Math Handbook of Formulas, Processes and Tricks

Math Handbook

of Formulas, Processes and Tricks

(mathguy.us)

Algebra and PreCalculus

Prepared by: Earl L. Whitney, FSA, MAAA

Version 3.5

October 17, 2022

Copyright 2008©\22, Earl Whitney, Reno NV. All Rights Reserved

Algebra Handbook

Table of Contents

Page

Description

9

10

11

12

13

14

Chapter 1: Basics

Order of Operations (PEMDAS, Parenthetical Device)

Graphing with Coordinates (Coordinates, Plotting Points)

Linear Patterns (Recognition, Converting to an Equation)

Identifying Number Patterns

Completing Number Patterns

Real Number Sets (Sets of Numbers, Real Number Set Tree)

15

16

Chapter 2: Operations

Operating with Real Numbers (Absolute Value, Add, Subtract, Multiply, Divide)

Properties of Algebra (Addition & Multiplication, Zero, Equality)

18

19

Chapter 3: Solving Equations

Solving Multi©\Step Equations

Tips and Tricks in Solving Multi©\Step Equations

20

21

22

23

Chapter 4: Probability & Statistics

Probability and Odds

Probability with Dice

Combinations

Statistical Measures

24

25

26

27

28

29

30

31

32

33

34

Chapter 5: Functions

Introduction to Functions (Definitions, Line Tests)

Special Integer Functions

Operations with Functions

Composition of Functions

Inverses of Functions

Transformation ¨C Translation

Transformation ¨C Vertical Stretch and Compression

Transformation ¨C Horizontal Stretch and Compression

Transformation ¨C Reflection

Transformation ¨C Summary

Building a Graph with Transformations

Cover art by Rebecca Williams,

Twitter handle: @jolteonkitty

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Algebra Handbook

Table of Contents

Page

Description

35

36

37

38

39

40

41

Chapter 6: Linear Functions

Slope of a Line (Mathematical Definition)

Slope of a Line (Rise over Run)

Slopes of Various Lines (8 Variations)

Various Forms of a Line (Standard, Slope©\Intercept, Point©\Slope)

Slopes of Parallel and Perpendicular Lines

Parallel, Perpendicular or Neither

Parallel, Coincident or Intersecting

42

43

44

45

46

47

48

Chapter 7: Inequalities

Properties of Inequalities

Graphs of Inequalities in One Dimension

Compound Inequalities in One Dimension

Inequalities in Two Dimensions

Graphs of Inequalities in Two Dimensions

Absolute Value Functions (Equations)

Absolute Value Functions (Inequalities)

49

50

51

52

53

54

55

Chapter 8: Systems of Equations

Graphing a Solution

Substitution Method

Elimination Method

Classification of Systems of Equations

Linear Dependence

Systems of Inequalities in Two Dimensions

Parametric Equations

56

57

58

59

Chapter 9: Exponents (Basic) and Scientific Notation

Exponent Formulas

Scientific Notation (Format, Conversion)

Adding and Subtracting with Scientific Notation

Multiplying and Dividing with Scientific Notation

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Algebra Handbook

Table of Contents

Page

Description

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61

62

63

64

65

66

67

68

69

70

71

72

Chapter 10: Polynomials ¨C Basic

Introduction to Polynomials

Adding and Subtracting Polynomials

Multiplying Binomials (FOIL, Box, Numerical Methods)

Multiplying Polynomials

Dividing Polynomials

Factoring Polynomials

Special Forms of Quadratic Functions (Perfect Squares)

Special Forms of Quadratic Functions (Differences of Squares)

Factoring Trinomials ¨C Simple Case Method

Factoring Trinomials ¨C AC Method

Factoring Trinomials ¨C Brute Force Method

Factoring Trinomials ¨C Quadratic Formula Method

Solving Equations by Factoring

73

74

75

76

77

79

Chapter 11: Quadratic Functions

Introduction to Quadratic Functions

Completing the Square

Table of Powers and Roots

The Quadratic Formula

Quadratic Inequalities in One Variable

Fitting a Quadratic through Three Points

80

81

82

83

84

85

Chapter 12: Complex Numbers

Complex Numbers ©\ Introduction

Operations with Complex Numbers

The Square Root of i

Complex Numbers ¨C Graphical Representation

Complex Number Operations in Polar Coordinates

Complex Solutions to Quadratic Equations

86

87

88

89

Chapter 13: Radicals

Radical Rules

Simplifying Square Roots (Extracting Squares, Extracting Primes)

Solving Radical Equations

Solving Radical Equations (Positive Roots, The Missing Step)

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October 17, 2022

Algebra Handbook

Table of Contents

Page

Description

90

91

92

93

94

95

96

97

99

100

101

102

103

104

105

106

107

Chapter 14: Matrices

Matrix Addition and Scalar Multiplication

Matrix Multiplication

Identity Matrices and Inverse Matrices

Inverse of a 2x2 Matrix

Determinants ¨C The General Case

Matrix Division and Inverses

Calculating Inverses ¨C The General Case (Gauss©\Jordan Elimination)

Calculating Inverses Using Adjoint Matrices

Cramer¡¯s Rule ¨C 2 Equations

Cramer¡¯s Rule ¨C 3 Equations

Augmented Matrices

2x2 Augmented Matrix Examples

3x3 Augmented Matrix Example

Characteristic Equation and Eigenvalues

Eigenvectors

2x2 Eigenvalues and Eigenvectors ¨C General Case

Calculating Inverses Using Characteristic Equations

109

110

111

112

113

114

115

116

117

118

119

122

123

124

125

Chapter 15: Exponents and Logarithms

Exponent Formulas

Logarithm Formulas

e

Table of Exponents and Logs

Converting Between Exponential and Logarithmic Forms

Expanding Logarithmic Expressions

Condensing Logarithmic Expressions

Condensing Logarithmic Expressions ¨C More Examples

Graphing an Exponential Function

Four Exponential Function Graphs

Graphing a Logarithmic Function

Four Logarithmic Function Graphs

Graphs of Various Functions

Applications of Exponential Functions (Growth, Decay, Interest)

Solving Exponential and Logarithmic Equations

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