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Get ahead in pre-calculus
Pre-calculus courses have become increasingly popular with 35 percent of students in the U.S. taking the course in middle or high school. Often, completion of such a course is a prerequisite for calculus and other upper level mathematics courses.
Pre-Calculus For Dummies is an invaluable resource for students enrolled in pre-calculus courses. By presenting the essential topics in a clear and concise manner, the book helps students improve their understanding of pre-calculus and become prepared for upper level math courses.
Whether used as a classroom aid or as a refresher in preparation for an introductory calculus course, this book is one you’ll want to have on hand to perform your very best.
Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.
Introduction 1
About This Book 1
Foolish Assumptions 2
Icons Used in This Book 3
Beyond the Book 3
Where to Go from Here 3
Part 1: Getting Started with Pre-Calculus 5
Chapter 1: Pre-Pre-Calculus 7
Pre-Calculus: An Overview 8
All the Number Basics (No, Not How to Count Them!) 9
The multitude of number types: Terms to know 9
The fundamental operations you can perform on numbers 11
The properties of numbers: Truths to remember 11
Visual Statements: When Math Follows Form with Function 12
Basic terms and concepts 13
Graphing linear equalities and inequalities 14
Gathering information from graphs 15
Get Yourself a Graphing Calculator 16
Chapter 2: Playing with Real Numbers 19
Solving Inequalities 19
Recapping inequality how-tos 20
Solving equations and inequalities when absolute value is involved 20
Expressing solutions for inequalities with interval notation 22
Variations on Dividing and Multiplying: Working with Radicals and Exponents 24
Defining and relating radicals and exponents 24
Rewriting radicals as exponents (or, creating rational exponents) 25
Getting a radical out of a denominator: Rationalizing 26
Chapter 3: The Building Blocks of Pre-Calculus Functions 31
Qualities of Special Function Types and Their Graphs 32
Even and odd functions 32
One-to-one functions 32
Dealing with Parent Functions and Their Graphs 33
Linear functions 33
Quadratic functions 33
Square-root functions 34
Absolute-value functions 34
Cubic functions 35
Cube-root functions 36
Graphing Functions That Have More Than One Rule: Piece-Wise Functions 37
Setting the Stage for Rational Functions 38
Step 1: Search for vertical asymptotes 39
Step 2: Look for horizontal asymptotes 40
Step 3: Seek out oblique asymptotes 41
Step 4: Locate the x- and y-intercepts 42
Putting the Results to Work: Graphing Rational Functions 42
Chapter 4: Operating on Functions 49
Transforming the Parent Graphs 50
Stretching and flattening 50
Translations 52
Reflections 54
Combining various transformations (a transformation in itself!) 55
Transforming functions point by point 57
Sharpen Your Scalpel: Operating on Functions 58
Adding and subtracting 59
Multiplying and dividing 60
Breaking down a composition of functions 60
Adjusting the domain and range of combined functions (if applicable) 61
Turning Inside Out with Inverse Functions 63
Graphing an inverse 64
Inverting a function to find its inverse 66
Verifying an inverse 66
Chapter 5: Digging Out and Using Roots to Graph Polynomial Functions 69
Understanding Degrees and Roots 70
Factoring a Polynomial Expression 71
Always the first step: Looking for a GCF 72
Unwrapping the box containing a trinomial 73
Recognizing and factoring special polynomials 74
Grouping to factor four or more terms 77
Finding the Roots of a Factored Equation 78
Cracking a Quadratic Equation When It Won’t Factor 79
Using the quadratic formula 79
Completing the square 80
Solving Unfactorable Polynomials with a Degree Higher Than Two 81
Counting a polynomial’s total roots 82
Tallying the real roots: Descartes’s rule of signs 82
Accounting for imaginary roots: The fundamental theorem of algebra 83
Guessing and checking the real roots 84
Put It in Reverse: Using Solutions to Find Factors 90
Graphing Polynomials 91
When all the roots are real numbers 91
When roots are imaginary numbers: Combining all techniques 95
Chapter 6: Exponential and Logarithmic Functions 97
Exploring Exponential Functions 98
Searching the ins and outs of exponential functions 98
Graphing and transforming exponential functions 100
Logarithms: The Inverse of Exponential Functions 102
Getting a better handle on logarithms 102
Managing the properties and identities of logs 103
Changing a log’s base 105
Calculating a number when you know its log: Inverse logs 105
Graphing logs 106
Base Jumping to Simplify and Solve Equations 109
Stepping through the process of exponential equation solving 109
Solving logarithmic equations 112
Growing Exponentially: Word Problems in the Kitchen 113
Part 2: The Essentials of Trigonometry 117
Chapter 7: Circling in on Angles 119
Introducing Radians: Circles Weren’t Always Measured in Degrees 120
Trig Ratios: Taking Right Triangles a Step Further 121
Making a sine 121
Looking for a cosine 122
Going on a tangent 124
Discovering the flip side: Reciprocal trig functions 125
Working in reverse: Inverse trig functions 126
Understanding How Trig Ratios Work on the Coordinate Plane 127
Building the Unit Circle by Dissecting the Right Way 129
Familiarizing yourself with the most common angles 129
Drawing uncommon angles 131
Digesting Special Triangle Ratios 132
The 45er: 45 -45 -90 triangle 132
The old 30-60: 30 -60 -90 triangle 133
Triangles and the Unit Circle: Working Together for the Common Good 135
Placing the major angles correctly, sans protractor 135
Retrieving trig-function values on the unit circle 138
Finding the reference angle to solve for angles on the unit circle 142
Measuring Arcs: When the Circle Is Put in Motion 146
Chapter 8: Simplifying the Graphing and Transformation of Trig Functions 149
Drafting the Sine and Cosine Parent Graphs 150
Sketching sine 150
Looking at cosine 152
Graphing Tangent and Cotangent 154
Tackling tangent 154
Clarifying cotangent 157
Putting Secant and Cosecant in Pictures 159
Graphing secant 159
Checking out cosecant 161
Transforming Trig Graphs 162
Messing with sine and cosine graphs 163
Tweaking tangent and cotangent graphs 173
Transforming the graphs of secant and cosecant 176
Chapter 9: Identifying with Trig Identities: The Basics 181
Keeping the End in Mind: A Quick Primer on Identities 182
Lining Up the Means to the End: Basic Trig Identities 182
Reciprocal and ratio identities 183
Pythagorean identities 185
Even/odd identities 188
Co-function identities 190
Periodicity identities 192
Tackling Difficult Trig Proofs: Some Techniques to Know 194
Dealing with demanding denominators 195
Going solo on each side 199
Chapter 10: Advanced Identities: Your Keys to Success 201
Finding Trig Functions of Sums and Differences 202
Searching out the sine of a b 202
Calculating the cosine of a b 206
Taming the tangent of a b 209
Doubling an Angle and Finding Its Trig Value 211
Finding the sine of a doubled angle 212
Calculating cosines for two 213
Squaring your cares away 215
Having twice the fun with tangents 216
Taking Trig Functions of Common Angles Divided in Two 217
A Glimpse of Calculus: Traveling from Products to Sums and Back 219
Expressing products as sums (or differences) 219
Transporting from sums (or differences) to products 220
Eliminating Exponents with Power-Reducing Formulas 221
Chapter 11: Taking Charge of Oblique Triangles with the Laws of Sines and Cosines 223
Solving a Triangle with the Law of Sines 224
When you know two angle measures 225
When you know two consecutive side lengths 228
Conquering a Triangle with the Law of Cosines 235
SSS: Finding angles using only sides 236
SAS: Tagging the angle in the middle (and the two sides) 238
Filling in the Triangle by Calculating Area 240
Finding area with two sides and an included angle (for SAS scenarios) 241
Using Heron’s Formula (for SSS scenarios) 241
Part 3: Analytic Geometry and System Solving 243
Chapter 12: Plane Thinking: Complex Numbers and Polar Coordinates 245
Understanding Real versus Imaginary 246
Combining Real and Imaginary: The Complex Number System 247
Grasping the usefulness of complex numbers 247
Performing operations with complex numbers 248
Graphing Complex Numbers 250
Plotting Around a Pole: Polar Coordinates 251
Wrapping your brain around the polar coordinate plane 252
Graphing polar coordinates with negative values 254
Changing to and from polar coordinates 256
Picturing polar equations 259
Chapter 13: Creating Conics by Slicing Cones 263
Cone to Cone: Identifying the Four Conic Sections 264
In picture (graph form) 264
In print (equation form) 266
Going Round and Round: Graphing Circles 267
Graphing circles at the origin 267
Graphing circles away from the origin 268
Writing in center–radius form 269
Riding the Ups and Downs with Parabolas 270
Labeling the parts 270
Understanding the characteristics of a standard parabola 271
Plotting the variations: Parabolas all over the plane 272
The vertex, axis of symmetry, focus, and directrix 273
Identifying the min and max of vertical parabolas 276
The Fat and the Skinny on the Ellipse 278
Labeling ellipses and expressing them with algebra 279
Identifying the parts from the equation 281
Pair Two Curves and What Do You Get? Hyperbolas 284
Visualizing the two types of hyperbolas and their bits and pieces 284
Graphing a hyperbola from an equation 287
Finding the equations of asymptotes 287
Expressing Conics Outside the Realm of Cartesian Coordinates 289
Graphing conic sections in parametric form 290
The equations of conic sections on the polar coordinate plane 292
Chapter 14: Streamlining Systems, Managing Variables 295
A Primer on Your System-Solving Options 296
Algebraic Solutions of Two-Equation Systems 297
Solving linear systems 297
Working nonlinear systems 300
Solving Systems with More than Two Equations 304
Decomposing Partial Fractions 306
Surveying Systems of Inequalities 307
Introducing Matrices: The Basics 309
Applying basic operations to matrices 310
Multiplying matrices by each other 311
Simplifying Matrices to Ease the Solving Process 312
Writing a system in matrix form 313
Reduced row-echelon form 313
Augmented form 314
Making Matrices Work for You 315
Using Gaussian elimination to solve systems 316
Multiplying a matrix by its inverse 320
Using determinants: Cramer’s Rule 323
Chapter 15: Sequences, Series, and Expanding Binomials for the Real World 327
Speaking Sequentially: Grasping the General Method 328
Determining a sequence’s terms 328
Working in reverse: Forming an expression from terms 329
Recursive sequences: One type of general sequence 330
Difference between Terms: Arithmetic Sequences 331
Using consecutive terms to find another 332
Using any two terms 332
Ratios and Consecutive Paired Terms: Geometric Sequences 334
Identifying a particular term when given consecutive terms 334
Going out of order: Dealing with nonconsecutive terms 335
Creating a Series: Summing Terms of a Sequence 337
Reviewing general summation notation 337
Summing an arithmetic sequence 338
Seeing how a geometric sequence adds up 339
Expanding with the Binomial Theorem 342
Breaking down the binomial theorem 344
Expanding by using the binomial theorem 345
Chapter 16: Onward to Calculus 351
Scoping Out the Differences between Pre-Calculus and Calculus 352
Understanding Your Limits 353
Finding the Limit of a Function 355
Graphically 355
Analytically 356
Algebraically 357
Operating on Limits: The Limit Laws 361
Calculating the Average Rate of Change 362
Exploring Continuity in Functions 363
Determining whether a function is continuous 364
Discontinuity in rational functions 365
Part 4: The Part of Tens 367
Chapter 17: Ten Polar Graphs 369
Spiraling Outward 369
Falling in Love with a Cardioid 370
Cardioids and Lima Beans 370
Leaning Lemniscates 371
Lacing through Lemniscates 372
Roses with Even Petals 372
A rose Is a Rose Is a Rose 373
Limaçon or Escargot? 373
Limaçon on the Side 374
Bifolium or Rabbit Ears? 374
Chapter 18: Ten Habits to Adjust before Calculus 375
Figure Out What the Problem Is Asking 375
Draw Pictures (the More the Better) 376
Plan Your Attack — Identify Your Targets 377
Write Down Any Formulas 377
Show Each Step of Your Work 378
Know When to Quit 378
Check Your Answers 379
Practice Plenty of Problems 380
Keep Track of the Order of Operations 380
Use Caution When Dealing with Fractions 381
Index 383
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