Pre-Calculus For Dummies

Title Page
Copyright Page
Table of Contents
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
Graphing f(x)=3x-1/x2+4x-21
Graphing g(x)=6x+12/4-3x
Graphing h(x)=x2-9/x+2
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
Listing the possibilities with the rational root theorem
Testing roots by dividing polynomials
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:45deg-45deg-90deg
triangle
The old 30-60:30deg-60deg-90deg
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
EULA