# Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry, A Corequisite Solution, 4th edition

Published by Pearson (May 8, 2019) © 2019

**Michael Sullivan**Chicago State University

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Known for their ability to connect with students, acclaimed authors Sullivan and Sullivan focus on the fundamentals (preparing for class, practice with homework, reviewing key concepts), encouraging the conceptual understanding and mastery of basic skills you'll need for this and future courses. Part of the Concepts Through Functions Series, **Precalculus: Concepts through Functions, A Unit Circle Approach to Trigonometry** introduces functions at the start and maintains a continuous theme by introducing and developing a new function in every chapter. Graphing utility coverage is optional and can be included at the instructor's discretion. In the **4th Edition**, new Desmos^{©} screen captures have been added throughout, reflecting that graphing technology expands beyond graphing calculators; updated Retain Your Knowledge problems improve your recall of concepts learned earlier in the course; and much more.

**Study Skills for Mathematics**

SS1How Learning Math is Different

SS2The Growth Mindset and Grit

SS3Resources Available for Help

SS4Time Management

SS5How to be An Effective Listener and How to Take Notes

SS6How to do Math Homework the Right Way

SS7How to Read a Math Book

SS8How to Study for a Math Exam

SS9Overcoming Math and Test Anxiety

**Elementary Algebra Review**

R.1 Sets and Classifications of Numbers

- Use set notation
- Classify numbers
- Approximate decimals by rounding or truncating
- Plot points on the real number line
- Use inequalities to order real numbers

R.2 Properties of Real Numbers

- Compute the absolute value of a real number
- Add and subtract signed numbers
- Multiply and divide signed numbers
- State the associative and distributive properties

R.3 Perform Operations on Rational Numbers

- Write rational numbers written as fractions in lowest terms
- Multiply and divide rational numbers written as fractions
- Add or subtract rational numbers written as fractions

R.4 Order of Operations

- Evaluate real numbers with exponents
- Use the order of operations to evaluate expressions

R.5 Algebraic Expressions

- Translate English expressions into mathematical language
- Evaluate algebraic expressions
- Simplify algebraic expressions by combining like terms
- Determine the domain of a variable

R.6 Square Roots

- Evaluate square roots of perfect squares
- Determine whether a square root is rational, irrational, or not a real number
- Find square roots of variable expressions
- Use the product rule to simplify square roots

R.8 Geometry Essentials

- Use the Pythagorean Theorem and Its Converse
- Know Geometry Formulas
- Understand Congruent Triangles and Similar Triangles

R.9 Laws of Exponents

- Simplify Exponential Expressions Using the Product Rule
- Simplify Exponential Expressions Using the Quotient Rule
- Evaluate Exponential Expressions with a Zero or Negative Exponent
- Simplify Exponential Expressions Using the Power Rule
- Simplify Exponential Expressions Containing Products or Quotients
- Simplify Exponential Expressions Using the Laws of Exponents

R.10 Adding and Subtracting Polynomials

- Define monomial and determine the coefficient and degree of a monomial
- Define polynomial and determine the degree of a polynomial
- Simplify polynomials by combining like terms

R.11 Multiplying Polynomials

- Multiply a monomial by a polynomial
- Multiply two binomials
- Multiply two polynomials
- Multiply special products

**Preparing for Chapter F **

F.P1 Linear Equations in One Variable

- Determine whether a number is a solution to an equation
- Solve linear equations
- Determine whether an equation is a conditional equation, an identity, or a contradiction
- Solve for a variable in a formula

F.P2 Greatest Common Factor; Factoring by Grouping

- Factor out the greatest common factor
- Factor by grouping

F.P3 More Factoring

- Factor trinomials of the form
*x*^{2}+*bx*+*c* - Factor perfect square trinomials
- Factor the difference of two squares

F.P4 Polynomial Equations

- Solve polynomial equations using the zero-product property
- Solve quadratic equations using the square root property

F.P5 Solving Quadratic Equations by Completing the Square

- Complete the square in one variable
- Solve quadratic equations by completing the square

**Chapter F. Foundations: A Prelude to Functions**

F.1 The Distance and Midpoint Formulas

- Use the distance formula
- Use the midpoint formula

F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry

- Graph equations by plotting points
- Find intercepts from a graph
- Find intercepts from an equation
- Test an equation for symmetry
- Know how to graph key equations

F.3 Lines

F.4 Circles

**Preparing for Chapter 1**

1.P1 Linear Inequalities in One Variable

- Represent inequalities using the real number line and interval notation
- Understand the properties of inequalities
- Solve linear inequalities
- Solve problems involving linear inequalities

1.P2 nth Roots

- Evaluate
*n*th roots - Simplify expressions of the form

1.P3 An Introduction to Problem Solving

- Translate English sentences into mathematical statements
- Model and solve direct translation problems
- Model and solve mixture problems (optional)
- Model and solve uniform motion problems
- Use geometry formulas to solve problems

**Chapter 1. Functions and Their Graphs**

1.1 Functions

1.2 The Graph of a Function

1.3 Properties of Functions

1.4 Library of Functions; Piecewise-defined Functions

1.5 Graphing Techniques: Transformations

1.6 Mathematical Models: Building Functions

1.7 Building Mathematical Models Using Variation

**Preparing for Chapter 2**

2.P1 Factoring Trinomials Where the Leading Coefficient is Not One

Factor trinomials for the form *ax*^{2} + *bx* + *c*, *a* ≠ 1

2.P2 The Complex Number System

- Evaluate the square root of negative real numbers
- Add or subtract complex numbers
- Multiply complex numbers
- Divide complex numbers
- Evaluate the powers of
*i*

Solve quadratic equations using the Square Root Property

2.P3 Solving Quadratic Equations by the Quadratic Formula

- Solve quadratic equations using the quadratic formula
- Use the discriminant to determine the nature of solutions of a quadratic equation

2.P4 Solving Equations Quadratic in Form

- Solve equations that are quadratic in form

2.P5 Compound Inequalities

- Determine the intersection or union of two sets
- Solve compound inequalities involving “and”
- Solve compound inequalities involving “or”
- Solve problems using compound inequalities

**Chapter 2. Linear and Quadratic Functions**

2.1 Properties of Linear Functions and Linear Models

2.2 Building Linear Models from Data

2.3 Quadratic Functions and Their Zeros

2.4 Properties of Quadratic Functions

2.5 Inequalities Involving Quadratic Functions

2.6 Building Quadratic Models from Verbal Descriptions and from Data

2.7 Complex Zeros of a Quadratic Function

2.8 Equations and Inequalities Involving the Absolute Value Function

**Preparing for Chapter 3**

3.P1 Factoring

- Factoring trinomials using substitution
- Factoring the sum and difference of two cubes

3.P2 Dividing Polynomials; Synthetic Division

- Divide a polynomial by a monomial
- Divide polynomials using long division
- Divide polynomials using synthetic division

3.P3 Multiplying and Dividing Rational Expressions

- Determine the domain of a rational expression
- Simplify rational expressions
- Multiply rational expressions
- Divide rational expressions

3.P4 Adding and subtracting rational expressions

- Add or subtract rational expressions with a common denominator
- Find the least common denominator of two or more rational expressions
- Add or subtract rational expressions with different denominators

3.P5 Complex Rational Expressions

- Simplify a complex rational expression by simplifying the numerator and denominator separately (Method I)
- Simplify a complex rational expression using the least common denominator (Method II)

3.P6 Rational Equations

- Solve equations containing rational expressions
- Solve equations involving rational functions

**Chapter 3. Polynomial and Rational Functions**

3.1 Polynomial Functions and Models

3.2 The Real Zeros of a Polynomial Function

3.3 Complex Zeros; Fundamental Theorem of Algebra

3.4 Properties of Rational Functions

3.5 The Graph of a Rational Function

3.6 Polynomial and Rational Inequalities

**Preparing for Chapter 4**

4.P1 Rational Exponents

- Evaluate expressions of the form a
^{1/n} - Evaluate expressions of the form a
^{m/n}

4.P2 Simplifying Expressions Using the Laws of Exponents

- Simplify expressions involving rational exponents
- Simplify radical expressions
- Factor expressions containing rational exponents

4.P3 Simplifying Radical Expressions Using Properties of Radicals

- Use the product property to multiply radical expressions
- Use the product property to simplify radical expressions
- Use the quotient property to simplify radical expressions

4.P4 Adding, Subtracting, and Multiplying Radical Expressions

- Add or subtract radical expressions
- Multiply radical expressions

4.P5 Rationalizing Radical Expressions

- Rationalize a denominator containing one term
- Rationalize a denominator containing two terms

**Chapter 4. Exponential and Logarithmic Functions**

4.1 Composite Functions

4.2 One¿to¿One Functions; Inverse Functions

4.3 Exponential Functions

4.4 Logarithmic Functions

4.5 Properties of Logarithms

4.6 Logarithmic and Exponential Equations

4.7 Compound Interest

4.8 Exponential Growth and Decay; Newton’s Law; Logistic Growth and Decay

4.9 Building Exponential, Logarithmic, and Logistic Functions from Data

**5. Trigonometric Functions**

5.1 Angles and Their Measure

5.2 Trigonometric Functions: Unit Circle Approach

5.3 Properties of the Trigonometric Functions

5.4 Graphs of the Sine and Cosine Functions

5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

5.6 Phase Shift; Sinusoidal Curve Fitting

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**6. Analytic Trigonometry**

6.1 The Inverse Sine, Cosine, and Tangent Functions

6.2 The Inverse Trigonometric Functions (Continued)

6.3 Trigonometric Equations

6.4 Trigonometric Identities

6.5 Sum and Difference Formulas

6.6 Double-angle and Half-angle Formulas

6.7 Product-to-Sum and Sum-to-Product Formulas

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**7. Applications of Trigonometric Functions**

7.1 Right Triangle Trigonometry; Applications

7.2 The Law of Sines

7.3 The Law of Cosines

7.4 Area of a Triangle

7.5 Simple Harmonic Motion; Damped Motion; Combining Waves

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**8. Polar Coordinates; Vectors**

8.1 Polar Coordinates

8.2 Polar Equations and Graphs

8.3 The Complex Plane; De Moivre’s Theorem

8.4 Vectors

8.5 The Dot Product

8.6 Vectors in Space

8.7 The Cross Product

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**9. Analytic Geometry**

9.1 Conics

9.2 The Parabola

9.3 The Ellipse

9.4 The Hyperbola

9.5 Rotation of Axes; General Form of a Conic

9.6 Polar Equations of Conics

9.7 Plane Curves and Parametric Equations

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**10. Systems of Equations and Inequalities**

10.1 Systems of Linear Equations: Substitution and Elimination

10.2 Systems of Linear Equations: Matrices

10.3 Systems of Linear Equations: Determinants

10.4 Matrix Algebra

10.5 Partial Fraction Decomposition

10.6 Systems of Nonlinear Equations

10.7 Systems of Inequalities

10.8 Linear Programming

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**11. Sequences; Induction; the Binomial Theorem**

11.1 Sequences

11.2 Arithmetic Sequences

11.3 Geometric Sequences; Geometric Series

11.4 Mathematical Induction

11.5 The Binomial Theorem

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**12. Counting and Probability**

12.1 Counting

12.2 Permutations and Combinations

12.3 Probability

Chapter Review

Chapter Test

Cumulative Review

Chapter Projects

**13. A Preview of Calculus: The Limit, Derivative, and Integral of a Function**

13.1 Finding Limits Using Tables and Graphs

13.2 Algebra Techniques for Finding Limits

13.3 One-sided Limits; Continuous Functions

13.4 The Tangent Problem; The Derivative

13.5 The Area Problem; The Integral

Chapter Review

Chapter Test

Chapter Projects

**Appendix A: Review**

A.1 Algebra Essentials

A.2 Geometry Essentials

A.3 Polynomials

A.4 Factoring Polynomials

A.5 Synthetic Division

A.6 Rational Expressions

A.7 *n*th Roots; Rational Exponents

A.8 Solving Equations

A.9 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications

A.10 Interval Notation; Solving Inequalities

A.11 Complex Numbers

**Appendix B: Graphing Utilities**

B.1 The Viewing Rectangle B1

B.2 Using a Graphing Utility to Graph Equations B3

B.3 Using a Graphing Utility to Locate Intercepts and Check

for Symmetry B5

B.4 Using a Graphing Utility to Solve Equations B6

B.5 Square Screens B8

B.6 Using a Graphing Utility to Graph Inequalities B9

B.7 Using a Graphing Utility to Solve Systems of Linear Equations B9

B.8 Using a Graphing Utility to Graph a Polar Equation B11

B.9 Using a Graphing Utility to Graph Parametric Equations B11

*Answers*

*Photo Credits*

*Index*

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