
−1
1
(1, 0)
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)=a
x
Vertical asymptote: y-axis
Origin symmetry Decreasing on (−∞, ∞) Continuous
Vertical asymptote: y-axis for f(x)=a
−x
Reflection of graph of f(x)=a
x
Horizontal asymptote: x-axis Horizontal asymptote: x-axis in the line y=x
Continuous
Sine Function Cosine Function Tangent Function
f(x)=sin x f(x)=cos x f(x)=tan x
1
−2
−3
2
3
πππ
2
−
f(x) = sin x
x
2
y
π
−2
−3
2
3
π ππ
2
π
π
2
−
−
f(x) = cos x
x
2
y
2
1
3
ππ
2
π
2
−
f(x) = tan x
x
y
3π
2
Domain: (−∞, ∞)
Range: [−1, 1]
Period: 2π
x-intercepts: (nπ, 0)
y-intercept: (0, 0)
Odd function
Origin symmetry

Domain: (−∞, ∞)
Range: [−1, 1]
Period: 2π
x-intercepts: (
π
2
+nπ, 0)
y-intercept:
(0, 1)
Even function
y-axis symmetry
Domain: all x≠
π
2
+nπ
Range: (−∞, ∞)
Period: π
x-intercepts: (nπ, 0)
y-intercept: (0, 0)
Vertical asymptotes:
x=
π
2
+nπ
Odd function
Origin symmetryCopyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

Cosecant Function Secant Function Cotangent Function
f(x)=csc x f(x)=sec x f(x)=cot x
2
1
3
ππ
π
2
−
f(x) = csc x =
x
y
2
π
1
sin x
2
−2
−3
3
πππ
2
π
2
−
−
x
y
3
π2π
2
f(x) = sec x =
1
cos x
2
1
3
πππ
2
π
2
−
−
f(x) = cot x =
x
y
2
π
1
tan x
Domain: all x≠nπ Domain: all x≠
π
2
+nπ
Range: (−∞, −1]∪[1, ∞)
Period: 2π
y-intercept: (0, 1)
Vertical asymptotes:
x=
π
2
+nπ
Even function
y-axis symmetry
Domain: all x≠nπ
Range: (−∞, ∞)
Period: π
x-intercepts:
(
π
2
+nπ, 0)
Vertical asymptotes:
x=nπ
Odd function Origin symmetry
Range:
(−∞, −1]∪[1, ∞)
Period: 2π
No intercepts Vertical asymptotes:
x=nπ
Odd function Origin symmetry
Inverse Sine Function Inverse Cosine Function Inverse Tangent Function
f(x)=arcsin x

f(x)=arccos x f(x)=arctan x
x
y
−11
2
2
π
π
−
f(x) = arcsin x
x
y
−11
f(x) = arccos x
π
x
y
−1−2 12
f(x) = arctan x
2
π
−
2
π
Domain: [−1, 1] Domain: [−1, 1]
Range: [0, π]
y-intercept: (
0,
π
2)
Domain: (−∞, ∞)
Range: (
−
π
2
,
π
2)
Intercept: (0, 0)
Horizontal asymptotes:
y=±
π
2
Odd function
Origin symmetry
Range: [
−
π
2
,
π
2]
Intercept: (0, 0)
Odd function Origin symmetryCopyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

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 • Mexico • Singapore • United Kingdom • United StatesCopyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

Precalculus
with CalcChat and CalcView
Tenth Edition
Ron Larson
Product Director: Terry Boyle
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Compositor: Larson Texts, Inc.
© 2018, 2014 Cengage Learning
ALL RIGHTS RESERVED. No part of this work covered by the copyright
herein may be reproduced or distributed in any form or by any means,
except as permitted by U.S. copyright law, without the prior written
permission of the copyright owner.
Library of Congress Control Number: 2016944978
Student Edition:
ISBN: 978-1-337-27107-3
Loose-leaf Edition:
ISBN: 978-1-337-29158-3
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Printed in the United States of America
Print Number: 01 Print Year: 2016Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

iii
Contents
1 Functions and Their Graphs 1
1.1 Rectangular Coordinates 2
1.2 Graphs of Equations 11
1.3 Linear Equations in Two Variables 22
1.4 Functions 35
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
1.9 Inverse Functions 84
1.10 Mathematical Modeling and Variation 93
Chapter Summary 104
Review Exercises 106
Chapter Test 109
Proofs in Mathematics 110
P.S. Problem Solving 111
2
Polynomial and Rational Functions 113
2.1 Quadratic Functions and Models 114 2.2 Polynomial Functions of Higher Degree 123 2.3 Polynomial and Synthetic Division 136 2.4 Complex Numbers 145 2.5 Zeros of Polynomial Functions 152 2.6 Rational Functions 166 2.7 Nonlinear Inequalities 178 Chapter Summary 188
Review Exercises 190
Chapter Test 192
Proofs in Mathematics 193
P.S. Problem Solving 195
3
Exponential and Logarithmic Functions 197
3.1 Exponential Functions and Their Graphs 198 3.2 Logarithmic Functions and Their Graphs 209 3.3 Properties of Logarithms 219 3.4 Exponential and Logarithmic Equations 226 3.5 Exponential and Logarithmic Models 236 Chapter Summary 248
Review Exercises 250
Chapter Test 253
Cumulative Test for Chapters 1–3 254
Proofs in Mathematics 256
P.S. Problem Solving 257Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

iv Contents
4 Trigonometry 259
4.1 Radian and Degree Measure 260
4.2 Trigonometric Functions: The Unit Circle 270
4.3 Right Triangle Trigonometry 277
4.4 Trigonometric Functions of Any Angle 288
4.5 Graphs of Sine and Cosine Functions 297
4.6 Graphs of Other Trigonometric Functions 308
4.7 Inverse Trigonometric Functions 318
4.8 Applications and Models 328
Chapter Summary 338
Review Exercises 340
Chapter Test 343
Proofs in Mathematics 344
P.S. Problem Solving 345
5
Analytic Trigonometry 347
5.1 Using Fundamental Identities 348
5.2 Verifying Trigonometric Identities 355 5.3 Solving Trigonometric Equations 362 5.4 Sum and Difference Formulas 374 5.5 Multiple-Angle and Product-to-Sum Formulas 381 Chapter Summary 390
Review Exercises 392
Chapter Test 394
Proofs in Mathematics 395
P.S. Problem Solving 397
6
Additional Topics in Trigonometery 399
6.1 Law of Sines 400 6.2 Law of Cosines 409 6.3 Vectors in the Plane 416 6.4 Vectors and Dot Products 429 6.5 The Complex Plane 438 6.6 Trigonometric Form of a Complex Number 445 Chapter Summary 454
Review Exercises 456
Chapter Test 459
Cumulative Test for Chapters 4–6 460
Proofs in Mathematics 462
P.S. Problem Solving 465
7
Systems of Equations and Inequalities 467
7.1 Linear and Nonlinear Systems of Equations 468 7.2 Two-Variable Linear Systems 478
7.3 Multivariable Linear Systems 490 7.4 Partial Fractions 502 7.5 Systems of Inequalities 510 7.6 Linear Programming 520 Chapter Summary 529
Review Exercises 531
Chapter Test 535
Proofs in Mathematics 536
P.S. Problem Solving 537Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

Contents v
8 Matrices and Determinants 539
8.1 Matrices and Systems of Equations 540
8.2 Operations with Matrices 553
8.3 The Inverse of a Square Matrix 568
8.4 The Determinant of a Square Matrix 577
8.5 Applications of Matrices and Determinants 585
Chapter Summary 598
Review Exercises 600
Chapter Test 604
Proofs in Mathematics 605
P.S. Problem Solving 607
9
Sequences, Series, and Probability 609
9.1 Sequences and Series 610 9.2 Arithmetic Sequences and Partial Sums 620 9.3 Geometric Sequences and Series 629 9.4 Mathematical Induction 638 9.5 The Binomial Theorem 648 9.6 Counting Principles 656 9.7 Probability 666 Chapter Summary 678 Review Exercises 680 Chapter Test 683
Cumulative Test for Chapters 7–9 684
Proofs in Mathematics 686
P.S. Problem Solving 689
10
Topics in Analytic Geometry 691
10.1 Lines 692 10.2 Introduction to Conics: Parabolas 699 10.3 Ellipses 708 10.4 Hyperbolas 717 10.5 Rotation of Conics 727 10.6 Parametric Equations 735 10.7 Polar Coordinates 745 10.8 Graphs of Polar Equations 751 10.9 Polar Equations of Conics 759 Chapter Summary 766 Review Exercises 768 Chapter Test 771
Proofs in Mathematics 772
P.S. Problem Solving 775Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203

vi Contents
Appendices
Appendix A:Review of Fundamental Concepts of Algebra
A.1 Real Numbers and Their Properties A1
A.2 Exponents and Radicals A13
A.3 Polynomials and Factoring A25
A.4 Rational Expressions A35
A.5 Solving Equations A45
A.6 Linear Inequalities in One Variable A58
A.7 Errors and the Algebra of Calculus A67
Appendix B: Concepts in Statistics (online)*
B.1 Representing Data
B.2 Analyzing Data
B.3 Modeling Data
Answers to Odd-Numbered Exercises and Tests A75
Index A179
Index of Applications (onlin