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Review Notes for IB Standard Level Math© 2015-2020, Steve Muench
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@stevemuench
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February 3, 2020
1
,Contents
1 Algebra 8
1.1 Rules of Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Rules of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Allowed and Disallowed Calculator Functions During the Exam . . . . . . . . . . 8
1.5 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Sum of Finite Arithmetic Series (u1 + · · · + un ) . . . . . . . . . . . . . . . . . . . 9
1.8 Partial Sum of Finite Arithmetic Series (uj + · · · + un ) . . . . . . . . . . . . . . . 10
1.9 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10 Sum of Finite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.11 Sum of Infinite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.11.1 Example Involving Sum of Infinite Geometric Series . . . . . . . . . . . . 11
1.12 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.12.1 Sigma Notation for Arithmetic Series . . . . . . . . . . . . . . . . . . . . . 13
1.12.2 Sigma Notation for Geometric Series . . . . . . . . . . . . . . . . . . . . . 13
1.12.3 Sigma Notation for Infinite Geometric Series . . . . . . . . . . . . . . . . 14
1.12.4 Defining Functions Using Sigma Notation . . . . . . . . . . . . . . . . . . 14
1.13 Applications: Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.14 Applications: Population Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.15 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.16 Using Logarithms to Solve Equations . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.17 Using Exponentiation to Solve Equations . . . . . . . . . . . . . . . . . . . . . . 17
1.18 Logarithm Facts Involving 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.19 Laws of Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.20 Change of Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.21 Powers of Binomials and Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . 18
1.22 Expansion of (a + b)n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.23 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.23.1 Using The Binomial Theorem for a Single Term . . . . . . . . . . . . . . . 21
1.23.2 Example of Using Binomial Theorem . . . . . . . . . . . . . . . . . . . . . 22
1.24 Solving Systems of Three Linear Equations Using Substitution . . . . . . . . . . 23
1.25 Solving Systems of Three Linear Equations Using Technology . . . . . . . . . . . 24
2 Functions and Equations 25
2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Union and Intersection of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Common Sets of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Intervals of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Concept of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Graph of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Domain of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.8 Range of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 Composing One Function with Another . . . . . . . . . . . . . . . . . . . . . . . 29
2.10 Identity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 Inverse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.12 Determining the Inverse Function as Reflection in Line y =x . . . . . . . . . . . 30
2.13 Determining the Inverse Function Analytically . . . . . . . . . . . . . . . . . . . 31
2.14 Drawing and Analyzing Graphs with Your Calculator . . . . . . . . . . . . . . . 32
2.14.1 Drawing the Graph of a Function . . . . . . . . . . . . . . . . . . . . . . . 32
2
, 2.14.2 Restricting the Domain of a Graph . . . . . . . . . . . . . . . . . . . . . . 32
2.14.3 Zooming Graph to See Exactly What You Want . . . . . . . . . . . . . . 33
2.14.4 Finding a Maximum Value in an Interval . . . . . . . . . . . . . . . . . . 33
2.14.5 Finding a Minimum Value Value in an Interval . . . . . . . . . . . . . . . 34
2.14.6 Finding the x-Intercepts or “Zeros” of a Graph in an Interval . . . . . . . 34
2.14.7 Finding the y-Intercept of a Graph . . . . . . . . . . . . . . . . . . . . . . 35
2.14.8 Vertical Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.14.9 Graphing Vertical Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.14.10 Horizontal Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.14.11 Tips to Compute Horizontal Asymptotes of Rational Functions . . . . . . 37
2.14.12 Graphing Horizontal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.14.13 Symmetry: Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.14.14 Symmetry: Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.14.15 Solving Equations Graphically . . . . . . . . . . . . . . . . . . . . . . . . 38
2.15 Transformations of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.15.1 Horizontal and Vertical Translations . . . . . . . . . . . . . . . . . . . . . 40
2.15.2 Vertical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.15.3 Horizontal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.15.4 Vertical Stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.15.5 Horizontal Stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.15.6 Order Matters When Doing Multiple Transformations in Sequence . . . . 43
2.15.7 Graphing the Result of a Sequence of Transformations . . . . . . . . . . . 44
2.15.8 Determining Point Movement Under a Sequence of Transformations . . . 45
2.15.9 Vector Notation for Function Translation . . . . . . . . . . . . . . . . . . 47
2.16 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.16.1 Using the Quadratic Formula to Find Zeros of Quadratic Function . . . . 49
2.16.2 Finding the Vertex If You Know the Zeros . . . . . . . . . . . . . . . . . . 49
2.16.3 Graph and Axis of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.16.4 Computing the Vertex From the Coefficients . . . . . . . . . . . . . . . . 50
2.16.5 Using the Discriminant to Find the Number of Zeros . . . . . . . . . . . . 51
2.16.6 Y-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.16.7 X-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.16.8 Completing the Square to Get Binomial Squared Form . . . . . . . . . . . 53
2.16.9 Vertex (h, k) Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.17 Reciprocal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.18 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.19 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.20 Continuously Compounded Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.21 Continuous Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.22 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Circular Functions and Trigonometry 59
3.1 Understanding Radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 Degrees Represent a Part of a Circular Path . . . . . . . . . . . . . . . . . 59
3.1.2 Computing the Fraction of a Complete Revolution an Angle Represents . 59
3.1.3 Attempting to Measure an Angle Using Distance . . . . . . . . . . . . . . 59
3.1.4 Arc Distance on the Unit Circle Uniquely Identifies an Angle θ . . . . . . 61
3.1.5 Computing the Fraction of a Complete Revolution for Angle in Radians . 61
3.2 Converting Between Radians and Degrees . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 Converting from Degrees to Radians . . . . . . . . . . . . . . . . . . . . . 61
3.2.2 Converting from Radians to Degrees . . . . . . . . . . . . . . . . . . . . . 62
3
, 3.3 Length of an Arc Subtended by an Angle . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Inscribed and Central Angles that Subtend the Same Arc . . . . . . . . . . . . . 63
3.5 Area of a Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6 Definition of cos θ and sin θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7 Interpreting cos θ and sin θ on the Unit Circle . . . . . . . . . . . . . . . . . . . . 65
3.8 Radian Angle Measures Can Be Both Positive and Negative . . . . . . . . . . . . 65
3.9 Remembering the Exact Values of Key Angles on Unit Circle . . . . . . . . . . . 66
3.10 The Pythagorean Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.11 Double Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.12 Definition of tan θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.13 Using a Right Triangle to Solve Trigonometric Problems . . . . . . . . . . . . . . 68
3.13.1 Using Right Triangle with an Acute Angle . . . . . . . . . . . . . . . . . . 68
3.13.2 Using Right Triangle with an Obtuse Angle . . . . . . . . . . . . . . . . . 69
3.14 Using Inverse Trigonometric Functions on Your Calculator . . . . . . . . . . . . . 70
3.15 Circular Functions sin, cos, and tan . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.15.1 The Graph of sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.15.2 The Graph of cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.15.3 The Graph of tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.15.4 Transformations of Circular Functions . . . . . . . . . . . . . . . . . . . . 72
3.15.5 Using Transformation to Highlight Additional Identities . . . . . . . . . . 73
3.15.6 Determining Period from Minimum and Maximum . . . . . . . . . . . . . 73
3.16 Applications of the sin Function: Tide Example . . . . . . . . . . . . . . . . . . . 74
3.17 Applications of the cos Function: Ferris Wheel Example . . . . . . . . . . . . . . 75
3.18 Solving Trigonometric Equations in a Finite Interval . . . . . . . . . . . . . . . . 77
3.19 Solving Quadratic Equations in sin, cos, and tan . . . . . . . . . . . . . . . . . . 78
3.20 Solutions of Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.21 The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.22 The Sine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.23 Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Vectors 82
4.1 Vectors as Displacements in the Plane . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Vectors as Displacements in Three Dimensions . . . . . . . . . . . . . . . . . . . 82
4.3 Terminology: Tip and Tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Representation of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Magnitude of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 Multiplication of a Vector by a Scalar . . . . . . . . . . . . . . . . . . . . . . . . 85
4.7 Negating a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.8 Sum of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.9 Difference of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.10 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.11 Scaling Any Vector to Produce a Parallel Unit Vector . . . . . . . . . . . . . . . 88
4.12 Position Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.13 Determining Whether Vectors are Parallel . . . . . . . . . . . . . . . . . . . . . . 89
4.14 Finding Parallel Vector with Certain Fixed Length . . . . . . . . . . . . . . . . . 90
4.15 Scalar (or “Dot”) Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . 90
4.16 Perpendicular Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.17 Base Vectors for Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.18 Base Vectors for Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.19 The Angle Between Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.20 Vector Equation of a Line in Two and Three Dimensions . . . . . . . . . . . . . . 92
4
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