Review Notes for IB Standard Level Math
© 2015-2020, Steve Muench
@stevemuench
These notes are free of charge. If you paid to obtain them,
please send me an email to let me know about it.
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or my worked solutions to the May 2015 (Timezone 2) exam
http://bit.ly/sm-ib-sl-maths-may-2015-tz2
or my worked solutions to the November 2015 exam
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with any student you believe might benefit from them.
If you downloaded these notes from a source other than
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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
© 2015-2020, Steve Muench
@stevemuench
These notes are free of charge. If you paid to obtain them,
please send me an email to let me know about it.
Feel free to share the link to these notes
http://bit.ly/sm-ib-sl-maths-review-notes
or my worked solutions to the November 2014 exam
http://bit.ly/sm-ib-sl-maths-nov-2014
or my worked solutions to the May 2015 (Timezone 2) exam
http://bit.ly/sm-ib-sl-maths-may-2015-tz2
or my worked solutions to the November 2015 exam
https://bit.ly/sm-ib-sl-maths-nov-2015
with any student you believe might benefit from them.
If you downloaded these notes from a source other than
the bit.ly link above, please check there to make sure
you are reading the latest version. It may contain
additional content and important corrections!
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
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, 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
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