INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC

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1 INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions Page 5 General Approach Page 5 Number Systems Page 6 The Sum or Difference of Two Cubes Page 7 Disguised Quadratic Expressions Page 8 Completing the Square Page 9 Grouping Terms Page 10 Long Division of Polynomial Expressions Page 11 The Remainder Theorem Page 15 The Factor Theorem Page 16 Writing Expressions in Specific Forms Page 17 Solving Algebraic Equations Page 18 The Quadratic Formula Page 19 Verifying Solutions Page 20 Literal Equations Page 22 Equating Coefficients Page 23 Points of Intersection Page 24 Simultaneous Equations Page 25 Finding Solutions of Literal Simultaneous Equations Page 28 The Discriminant and Its Applications Page 31 The Binomial Theorem Page 33 Expanding Terms Page 33 Pascal s Triangle Page 33 Finding a Specific Term Page 34 Finding the Coefficient of a Specific Term Page 35

2 Matrices Page 36 Matrix Arithmetic Page 36 Matrix Multiplication Page 38 The Identity Matrix Page 39 The Determinant & Inverse Matrix Page 40 Solving Matrix Equations Page 42 Solving Linear Equations in Terms of Two Variables Page 43 Solutions of Simultaneous Linear Equations Page 44 Simultaneous Equations With More than Two Variables Page 47 Algebra of Indicial & Exponential Functions Page 48 Index Laws Page 48 Simplifying Indicial Expressions Page 50 Solving Indicial Equations Page 51 Algebra of Logarithmic Functions Page 55 Simplifying Logarithmic Expressions Page 55 Logarithmic Laws Page 56 Change of Base Rule Page 58 Solving Logarithmic Equations Page 59 Algebra of Trigonometric Functions Page 63 Measuring Angles & Angle Conversions Page 63 The Unit Circle Page 64 Reciprocal Functions Page 65 Important Exact Values Page 66 Symmetry Properties Supplementary Rules Page 67 Symmetry Properties Complementary Rules Page 67 Finding Exact Values of Trigonometric Expressions Method Page 69 Trigonometric Identities Page 70 Compound Angle Formulae Page 71 Double Angle Formulae Page 72 Solving Trigonometric Equations Page 73 Solving Complex Trigonometric Equations Page 75 General Solutions for Trigonometric Equations Page 76 Solving Inequations Page 80 Functional Equations Page 82

3 RELATIONS, FUNCTIONS AND THEIR GRAPHS Relations, Functions and Their Graphs Page 83 The Horizontal Line Test Page 83 The Vertical Line Test Page 83 Types of Relations Page 84 Functions and Inverses Page 85 Function Notation and Number Systems Page 86 The Domain Page 87 The Range Page 88 Notations Used to Describe Domains and Ranges Page 88 The Largest Possible Domain Page 90 Continuity and Discontinuity Page 91 Curve Sketching Page 93 Common Curve Shapes Page 93 Sketching Curves by Considering the Essential Features of a Graph Page 94 X and Y Intercepts Page 94 Stationary Points Page 95 Asymptotes Page 96 Asymptotic Behaviour Page 98 Sketching Inequations Page 100 Polynomial Functions Page 101 Linear Functions and Their Graphs Page 102 Quadratic Functions and Their Graphs Page 104 Cubic Functions and Their Graphs Page 108 Quartic Functions and Their Graphs Page 110 Higher Order Polynomial Functions Page 113 Analysing Technology Neutral Questions Page 116 Negative Power Functions Page 117 The Rectangular Hyperbola and its Graphs Page 117 The Truncus and its Graphs Page 120 Higher Order Negative Power Functions Page 122

4 Fractional Power Functions Page 123 Square Root Functions and Their Graphs Page 123 Other Fractional Power Functions Page 125 The Exponential Graph Page 126 The Logarithmic Graph Page 127 Graphs of Trigonometric Functions Page 129 The Amplitude Page 130 The Period Page 131 Horizontal Translations Page 132 Vertical Translations Page 133 Asymptotes Page 134 The Range or Maximum/Minimum Values Page 135 Functions, Relations and Transformations Page 136 Dilations Page 137 Reflections Page 139 Translations Page 141 Summary of Transformation Notations Page 143 The Order of Transformations Page 143 Identifying Transformations Page 144 Transformations Involving Power Functions Page 145 Transformations Involving Hyperbolae and Higher Order Functions Page 147 Transformations Involving the Truncus and Higher Order Functions Page 148 Transformations Involving the Square Root Function and Related Functions Page 149 Transformations Involving Exponential Functions Page 150 Transformations Involving Logarithmic Functions Page 151 Transformations Involving Trigonometric Functions Page 152 Matrix Representation of Transformations Page 153 Finding the Image of a Point Page 154 Finding the Image of a Function or Relation Page 156

5 SPECIAL RELATIONS The Modulus (Absolute Value) Function Page 157 Properties of the Modulus Function Page 158 Intervals and Functions Page 160 Sketching Absolute Value Functions in the Form y f (x) Page 161 Transformations Involving the Absolute Value Function Page 162 Sketching Absolute Value Functions in the Form y a f( x) b Page 163 Absolute Value Functions in the Form f x Sketching Absolute Value Functions in the Form y f x and y f g( x) y Page 164 Page 165 Defining Absolute Value Functions Over Relevant Intervals Page 167 Sketching Absolute Value Functions Using Algebra Page 168 Solving Modulus Equations and Inequations Page 168 The Algebra of Functions Page 170 Addition and Subtraction of Ordinates Page 171 Multiplication of Ordinates Page 172 Composite Functions Page 173 Common Operations Page 174 Sketching Composite Functions Page 175 Existence of Composite Functions Page 176 Domains and Ranges of Composite Functions Page 176 Inverse Relations Page 177 Inverse Functions Page 177 Identifying Inverse Pairs Page 178 Sketching Inverse Relations Page 178 Writing Equations Describing Inverse Functions Page 179

6 Finding Equations of Relations and Functions Page 182 Finding Equations of Linear Functions Page 183 Determining Rules for Quadratic Functions Page 184 Determining Rules for Cubic Functions Page 185 Determining Rules for Quartic Functions Page 186 Finding Equations Describing Exponential Functions Page 187 Finding Equations Describing Logarithmic Functions Page 188 Determining Rules for Trigonometric Functions Page 189 Modelling Page 192 Polynomial Models Page 192 Modelling With Exponential Functions Page 196 CALCULUS Limits and Derivatives Page 198 Conditions for the Existence of a Limit Page 198 Evaluating Limits Graphically Page 199 Evaluating Limits Algebraically Page 200 Limit Theorems Page 201 Differentiation Page 202 Derivatives from First Principles Page 202 Differentiation by Rule Page 204 Derivatives of Polynomial and Rational Functions Page 204 Summary of Differentiation Techniques Page 205 Finding Derivatives Method Page 206 Important Derivatives Page 208 The Chain Rule Page 209 The Product Rule Page 214 The Quotient Rule Page 215