Table of Contents

About This Book

Conventions Used in This Book

Foolish Assumptions

How This Book Is Organized

Part I: Set It Up, Solve It, Graph It

Part II: The Essentials of Trigonometry

Part III: Analytic Geometry and System Solving

Part IV: The Part of Tens

Icons Used in This Book

Where to Go from Here

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-Calc: Functions

Qualities of Even and Odd Functions and Their Graphs

Dealing with Parent Functions and Their Graphs

Quadratic functions

Square-root functions

Absolute-value functions

Cubic functions

Cube-root functions

Transforming the Parent Graphs

Vertical transformations

Horizontal transformations

Translations

Reflections

Combining various transformations (a transformation in itself!)

Transforming functions point by point

Graphing Functions that Have More than One Rule: Piece-Wise Functions

Calculating Outputs 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 Output to Work: Graphing Rational Functions

The denominator has the greater degree

The numerator and denominator have equal degrees

The numerator has the greater degree

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 4: 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 FOIL method for trinomials

Recognizing and factoring special types of 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 5: Exponential and Logarithmic Functions

Exploring Exponential Functions

Searching the ins and outs of an exponential function

Graphing and transforming an exponential function

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 logarithm equations

Growing Exponentially: Word Problems in the Kitchen

Chapter 6: 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° triangles

The old 30-60: 30°-60°-90° triangles

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 7: Simplifying the Graphing and Transformation of Trig Functions

Drafting the Sine and Cosine Parent Graphs

Sketching sine

Looking at cosine

Graphing Tangent and Cotangent

Tacking tangent

Clarifying cotangent

Putting Secant and Cosecant in Pictures

Graphing secant

Checking out cosecant

Transforming Trig Graphs

Screwing with sine and cosine graphs

Tweaking tangent and cotangent graphs

Transforming the graphs of secant and cosecant

Chapter 8: 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 identities

Pythagorean identities

Even/odd identities

Co-function identities

Periodicity identities

Tackling Difficult Trig Proofs: Some Techniques to Know

Dealing with dreaded denominators

Going solo on each side

Chapter 9: Advanced Identities: Your Keys to Pre-Calc 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’s Trig Value without Knowing the Angle

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 10: 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)

Chapter 11: Plane Thinking: Complex Numbers and Polar Coordinates

Understanding Real versus Imaginary (According to Mathematicians)

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 12: Slicing Cones or Measuring the Path of a Comet — with Confidence

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

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 (Not at the origin)

Finding the vertex, axis of symmetry, focus, and directrix

Identifying the min and max on vertical parabolas

The Fat and the Skinny on the Ellipse (A Fancy Word for Oval)

Labeling ellipses and expressing them with algebra

Identifying the parts of the oval: Vertices, co-vertices, axes, and foci

Pair Two Parabolas 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 equation 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 13: Streamlining Systems, Managing Variables

A Primer on Your System-Solving Options

Finding Solutions of Two-Equation Systems Algebraically

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

Finding reduced row echelon form

Augmented form

Conquering Matrices

Using Gaussian elimination to solve systems

Multiplying a matrix by its inverse

Using determinants: Cramer’s rule

Chapter 14: Sequences, Series, and Expanding Binomials for the Real World

Speaking Sequentially: Grasping the General Method

Calculating a sequence’s terms by using the sequence expression

Working in reverse: Forming an expression from terms

Recursive sequences: One type of general sequence

Covering the Distance between Terms: Arithmetic Sequences

Using consecutive terms to find another in an arithmetic sequence

Using any two terms

Sharing Ratios with Consecutive Paired Terms: Geometric Sequences

Identifying a term when you know consecutive terms

Going out of order: Finding a term when the terms are nonconsecutive

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

Starting at the beginning: Binomial coefficients

Expanding by using the binomial theorem

Chapter 15: Scouting the Road Ahead to Calculus

Scoping Out the Differences between Pre-Calc and Calc

Understanding and Communicating about Limits

Finding the Limit of a Function

Graphically

Analytically

Algebraically

Operating on Limits: The Limit Laws

Exploring Continuity in Functions

Determining whether a function is continuous

Dealing with discontinuity

Chapter 16: Ten Habits to Develop as You Prepare for Calculus

Figure Out What the Problem Is Asking

Draw Pictures (And Plenty of ’Em)

Plan Your Attack

Write Down Any Formulas

Show Each Step of Your Work

Know When to Quit

Check Your Answers

Practice Plenty of Problems

Make Sure You Understand the Concepts

Pepper Your Teacher with Questions

Chapter 17: Ten Habits to Quit Before Calculus

Operating Out of Order

Squaring without FOILing

Splitting Up Terms in Denominators

Combining the Wrong Terms

Forgetting the Reciprocal

Losing Track of Minus Signs

Oversimplifying Radicals

Erring in Exponential Dealings

Canceling Out Too Quickly

Distributing Improperly