Easy Pre-Calculus Step-by-Step, Second Edition

By Carolyn Wheater

Get the knowledge and skills you need to solve pre-calculus problems with confidence!The quickest route to learning a subject is through a solid grounding in the basics. Rather than endless drills, this accessible guide presents an original, step-by-step approach to help you develop a better understanding of pre-calculus topics. You’ll find important concepts linked together by clear explanations, invaluable exercises, and helpful worked-out problems. Once you’ve mastered the topics in this book, you will find yourself well-equipped to begin your calculus studies. This book features:•A new Trigonometry chapter that will round out your pre-calculus studies•Clear explanations that break down concepts into easy-to-understand steps•Stay-in-step "pop-ups" offering helpful advice and cautions against common errors•Step-it-up skill-building exercises linking practice to the core steps already presented•Worked-out solutions to all exercises that reinforce understanding of concepts

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Dec 28, 2018
• ISBN: 9781260135121

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Other titles in the series:

Easy Mathematics Step-by-Step, William D. Clark, PhD, and Sandra Luna McCune, PhD

Easy Algebra Step-by-Step, Sandra Luna McCune, PhD, and William D. Clark, PhD

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Contents

Preface

1   Graphs and the Graphing Calculator

Step 1. Review Linear Equations

Step 2. Meet the Parents

Step 3. Master the Fundamentals of Graphing

Step 4. Transform the Graphs

Step 5. Master Basic Calculator Skills

2   Functions

Step 1. Analyze Relations and Functions

Step 2. Perform Arithmetic of Functions

Step 3. Compose Functions

Step 4. Find Inverse Functions

3   Quadratic Functions

Step 1. Solve Quadratic Functions

Step 2. Explore Complex Numbers

Step 3. Graph Parabolas

Step 4. Apply What You’ve Learned

4   Polynomial Functions

Step 1. Factor Polynomials

Step 2. Use Synthetic Division to Find Zeros

Step 3. Use Long Division to Factor Polynomials

Step 4. Find the Zeros of Polynomial Functions

Step 5. Graph Polynomial Functions

5   Rational Functions

Step 1. Operate with Rational Expressions

Step 2. Deal with Discontinuities

Step 3. Examine End Behavior

Step 4. Find Intercepts

Step 5. Graph Rational Functions

6   Conic Sections

Step 1. Analyze and Graph Parabolas

Step 2. Analyze and Graph Ellipses

Step 3. Analyze and Graph Circles

Step 4. Analyze and Graph Hyperbolas

Step 5. Graph and Solve Quadratic Systems

7   Exponential and Logarithmic Functions

Step 1. Get to Know the Exponential Function

Step 2. Define the Logarithmic Function

Step 3. Learn to Use Properties of Logarithms

Step 4. Solve Exponential and Logarithmic Equations

Step 5. Explore Exponential Growth and Decay

8   Radical Functions

Step 1. Review Rules for Exponents

Step 2. Simplify Radicals and Rationalize Denominators

Step 3. Explore the Square Root and Cube Root Functions

9   Systems of Equations

Step 1. Review Two-Variable Linear Systems

Step 2. Solve Systems of Linear Equations in Three Variables

Step 3. Solve Nonlinear Systems

10   Matrices and Determinants

Step 1. Master Matrix Arithmetic

Step 2. Find Determinants

Step 3. Apply Cramer’s Rule

Step 4. Find Inverse Matrices

Step 5. Solve Systems by Inverse Matrices

11   Triangle Trigonometry

Step 1. Review Right Triangle Trigonometry

Step 2. Use Trigonometry to Find Areas

Step 3. Use the Law of Cosines to Extend to Nonright Triangles

Step 4. Apply the Law of Sines

12   Trigonometric Functions

Step 1. Expand Your Concept of Angles

Step 2. Learn the Unit Circle

Step 3. Define Trigonometric Functions

Step 4. Explore Inverse Trigonometric Functions

Step 5. Verify Identities

Step 6. Use Identities

Step 7. Solve Trigonometric Equations

13   Polar and Parametric Equations

Step 1. Plot Points in the Polar Coordinate System

Step 2. Convert Coordinates Between Systems

Step 3. Graph Polar Equations

Step 4. Convert Equations Between Polar and Rectangular Forms

Step 5. Convert Equations Between Parametric and Function Forms

Step 6. Graph Parametric Equations

14   Transformations

Step 1. Understand the Geometry of Reflections

Step 2. Translate by Reflecting

Step 3. Rotate by Reflecting

Step 4. Dilate Geometrically

Step 5. Use Matrices for Transformations

15   Rotating Conics

Step 1. Create the Equation that Rotates a Conic

Step 2. Recognize Rotation in Equations

Step 3. Graph a Rotated Conic

Step 4: Classify Rotated Conics without Graphing

16   Complex Numbers

Step 1. Change Between Forms

Step 2. Add and Subtract Complex Numbers

Step 3. Multiply and Divide Complex Numbers

Step 4. Raise a Complex Number to a Power

Step 5. Find the Roots of a Complex Number

17   Limits

Step 1. Evaluate Limits

Step 2. Deal with Problems

Step 3. Use Properties of Limits

Step 4. Evaluate Infinite Limits and Limits at Infinity

18   Sequences and Series

Step 1. Find Terms of Sequences

Step 2. Find Limits of Sequences

Step 3. Find Sums of Series

Step 4. Find Sums of Infinite Series

Answer Key

Index

Preface

Easy Pre-Calculus Step-by-Step is an interactive approach to the mathematics necessary for pre-calculus. It contains completely detailed, step-by-step instructions for the skills and concepts that are the foundation for advanced mathematics. Moreover, it features guiding principles, cautions against common errors, and offers other helpful advice as pop-ups in the margins. The book leads you from basic algebra to a clear understanding of functions, explores fundamental trigonometry, and introduces you to the notion of limits. Concepts are broken into basic components to provide ample practice of fundamental skills.

Pre-Calculus, as its name implies, is meant to prepare you for calculus, but what does that mean? To succeed in calculus, or any advanced mathematics, you will need strong algebraic skills as well as a clear understanding of functions in general and the many different types of functions specifically: polynomial, rational, exponential, logarithmic, and more. You will want to build skill in graphing quickly by means of transformations, and you’ll explore the polar coordinate system. From the construction of the trigonometric ratios in right triangles to the definition of the trigonometric functions in the unit circle, you will extend your knowledge of relationships and how to apply them. You may even investigate sequences and series and peek into the future by exploring limits.

Pre-Calculus may seem like too much to master, and the wide variety of topics may feel difficult to organize. With this step-by-step system, success will come. Learning pre-calculus, as with any mathematics, requires lots of practice. It requires courage to admit what you do not know, willingness to work on the problem, and a calm, orderly attempt to use what you do know. Most of all, it requires a true confidence in yourself and in the fact that with practice and persistence, you will be able to say, I can do this!

In addition to the step-by-step explanations and sample problems, this book presents a variety of exercises to let you assess your understanding. After working a set of exercises, use the solutions in the Answer Key to check your progress.

We sincerely hope Easy Pre-Calculus Step-by-Step will help you acquire the competence and the confidence you need to succeed in pre-calculus, calculus, and all your mathematical undertakings.

1

Graphs and the Graphing Calculator

A great deal of work in pre-calculus is centered on functions and their graphs. While the use of the graphing calculator has made it possible to explore many more, and many more complicated, functions than in the past, the ability to sketch the graph of a function by hand quickly is still an essential skill.

Step 1. Review Linear Equations

Your first introduction to graphing was linear functions, and they will always be important. Many of the skills you developed with linear graphs will carry over to other functions.

Recognize Horizontal and Vertical Lines

Recognizing the linear equations that don’t behave in a typical fashion will save you time. Horizontal lines, because they have a zero slope, have an equation of the form y = c, where c is some constant. Vertical lines are the real oddity. They are not functions, and their equations don’t fit the y = mx + b standard. The equation of a vertical line is x = c, where c is a constant.

Graph Quickly

While constructing a table of values is always available as a graphing method, it is time consuming, and linear equations in particular allow for quick sketching methods.

•   Slope-intercept. If the equation is in slope-intercept, or y = mx + b form, or can easily be converted to that form, plot the y-intercept (0, b), and then count the slope to locate other points on the line. Connect the points to create the line.

•   Intercept-intercept. If the equation is in standard form, the quickest method is to determine the x- and y-intercepts, plot them, and connect. To find the x-intercept, let y equal 0 and simplify. To find the y-intercept, replace x with 0 and simplify.

Write Equations

Given a graph, or information about a graph, you may be asked to write the equation of the line. The simplest way to do that is to use the point-slope form: y – y1 = m(x – x1)

•   Point-slope form. To write the equation of a line using point-slope form, you’ll need to know a point on the line (x1, y1) and the slope m, or two points (x1, y1) and (x2, y. Then the equation can be written by replacing x1, y1, and m in the point-slope form y – y1 = m(x – x1) with the known values. The point-slope form can be simplified and transformed to slope-intercept or standard form if desired.

•   Parallel and perpendicular lines. To write the equation of a line parallel to a given line, determine the slope of the given line, usually by putting the given equation in slope-intercept form. Write the equation of the desired line by using point-slope form with the same slope as the given line and the point you want the desired line to pass through.

To write the equation of a line perpendicular to a given line, you’ll also want to determine the slope of the given line, but you’ll use the negative reciprocal of the slope of the given line as the slope of the perpendicular line. Use point-slope form with that negative reciprocal slope and the point you want the line to pass through.

Exercise 1.1

Test your understanding by doing the following exercises.

1. Sketch the graph of x = 6.

2. Sketch the graph of y = -3.

.

4. Sketch the graph of 2x – 4y = 12.

5. Write the equation of a line with slope of –3 and a y-intercept of 5.

through the point (2,–1).

7. Write the equation of a line through the points (3,9) and (–4,2).

8. Write the equation of a line parallel to 5x – 7y = 35 through the point (–7,4).

9. Write the equation of a line perpendicular to 4x – 2y = 11 through the point (–2,5).

10. Find the equation of the perpendicular bisector of the segment that connects (3,–2) and (–5,6).

Step 2. Meet the Parents

If you are acquainted with the simplest function typical of a class, you’ll find it easier to predict the behavior of functions you’re trying to graph. These parent functions give you a place to start, and knowing a few key points on each parent graph will help you apply transformations.

Constant Function

The constant function f(x) = c, for some constant c, has a graph that is a horizontal line. Its domain is (−∞, ∞), and the range contains the single value, c.

Linear Function

Linear functions have the form f(x) = mx + b, where m is the slope and b is the y-intercept. The parent function is the function f(x) = x, with a slope of 1 and a y-intercept of 0. The domain is (−∞, ∞), and the range is (−∞, ∞). Key points on the parent graph are (−1,−1), (0,0), and (1,1).

Constant function

Linear function

Quadratic Function

The quadratic function has the form f(x) = ax² + bx + c, for real numbers a, b, and c, with a ≠ 0, but is easier to graph when the equation is in vertex form, y = a(x – h)² + k. In this form, the vertex is (h, k). The parent function for the quadratic family is the function f(x) = x², with the vertex at the origin. The domain of the parent function is (−∞, ∞), and the range is (0, ∞). Key points on the parent graph are (−1,1), (0,0), and (1,1).

Quadratic function

Square Root Function

has a domain of [0, ∞) and a range of [0, ∞). Key points on the graph are (0,0), (1,1), and (4,2).

Cubic Function

Cubic functions have the form f(x) = ax³ + bx² + cx + d, for real numbers a, b, c, and d, with a ≠ 0, but here, too, the graph can be sketched more easily if expressed as f(x) = a (x – h)³ + k. The parent function is f (x) = x³ with a domain of (−∞, ∞) and a range of (−∞, ∞). The key points to remember are (−1,−1), (0,0), and (1,1).

Square root function

Cubic function

Cube Root Function

has a domain of (−∞, ∞) and a range of (−∞, ∞). Key points on the graph are (−1,−1), (0,0), and (1,1).

Cube root function

Exponential Function

The parent function for the exponential family, f(x) = bx, for some constant base b, with b > 0, has a domain of (−∞, ∞) and a range of (0, ∞). The graph has a horizontal asymptote of y , (0,1), and (1,b).

Logarithmic Function

The logarithmic function is the inverse of the exponential function. The parent graph is f(x) = logbx. The domain is (0, ∞), and the range is (−∞, ∞). The graph has a vertical asymptote of x , (1,0), and (b,1).

Exponential function

Logarithmic function

Rational Function

. The domain of the function is (−∞,0) ∪ (0, ∞), and the range is (−∞,0) ∪ (0,∞). The graph has a vertical asymptote of x = 0 and a horizontal asymptote of y = 0. Key points on the graph are

.

Rational function

Exercise 1.2

Identify the family from which each graph comes.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Step 3. Master the Fundamentals of Graphing

When you begin to sketch the graph of any function, it’s wise to take a moment to think about essential information.

Identify the Domain

Begin with a domain of all real numbers, and then eliminate values, if necessary, according to the following checklist.

•   Is there a denominator? If so, eliminate any values that make the denominator equal to 0.

•   Is there a root with an even index? If so, eliminate any values that make the radicand negative.

•   Is there a logarithm? If so, eliminate any values that make the argument of the log equal to 0 or a negative number.

•   Is this function modeling a real situation? If so, consider what values make sense in that model. A function representing volume as