Easy Precalculus Step-by-Step
5/5
()
About this ebook
Take it step-by-step for pre-calculus success!
The quickest route to learning a subject is through a solid grounding in the basics. So what you won’t find in Easy Pre-calculus Step-by-Step is a lot of endless drills. Instead, you get a clear explanation that breaks down complex concepts into easy-to-understand steps, followed by highly focused exercises that are linked to core skills--enabling learners to grasp when and how to apply those techniques.
This book features:- Large step-by-step charts breaking down each step within a process and showing clear connections between topics and annotations to clarify difficulties
- Stay-in-step panels show how to cope with variations to the core steps
- Step-it-up exercises link practice to the core steps already presented
- Missteps and stumbles highlight common errors to avoid
You can master pre-calculus as long as you take it Step-by-Step!
Read more from Carolyn Wheater
Practice Makes Perfect Trigonometry Rating: 5 out of 5 stars5/5Practice Makes Perfect Basic Math Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect: Basic Math Review and Workbook, Third Edition Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect Basic Math Review and Workbook, Second Edition Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect Geometry Rating: 5 out of 5 stars5/5Easy Pre-Calculus Step-by-Step, Second Edition Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect: Algebra I Review and Workbook, Third Edition Rating: 2 out of 5 stars2/5Practice Makes Perfect Algebra I Review and Workbook, Second Edition Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect Algebra Rating: 0 out of 5 stars0 ratings
Related to Easy Precalculus Step-by-Step
Related ebooks
Easy Pre-Calculus Step-by-Step, Second Edition Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect Algebra Rating: 0 out of 5 stars0 ratingsAlgebra & Trigonometry Super Review - 2nd Ed. Rating: 0 out of 5 stars0 ratingsSchaum's Outline of Precalculus, Fourth Edition Rating: 5 out of 5 stars5/5Schaum's Outline of Intermediate Algebra, Third Edition Rating: 3 out of 5 stars3/5Practice Makes Perfect Algebra I Review and Workbook, Second Edition Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect Pre-Algebra Rating: 5 out of 5 stars5/5Practice Makes Perfect Algebra II Rating: 0 out of 5 stars0 ratingsPractice Makes Perfect: Algebra II Review and Workbook, Third Edition Rating: 0 out of 5 stars0 ratingsMath Word Problems Demystified 2/E Rating: 3 out of 5 stars3/5Schaum's Outline of Precalculus, 3rd Edition: 738 Solved Problems + 30 Videos Rating: 4 out of 5 stars4/5Practice Makes Perfect: Algebra I Review and Workbook, Third Edition Rating: 2 out of 5 stars2/5Practice Makes Perfect Algebra II Review and Workbook, Second Edition Rating: 4 out of 5 stars4/5Easy Physics Step-by-Step: With 95 Solved Problems Rating: 0 out of 5 stars0 ratingsSchaum's Outline of Probability, Third Edition Rating: 5 out of 5 stars5/5Trigonometry Workbook: Teacher Guide Rating: 0 out of 5 stars0 ratings2000 Solved Problems in Discrete Mathematics Rating: 4 out of 5 stars4/5The Practically Cheating Calculus Handbook Rating: 4 out of 5 stars4/5Schaum's Outline of Linear Algebra, Sixth Edition Rating: 4 out of 5 stars4/5Calculus DeMYSTiFieD, Second Edition Rating: 3 out of 5 stars3/5Algebra I Essentials For Dummies Rating: 2 out of 5 stars2/5Easy Algebra Step-by-Step Rating: 0 out of 5 stars0 ratingsLinear Algebra Demystified Rating: 0 out of 5 stars0 ratingsSchaum's Outline of College Algebra, Fourth Edition Rating: 3 out of 5 stars3/5Practice Makes Perfect Linear Algebra (EBOOK): With 500 Exercises Rating: 0 out of 5 stars0 ratingsPre-Algebra DeMYSTiFieD, Second Edition Rating: 0 out of 5 stars0 ratingsEasy Mathematics Step-by-Step Rating: 4 out of 5 stars4/5Algebra II Workbook For Dummies Rating: 4 out of 5 stars4/5Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice) Rating: 0 out of 5 stars0 ratingsAlgebra Know-It-ALL: Beginner to Advanced, and Everything in Between Rating: 5 out of 5 stars5/5
Training For You
Positioning: The Battle for Your Mind Rating: 4 out of 5 stars4/5Communicating in Business English: Business English Rating: 4 out of 5 stars4/5The English You Need for the Office: Business English Rating: 5 out of 5 stars5/5The McKinsey Way Rating: 4 out of 5 stars4/5Electronic Shorthand: An Easy-To-Learn Method Of Rapid Digital Note-Taking Rating: 5 out of 5 stars5/5Crucial Conversations Tools for Talking When Stakes Are High, Second Edition Rating: 4 out of 5 stars4/5Crucial Conversations: Tools for Talking When Stakes are High, Third Edition Rating: 4 out of 5 stars4/5Practice Makes Perfect Mastering Writing Rating: 5 out of 5 stars5/5HBR Guide to Coaching Employees (HBR Guide Series) Rating: 5 out of 5 stars5/5How to Make Money in Stocks Complete Investing System (EBOOK) Rating: 4 out of 5 stars4/5The Job Interview Phrase Book: The Things to Say to Get You the Job You Want Rating: 4 out of 5 stars4/5How to Trade In Stocks Rating: 4 out of 5 stars4/5The Disease to Please: Curing the People-Pleasing Syndrome Rating: 4 out of 5 stars4/5The 2-Hour Workshop Blueprint: Design Fast. Deliver Strong. Without Stress. Rating: 4 out of 5 stars4/5Team Building inside 0 - icebreakers: Create and live the team spirit! Rating: 3 out of 5 stars3/5Go It Alone!: The Secret to Building a Successful Business on Your Own Rating: 4 out of 5 stars4/5You Can't Lie to Me: The Revolutionary Program to Supercharge Your Inner Lie Detector and Get to the Truth Rating: 4 out of 5 stars4/5The Hero and the Outlaw: Building Extraordinary Brands Through the Power of Archetypes Rating: 5 out of 5 stars5/5The McGraw-Hill 36-Hour Course: Operations Management Rating: 5 out of 5 stars5/5McKinsey Mind Rating: 3 out of 5 stars3/5How to Hug a Porcupine: Negotiating the Prickly Points of the Tween Years Rating: 4 out of 5 stars4/5
Reviews for Easy Precalculus Step-by-Step
1 rating0 reviews
Book preview
Easy Precalculus Step-by-Step - Carolyn Wheater
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.
ISBN: 978-0-07-176768-2
MHID: 0-07-176768-1
The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-176767-5, MHID: 0-07-176767-3.
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps.
McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected].
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
TERMS OF USE
This is a copyrighted work and The McGraw-Hill Companies, Inc. (McGraw-Hill
) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms.
THE WORK IS PROVIDED AS IS.
McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.
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 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
15 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 Precalculus Step-by-Step is an interactive approach to the mathematics necessary for precalculus. 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.
Precalculus, 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.
Precalculus 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 precalculus, 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 Precalculus Step-by-Step will help you acquire the competence and the confidence you need to succeed in precalculus, calculus, and all your mathematical undertakings.
1
Graphs and the Graphing Calculator
A great deal of work in precalculus 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 , where c is some constant. Vertical lines are the real oddity. They are not functions, and their equations don’t fit the standard. The equation of a vertical line is , 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 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: .
• 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, y2) from which you can calculate the slope, using the slope formula . Then the equation can be written by replacing x1, y1, and m in the point-slope form 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 .
2. Sketch the graph of .
3. Sketch the graph of .
4. Sketch the graph of .
5. Write the equation of a line with slope of –3 and a y-intercept of 5.
6. Write the equation of a line with slope of 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 through the point (–7,4).
9. Write the equation of a line perpendicular to 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 , 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 , where m is the slope and b is the y-intercept. The parent function is the function , 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).
Quadratic Function
The quadratic function has the form , for real numbers a, b, and c, with , but is easier to graph when the equation is in vertex form, . In this form, the vertex is (h, k). The parent function for the quadratic family is the function , with 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).
Square Root Function
The 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 , for real numbers a, b, c, and d, with , but here, too, the graph can be sketched more