Precalculus
, by Young, Cynthia Y.
- ISBN: 9781119869405 | 1119869404
- Cover: Loose-leaf
- Copyright: 5/16/2023
Cynthia Young's Precalculus, 4th edition helps students take the guesswork out of studying by offering them an easy to read and clear roadmap that tells them what to do, how to do it, and whether they did it right. With this revision, the author focuses on the most difficult topics in precalculus, bringing clarity to challenging learning objectives.
0 Review: Equations and
Inequalities 1
0.1 Linear Equations 2
0.1.1 Solving Linear Equations in One Variable 2
0.1.2 Applications Involving Linear Equations 6
0.1.3 Interest Problems 8
0.1.4 Mixture Problems 10
0.1.5 Distance–Rate–Time Problems 12
0.2 Quadratic Equations 17
0.2.1 Factoring 17
0.2.2 Square Root Method 19
0.2.3 Completing the Square 21
0.2.4 Quadratic Formula 23
0.2.5 Applications Involving Quadratic
Equations 26
0.3 Other Types of Equations 30
0.3.1 Rational Equations 30
0.3.2 Radical Equations 33
0.3.3 Equations Quadratic in Form:
u-Substitution 35
0.3.4 Factorable Equations 37
0.3.5 Equations Involving Absolute Value 38
0.4 Inequalities 43
0.4.1 Graphing Inequalities and Interval
Notation 43
0.4.2 Linear Inequalities 46
0.4.3 Polynomial Inequalities 48
0.4.4 Rational Inequalities 51
0.4.5 Absolute Value Inequalities 53
0.5 Graphing Equations 58
0.5.1 Cartesian Plane 59
0.5.2 The Distance and Midpoint Formulas 59
0.5.3 Point-Plotting
61
0.5.4 Using Intercepts as Graphing Aids 62
0.5.5 Using Symmetry as a Graphing Aid 64
0.5.6 Circles 67
0.6 Lines 74
0.6.1 General Form of a Line and Slope 74
0.6.2 Equations of Lines 77
0.6.3 Parallel and Perpendicular Lines 80
0.7 Modeling Variation 86
0.7.1 Direct Variation 86
0.7.2 Inverse Variation 88
0.7.3 Joint Variation and Combined Variation 90
0.8* Linear Regression: Best Fit 0.8-1
0.8.1 Scatterplots 0.8-1
0.8.2 Identifying Patterns 0.8-5
0.8.3 Linear Regression 0.8-12
Review 95 | Review Exercises 97 | Practice Test 99
1 Functions and Their Graphs 100
1.1 Functions 101
1.1.1 Relations and Functions 101
1.1.2 Functions Defined by Equations 104
1.1.3 Function Notation 106
1.1.4 Domain of a Function 110
1.2 Graphs of Functions; Piecewise-Defined Functions;
Increasing and Decreasing Functions; Average Rate
of Change 118
1.2.1 Recognizing and Classifying Functions 118
1.2.2 Increasing and Decreasing Functions 122
1.2.3 Average Rate of Change 125
1.2.4 Piecewise-Defined Functions 128
1.3 Graphing Techniques: Transformations 138
1.3.1 Horizontal and Vertical Shifts 138
1.3.2 Reflection About the Axes 143
1.3.3 Stretching and Compressing 146
1.4 Operations on Functions and Composition of
Functions 152
1.4.1 Adding, Subtracting, Multiplying, and
Dividing Functions 153
1.4.2 Composition of Functions 154
1.5 One-to-One Functions and Inverse Functions 163
1.5.1 Determine Whether a Function Is
One-to-One 163
1.5.2 Inverse Functions 166
1.5.3 Graphical Interpretation of Inverse
Functions 168
1.5.4 Finding the Inverse Function 170
Review 179 | Review Exercises 181 | Practice Test 184
2 Polynomial and Rational
Functions 185
2.1 Quadratic Functions 186
2.1.1 Graphs of Quadratic Functions:
Parabolas 186
2.1.2 Finding the Equation of a Parabola 195
2.2 Polynomial Functions of Higher Degree 203
2.2.1 Identifying Polynomial Functions 203
2.2.2 Graphing Polynomial Functions Using
Transformations of Power Functions 206
2.2.3 Real Zeros of a Polynomial Function 207
2.2.4 Graphing General Polynomial
Functions 210
2.3 Dividing Polynomials: Long Division and Synthetic
Division 219
2.3.1 Long Division of Polynomials 219
2.3.2 Synthetic Division of Polynomials 223
2.4 The Real Zeros of a Polynomial Function 227
2.4.1 The Remainder Theorem and the Factor
Theorem 227
2.4.2 The Rational Zero Theorem and Descartes’
Rule of Signs 230
2.4.3 Factoring Polynomials 234
2.4.4 The Intermediate Value Theorem 236
2.4.5 Graphing Polynomial Functions 238
2 .5 Complex Zeros: The Fundamental Theorem of
Algebra 242
2.5.1 Complex Zeros 243
2.5.2 Factoring Polynomials 247
2.6 Rational Functions 251
2.6.1 Domain of Rational Functions 251
2.6.2 Vertical, Horizontal, and Slant
Asymptotes 253
2.6.3 Graphing Rational Functions 259
Review 271 | Review Exercises 273 | Practice
Test 276 | Cumulative Test 277
3 Exponential and Logarithmic
Functions 278
3.1 Exponential Functions and Their Graphs 279
3.1.1 Evaluating Exponential Functions 279
3.1.2 Graphs of Exponential Functions 281
3.1.3 The Natural Base e 285
3.1.4 Applications of Exponential Functions 287
3.2 Logarithmic Functions and Their Graphs 295
3.2.1 Evaluating Logarithms 295
3.2.2 Common and Natural Logarithms 298
3.2.3 Graphs of Logarithmic Functions 298
3.2.4 Applications of Logarithms 303
3.3 Properties of Logarithms 311
3.3.1 Properties of Logarithmic Functions 311
3.3.2 Change-of-
Base
Formula 316
3.4 Exponential and Logarithmic Equations 320
3.4.1 Exponential Equations 320
3.4.2 Solving Logarithmic Equations 324
3.4.3 Applications 326
3.5 Exponential and Logarithmic Models 331
3.5.1 Exponential Growth Models 332
3.5.2 Exponential Decay Models 333
3.5.3 Gaussian (Normal) Distribution Models 335
3.5.4 Logistic Growth Models 336
3.5.5 Logarithmic Models 337
Review 343 | Review Exercises 345 | Practice
Test 348 | Cumulative Test 349
4 Trigonometric Functions of
Angles 350
4.1 Angle Measure 351
4.1.1 Angles and their Measure 352
4.1.2 Radian Measure 354
4.1.3 Angles in Standard Position 357
4.1.4 Coterminal Angles 358
4.1.5 Arc Length 359
4.1.6 Area of a Circular Sector 360
4.1.7 Linear and Angular Speed 361
4.1.8 Relationship Between Linear and
Angular Speeds 363
4.2 Right Triangle Trigonometry 369
4.2.1 Right Triangle Ratios 370
4.2.2 Trigonometric Functions: Right
Triangle Ratios 371
4.2.3 Reciprocal Identities 372
4.2.4 Evaluating Trigonometric Functions
Exactly for Special Angle Measures 373
4.2.5 Using Calculators to Evaluate (Approximate)
Trigonometric Function Values 377
4.2.6 Solving a Right Triangle Given an Acute Angle
Measure and a Side Length 378
4.3 Trigonometric Functions of Angles 387
4.3.1 Trigonometric Functions: The Cartesian
Plane 387
4.3.2 Algebraic Signs of the Trigonometric
Functions 390
4.3.3 Ranges of the Trigonometric Functions 393
4.3.4 Reference Angles and Reference Right
Triangles 395
4.3.5 Evaluating Trigonometric Functions for
Nonacute Angles 398
4.4 The Law of Sines 405
4.4.1 Solving Oblique Triangles 405
4.5 The Law of Cosines 418
4.5.1 Solving Oblique Triangles Using the
Law of Cosines 419
4.5.2 The Area of a Triangle 423
Review 432 | Review Exercises 436 | Practice
Test 438 | Cumulative Test 439
5 Trigonometric Functions of Real
Numbers 440
5.1 Trigonometric Functions: The Unit Circle
Approach 441
5.1.1 Trigonometric Functions and the Unit Circle
(Circular Functions) 441
5.1.2 Properties of Trigonometric
(Circular) Functions 444
5.2 Graphs of Sine and Cosine Functions 451
5.2.1 The Graphs of Sinusoidal Functions 451
5.2.2 Graphing a Shifted Sinusoidal Function:
y = A sin(Bx + C) + D and y =
A cos(Bx + C) + D 464
5.2.3 Harmonic Motion 467
5.2.4 Graphing Sums of Functions: Addition of
Ordinates 471
5.3 Graphs of Other Trigonometric Functions 481
5.3.1 Graphing the Tangent, Cotangent, Secant,
and Cosecant Functions 481
5.3.2 Translations of Trigonometric
Functions 492
Review 500 | Review Exercises 504 | Practice
Test 505 | Cumulative Test 506
6 Analytic Trigonometry 507
6.1 Trigonometric Identities 508
6.1.1 Fundamental Identities 509
6.1.2 Simplifying Trigonometric Identities 511
6.1.3 Verifying Identities 513
6.2 Sum and Difference Identities 519
6.2.1 Sum and Difference Identities
for the Cosine Function 520
6.2.2 Sum and Difference Identities
for the Sine Function 523
6.2.3 Sum and Difference Identities for the
Tangent Function 526
6.3 Double-Angle and Half-Angle Identities 531
6.3.1 Applying Double-Angle Identities 531
6.3.2 Applying Half-Angle Identities 536
6.4 Product-to-Sum and Sum-to-Product
Identities 547
6.4.1 Product-to-Sum Identities 547
6.4.2 Sum-to-Product Identities 549
6.5 Inverse Trigonometric Functions 555
6.5.1 Inverse Sine Function 556
6.5.2 Inverse Cosine Function 560
6.5.3 Inverse Tangent Function 563
6.5.4 Remaining Inverse Trigonometric
Functions 565
6.5.5 Finding Exact Values for Expressions
Involving Inverse Trigonometric
Functions 568
6.6 Trigonometric Equations 576
6.6.1 Solving Trigonometric Equations by
Inspection 577
6.6.2 Solving Trigonometric Equations Using
Algebraic Techniques 580
6.6.3 Solving Trigonometric Equations That
Require the Use of Inverse Functions 582
6.6 .4 Using Trigonometric Identities to Solve
Trigonometric Equations 584
Review 593 | Review Exercises 597 | Practice
Test 600 | Cumulative Test 601
7 Vectors, the Complex Plane, and
Polar Coordinates 602
7.1 Vectors 603
7.1.1 Vectors: Magnitude and Direction 603
7.1.2 Vector Operations 607
7.1.3 Horizontal and Vertical Components of a
Vector 609
7.1.4 Unit Vectors 610
7.1.5 Resultant Vectors 611
7.2 The Dot Product 619
7.2.1 Multiplying Two Vectors: The Dot
Product 619
7.2.2 Angle Between Two Vectors 620
7.2.3 Work 623
7.3 Polar (Trigonometric) Form of Complex
Numbers 628
7.3.1 Complex Numbers in
Rectangular Form 629
7.3.2 Complex Numbers in Polar Form 630
7.4 Products, Quotients, Powers, and Roots of
Complex Numbers 638
7.4.1 Products of Complex Numbers 638
7.4.2 Quotients of Complex Numbers 640
7.4.3 Powers of Complex Numbers 641
7.4.4 Roots of Complex Numbers 643
7.5 Polar Equations and Graphs 649
7.5.1 Polar Coordinates 650
7.5.2 Converting Between Polar and
Rectangular Coordinates 651
7.5.3 Graphs of Polar Equations 653
Review 667 | Review Exercises 670 | Practice
Test 672 | Cumulative Test 673
8 Systems of Linear Equations and
Inequalities 674
8.1 Systems of Linear Equations in Two Variables 675
8.1.1 Solving Systems of Linear Equations 676
8.1.2 Three Methods and Three Types
of Solutions 685
8.2 Systems of Linear Equations in
Three Variables 691
8.2.1 Solving Systems of Linear Equations
in Three Variables 691
8.2.2 Types of Solutions 694
8.3 Systems of Linear Equations and Matrices 703
8.3.1 Matrices 703
8.3.2 Augmented Matrices 705
8.3.3 Row Operations on a Matrix 707
8.3.4 Row–Echelon
Form of a Matrix 708
8.3.5 Gaussian Elimination with Back-Substitution
709
8.3.6 Gauss–Jordan
Elimination 711
8.3.7 Inconsistent and Dependent Systems 714
8.4 Matrix Algebra 726
8.4.1 Equality of Matrices 726
8.4.2 Matrix Addition and Subtraction 727
8.4.3 Scalar and Matrix Multiplication 729
8.4.4 Matrix Equations 735
8.4.5 Finding the Inverse of a Matrix 737
8.4.6 Solving Systems of Linear Equations Using
Matrix Algebra and Inverses of Square
Matrices 742
8.5 The Determinant of a Square Matrix and Cramer’s
Rule 751
8.5.1 Determinant of a 2 × 2 Matrix 752
8.5.2 Determinant of an n × n Matrix 753
8.5.3 Cramer’s Rule: Systems of Linear Equations
in Two Variables 756
8.5.4 Cramer’s Rule: Systems of Linear Equations
in Three Variables 758
8.6 Partial Fractions 765
8.6.1 Performing Partial-Fraction
Decomposition 765
8.7 Systems of Linear Inequalities
in Two Variables 776
8.7.1 Linear Inequalities in Two Variables 776
8.7.2 Systems of Linear Inequalities in Two
Variables 779
8.7.3 The Linear Programming Model 784
Review 792 | Review Exercises 795 | Practice
Test 798 | Cumulative Test 799
9 Conics and Systems of Nonlinear
Equations and Inequalities 800
9.1 Conic Basics 801
9.1.1 Three Types of Conics 801
9.2 The Parabola 804
9.2.1 Parabola with a Vertex at the Origin 804
9.2.2 Parabola with a Vertex at the Point
(h, k) 809
9.2.3 Applications 812
9.3 The Ellipse 818
9.3.1 Ellipse Centered at the Origin 818
9.3.2 Ellipse Centered at the Point (h, k) 823
9.3.3 Applications 825
9.4 The Hyperbola 832
9.4.1 Hyperbola Centered at the Origin 832
9.4.2 Hyperbola Centered at the Point (h, k) 837
9.4.3 Applications 839
9.5 Systems of Nonlinear Equations 844
9.5.1 Solving a System of Nonlinear
Equations 844
9.6 Systems of Nonlinear Inequalities 854
9.6.1 Nonlinear Inequalities in Two Variables 854
9.6.2 Systems of Nonlinear Inequalities 857
9.7 Rotation of Axes 864
9.7.1 Transforming Second-Degree Equations
Using Rotation of Axes 864
9.7.2 Determine the Angle of Rotation Necessary
to Transform a General Second-Degree
Equation into an Equation of a Conic 867
9.8 Polar Equations of Conics 872
9.8.1 Equations of Conics in Polar
Coordinates 872
9.9 Parametric Equations and Graphs 881
9.9.1 Parametric Equations of a Curve 881
9.9.2 Applications of Parametric Equations 885
Review 890 | Review Exercises 893 | Practice
Test 895 | Cumulative Test 896
10 Sequences and Series 897
10.1 Sequences and Series 898
10.1.1 Sequences 898
10.1.2 Factorial Notation 901
10.1.3 Recursion Formulas 902
10.1.4 Sums and Series 903
10.2 Arithmetic Sequences and Series 909
10.2.1 Arithmetic Sequences 910
10.2.2 The General (nth) Term of an Arithmetic
Sequence 911
10.2.3 The Sum of an Arithmetic Sequence 912
10.3 Geometric Sequences and Series 918
10.3.1 Geometric Sequences 918
10.3.2 The General (nth) Term of a
Geometric Sequence 920
10.3.3 Geometric Series 921
10.4 Mathematical Induction 929
10.4.1 Mathematical Induction 929
10.5 The Binomial Theorem 933
10.5.1 Binomial Coefficients 934
10.5.2 Binomial Expansion 936
10.5.3 Pascal’s Triangle 937
10.5.4 Finding a Particular Term of a Binomial
Expansion 939
Review 943 | Review Exercises 944 | Practice
Test 946 | Cumulative Test 947
11 Limits: A Preview to Calculus 948
11.1 Introduction to Limits: Estimating Limits
Numerically and Graphically 949
11.1.1 Definition of a Limit 949
11.1.2 Estimating Limits Numerically and
Graphically 951
11.1.3 Limits That Fail to Exist 952
11.1.4 One-Sided Limits 955
11.2 Techniques for Finding Limits 961
11.2.1 Limit Laws 962
11.2.2 Finding Limits Using Limit Laws 963
11.2.3 Finding Limits Using Direct
Substitution 967
11.2.4 Finding Limits Using Algebraic
Techniques 969
11.2.5 Finding Limits Using Left-Hand
and Right-Hand Limits 971
11.3 Tangent Lines and Derivatives 975
11.3.1 Tangent Lines 975
11.3.2 The Derivative of a Function 980
11.3.3 Instantaneous Rates of Change 983
11.4 Limits at Infinity; Limits of Sequences 986
11.4.1 Limits at Infinity 986
11.4.2 Limits of Sequences 991
11.5 Finding the Area Under a Curve 995
11.5.1 Limits of Summations 996
11.5.2 The Area Problem 998
Review 1010 | Review Exercises 1012 | Practice
Test 1014 | Cumulative Test 1015
APPENDIX 1016
ANSWERS TO ALL EXERCISES 1111
APPLICATION INDEX XXX
SUBJECT INDEX XXX
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