Pre-Calculus For Dummies, 3ed

Introduction 

About This Book

Foolish Assumptions

Icons Used in This Book

Beyond the Book

Where to Go from Here

Part 1: Getting Started with Pre-Calculus 

Chapter 1: Pre-Pre-Calculus 

Pre-Calculus: An Overview

All the Number Basics (No, Not How to Count Them!)

The multitude of number types: Terms to know

The fundamental operations you can perform on numbers

The properties of numbers: Truths to remember

Visual Statements: When Math Follows Form with Function

Basic terms and concepts

Graphing linear equalities and inequalities

Gathering information from graphs

Get Yourself a Graphing Calculator

Chapter 2: Playing with Real Numbers

Solving Inequalities

Recapping inequality how-tos

Solving equations and inequalities when absolute value is involved

Expressing solutions for inequalities with interval notation

Variations on Dividing and Multiplying: Working with Radicals and Exponents

Defining and relating radicals and exponents

Rewriting radicals as exponents (or, creating rational exponents)

Getting a radical out of a denominator: Rationalizing

Chapter 3: The Building Blocks of Pre-Calculus Functions 

Qualities of Special Function Types and Their Graphs

Even and odd functions

One-to-one functions

Dealing with Parent Functions and Their Graphs

Linear functions

Quadratic functions

Square-root functions

Absolute-value functions

Cubic functions

Cube-root functions

Graphing Functions That Have More Than One Rule: Piece-Wise Functions

Setting the Stage for Rational Functions

Step 1: Search for vertical asymptotes

Step 2: Look for horizontal asymptotes

Step 3: Seek out oblique asymptotes

Step 4: Locate the x- and y-intercepts

Putting the Results to Work: Graphing Rational Functions

Chapter 4: Operating on Functions 

Transforming the Parent Graphs

Stretching and flattening

Translations

Reflections

Combining various transformations (a transformation in itself!)

Transforming functions point by point

Sharpen Your Scalpel: Operating on Functions

Adding and subtracting

Multiplying and dividing

Breaking down a composition of functions

Adjusting the domain and range of combined functions (if applicable)

Turning Inside Out with Inverse Functions

Graphing an inverse

Inverting a function to find its inverse

Verifying an inverse

Chapter 5: Digging Out and Using Roots to Graph Polynomial Functions 

Understanding Degrees and Roots

Factoring a Polynomial Expression

Always the first step: Looking for a GCF

Unwrapping the box containing a trinomial

Recognizing and factoring special polynomials

Grouping to factor four or more terms

Finding the Roots of a Factored Equation

Cracking a Quadratic Equation When It Won’t Factor

Using the quadratic formula

Completing the square

Solving Unfactorable Polynomials with a Degree Higher Than Two

Counting a polynomial’s total roots

Tallying the real roots: Descartes’s rule of signs

Accounting for imaginary roots: The fundamental theorem of algebra

Guessing and checking the real roots

Put It in Reverse: Using Solutions to Find Factors

Graphing Polynomials

When all the roots are real numbers

When roots are imaginary numbers: Combining all techniques

Chapter 6: Exponential and Logarithmic Functions 

Exploring Exponential Functions

Searching the ins and outs of exponential functions

Graphing and transforming exponential functions

Logarithms: The Inverse of Exponential Functions

Getting a better handle on logarithms

Managing the properties and identities of logs

Changing a log’s base

Calculating a number when you know its log: Inverse logs

Graphing logs

Base Jumping to Simplify and Solve Equations

Stepping through the process of exponential equation solving

Solving logarithmic equations

Growing Exponentially: Word Problems in the Kitchen

Part 2: The Essentials of Trigonometry 

Chapter 7: Circling in on Angles 

Introducing Radians: Circles Weren’t Always Measured in Degrees

Trig Ratios: Taking Right Triangles a Step Further

Making a sine

Looking for a cosine

Going on a tangent

Discovering the flip side: Reciprocal trig functions

Working in reverse: Inverse trig functions

Understanding How Trig Ratios Work on the Coordinate Plane

Building the Unit Circle by Dissecting the Right Way

Familiarizing yourself with the most common angles

Drawing uncommon angles

Digesting Special Triangle Ratios

The 45er: 45 -45 -90 triangle

The old 30-60: 30 -60 -90 triangle

Triangles and the Unit Circle: Working Together for the Common Good

Placing the major angles correctly, sans protractor

Retrieving trig-function values on the unit circle

Finding the reference angle to solve for angles on the unit circle

Measuring Arcs: When the Circle Is Put in Motion

Chapter 8: Simplifying the Graphing and Transformation of Trig Functions 

Drafting the Sine and Cosine Parent Graphs

Sketching sine

Looking at cosine

Graphing Tangent and Cotangent

Tackling tangent

Clarifying cotangent

Putting Secant and Cosecant in Pictures

Graphing secant

Checking out cosecant

Transforming Trig Graphs

Messing with sine and cosine graphs

Tweaking tangent and cotangent graphs

Transforming the graphs of secant and cosecant

Chapter 9: Identifying with Trig Identities: The Basics 

Keeping the End in Mind: A Quick Primer on Identities

Lining Up the Means to the End: Basic Trig Identities

Reciprocal and ratio identities

Pythagorean identities

Even/odd identities

Co-function identities

Periodicity identities

Tackling Difficult Trig Proofs: Some Techniques to Know

Dealing with demanding denominators

Going solo on each side

Chapter 10: Advanced Identities: Your Keys to Success 

Finding Trig Functions of Sums and Differences

Searching out the sine of a b 

Calculating the cosine of a b 

Taming the tangent of a b

Doubling an Angle and Finding Its Trig Value

Finding the sine of a doubled angle

Calculating cosines for two

Squaring your cares away

Having twice the fun with tangents

Taking Trig Functions of Common Angles Divided in Two

A Glimpse of Calculus: Traveling from Products to Sums and Back

Expressing products as sums (or differences)

Transporting from sums (or differences) to products

Eliminating Exponents with Power-Reducing Formulas

Chapter 11: Taking Charge of Oblique Triangles with the Laws of Sines and Cosines 

Solving a Triangle with the Law of Sines

When you know two angle measures

When you know two consecutive side lengths

Conquering a Triangle with the Law of Cosines

SSS: Finding angles using only sides

SAS: Tagging the angle in the middle (and the two sides)

Filling in the Triangle by Calculating Area

Finding area with two sides and an included angle (for SAS scenarios)

Using Heron’s Formula (for SSS scenarios)

Part 3: Analytic Geometry and System Solving 

Chapter 12: Plane Thinking: Complex Numbers and Polar Coordinates 

Understanding Real versus Imaginary

Combining Real and Imaginary: The Complex Number System

Grasping the usefulness of complex numbers

Performing operations with complex numbers

Graphing Complex Numbers

Plotting Around a Pole: Polar Coordinates

Wrapping your brain around the polar coordinate plane

Graphing polar coordinates with negative values

Changing to and from polar coordinates

Picturing polar equations

Chapter 13: Creating Conics by Slicing Cones 

Cone to Cone: Identifying the Four Conic Sections

In picture (graph form)

In print (equation form)

Going Round and Round: Graphing Circles

Graphing circles at the origin

Graphing circles away from the origin

Writing in center–radius form

Riding the Ups and Downs with Parabolas

Labeling the parts

Understanding the characteristics of a standard parabola

Plotting the variations: Parabolas all over the plane

The vertex, axis of symmetry, focus, and directrix

Identifying the min and max of vertical parabolas

The Fat and the Skinny on the Ellipse

Labeling ellipses and expressing them with algebra

Identifying the parts from the equation

Pair Two Curves and What Do You Get? Hyperbolas

Visualizing the two types of hyperbolas and their bits and pieces

Graphing a hyperbola from an equation

Finding the equations of asymptotes

Expressing Conics Outside the Realm of Cartesian Coordinates

Graphing conic sections in parametric form

The equations of conic sections on the polar coordinate plane

Chapter 14: Streamlining Systems, Managing Variables 

A Primer on Your System-Solving Options

Algebraic Solutions of Two-Equation Systems

Solving linear systems

Working nonlinear systems

Solving Systems with More than Two Equations

Decomposing Partial Fractions

Surveying Systems of Inequalities

Introducing Matrices: The Basics

Applying basic operations to matrices

Multiplying matrices by each other

Simplifying Matrices to Ease the Solving Process

Writing a system in matrix form

Reduced row-echelon form

Augmented form

Making Matrices Work for You

Using Gaussian elimination to solve systems

Multiplying a matrix by its inverse

Using determinants: Cramer’s Rule

Chapter 15: Sequences, Series, and Expanding Binomials for the Real World 

Speaking Sequentially: Grasping the General Method

Determining a sequence’s terms

Working in reverse: Forming an expression from terms

Recursive sequences: One type of general sequence

Difference between Terms: Arithmetic Sequences

Using consecutive terms to find another

Using any two terms

Ratios and Consecutive Paired Terms: Geometric Sequences

Identifying a particular term when given consecutive terms

Going out of order: Dealing with nonconsecutive terms

Creating a Series: Summing Terms of a Sequence

Reviewing general summation notation

Summing an arithmetic sequence

Seeing how a geometric sequence adds up

Expanding with the Binomial Theorem

Breaking down the binomial theorem

Expanding by using the binomial theorem

Chapter 16: Onward to Calculus 

Scoping Out the Differences between Pre-Calculus and Calculus

Understanding Your Limits

Finding the Limit of a Function

Graphically

Analytically

Algebraically

Operating on Limits: The Limit Laws

Calculating the Average Rate of Change

Exploring Continuity in Functions

Determining whether a function is continuous

Discontinuity in rational functions

Part 4: The Part of Tens 

Chapter 17: Ten Polar Graphs 

Spiraling Outward

Falling in Love with a Cardioid

Cardioids and Lima Beans

Leaning Lemniscates

Lacing through Lemniscates

Roses with Even Petals

A rose Is a Rose Is a Rose

Limaçon or Escargot?

Limaçon on the Side

Bifolium or Rabbit Ears?

Chapter 18: Ten Habits to Adjust before Calculus 

Figure Out What the Problem Is Asking

Draw Pictures (the More the Better)

Plan Your Attack — Identify Your Targets

Write Down Any Formulas

Show Each Step of Your Work

Know When to Quit

Check Your Answers

Practice Plenty of Problems

Keep Track of the Order of Operations

Use Caution When Dealing with Fractions

Index