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Algebraic Prerequisites



1.1 Graphing Basics

1.1.1 Using the Cartesian System

1.1.2 Thinking Visually




1.2 Relationships between Two Points

1.2.1 Finding the Distance between Two Points

1.2.2 Finding the Second Endpoint of a Segment




1.3 Relationships among Three Points

1.3.1 Collinearity and Distance

1.3.2 Triangles




1.4 Circles

1.4.1 Finding the Center-Radius Form of the Equation of a Circle

1.4.2 Finding the Center and Radius of a Circle

1.4.3 Decoding the Circle Formula

1.4.4 Solving Word Problems Involving Circles




1.5 Graphing Equations

1.5.1 Graphing Equations by Locating Points

1.5.2 Finding the x- and y-Intercepts of an Equation




1.6 Function Basics

1.6.1 Functions and the Vertical Line Test

1.6.2 Identifying Functions

1.6.3 Function Notation and Finding Function Values




1.7 Working with Functions

1.7.1 Determining Intervals Over Which a Function Is Increasing

1.7.2 Evaluating Piecewise-Defined Functions for Given Values

1.7.3 Solving Word Problems Involving Functions




1.8 Function Domain and Range

1.8.1 Finding the Domain and Range of a Function

1.8.2 Domain and Range: One Explicit Example

1.8.3 Satisfying the Domain of a Function




1.9 Linear Functions: Slope

1.9.1 An Introduction to Slope

1.9.2 Finding the Slope of a Line Given Two Points

1.9.3 Interpreting Slope from a Graph

1.9.4 Graphing a Line Using Point and Slope




1.10 Equations of a Line

1.10.1 Writing an Equation in Slope-Intercept Form

1.10.2 Writing an Equation Given Two Points

1.10.3 Writing an Equation in Point-Slope Form

1.10.4 Matching a Slope-Intercept Equation with Its Graph

1.10.5 Slope for Parallel and Perpendicular Lines




1.11 Graphing Functions

1.11.1 Graphing Some Important Functions

1.11.2 Graphing Piecewise-Defined Functions

1.11.3 Matching Equations with Their Graphs




1.12 Manipulating Graphs: Shifts and Stretches

1.12.1 Shifting Curves along Axes

1.12.2 Shifting or Translating Curves along Axes

1.12.3 Stretching a Graph

1.12.4 Graphing Quadratics Using Patterns




1.13 Manipulating Graphs: Symmetry and Reflections

1.13.1 Determining Symmetry

1.13.2 Reflections

1.13.3 Reflecting Specific Functions




1.14 Quadratic Functions: Basics

1.14.1 Deconstructing the Graph of a Quadratic Function

1.14.2 Nice-Looking Parabolas

1.14.3 Using Discriminants to Graph Parabolas

1.14.4 Maximum Height in the Real World




1.15 Quadratic Functions: The Vertex

1.15.1 Finding the Vertex by Completing the Square

1.15.2 Using the Vertex to Write the Quadratic Equation

1.15.3 Finding the Maximum or Minimum of a Quadratic

1.15.4 Graphing Parabolas




1.16 Composite Functions

1.16.1 Using Operations on Functions

1.16.2 Composite Functions

1.16.3 Components of Composite Functions

1.16.4 Finding Functions That Form a Given Composite

1.16.5 Finding the Difference Quotient of a Function




1.17 Rational Functions

1.17.1 Understanding Rational Functions

1.17.2 Basic Rational Functions




1.18 Graphing Rational Functions

1.18.1 Vertical Asymptotes

1.18.2 Horizontal Asymptotes

1.18.3 Graphing Rational Functions

1.18.4 Graphing Rational Functions: More Examples




1.19 Function Inverses

1.19.1 Understanding Inverse Functions

1.19.2 The Horizontal Line Test

1.19.3 Are Two Functions Inverses of Each Other?

1.19.4 Graphing the Inverse




1.20 Finding Function Inverses

1.20.1 Finding the Inverse of a Function

1.20.2 Finding the Inverse of a Function with Higher Powers

The Trigonometric Functions



2.1 Angles and Radian Measure

2.1.1 Finding the Quadrant in Which an Angle Lies

2.1.2 Finding Coterminal Angles

2.1.3 Finding the Complement and Supplement of an Angle

2.1.4 Converting between Degrees and Radians

2.1.5 Using the Arc Length Formula




2.2 Right Angle Trigonometry

2.2.1 An Introduction to the Trigonometric Functions

2.2.2 Evaluating Trigonometric Functions for an Angle in a Right Triangle

2.2.3 Finding an Angle Given the Value of a Trigonometric Function

2.2.4 Using Trigonometric Functions to Find Unknown Sides of Right Triangles

2.2.5 Finding the Height of a Building




2.3 The Trigonometric Functions

2.3.1 Evaluating Trigonometric Functions for an Angle in the Coordinate Plane

2.3.2 Evaluating Trigonometric Functions Using the Reference Angle

2.3.3 Finding the Value of Trigonometric Functions Given Information about the Values of Other Trigonometric Functions

2.3.4 Trigonometric Functions of Important Angles




2.4 Graphing Sine and Cosine Functions

2.4.1 An Introduction to the Graphs of Sine and Cosine Functions

2.4.2 Graphing Sine or Cosine Functions with Different Coefficients

2.4.3 Finding Maximum and Minimum Values and Zeros of Sine and Cosine

2.4.4 Solving Word Problems Involving Sine or Cosine Functions




2.5 Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts

2.5.1 Graphing Sine and Cosine Functions with Phase Shifts

2.5.2 Fancy Graphing: Changes in Period, Amplitude, Vertical Shift, and Phase Shift




2.6 Graphing Other Trigonometric Functions

2.6.1 Graphing the Tangent, Secant, Cosecant, and Cotangent Functions

2.6.2 Fancy Graphing: Tangent, Secant, Cosecant, and Cotangent

2.6.3 Identifying a Trigonometric Function from its Graph




2.7 Inverse Trigonometric Functions

2.7.1 An Introduction to Inverse Trigonometric Functions

2.7.2 Evaluating Inverse Trigonometric Functions

2.7.3 Solving an Equation Involving an Inverse Trigonometric Function

2.7.4 Evaluating the Composition of a Trigonometric Function and Its Inverse

2.7.5 Applying Trigonometric Functions: Is He Speeding?

Trigonometric Identities



3.1 Basic Trigonometric Identities

3.1.1 Fundamental Trigonometric Identities

3.1.2 Finding All Function Values




3.2 Simplifying Trigonometric Expressions

3.2.1 Simplifying a Trigonometric Expression Using Trigonometric Identities

3.2.2 Simplifying Trigonometric Expressions Involving Fractions

3.2.3 Simplifying Products of Binomials Involving Trigonometric Functions

3.2.4 Factoring Trigonometric Expressions

3.2.5 Determining Whether a Trigonometric Function Is Odd, Even, or Neither




3.3 Proving Trigonometric Identities

3.3.1 Proving an Identity

3.3.2 Proving an Identity: Other Examples




3.4 Solving Trigonometric Equations

3.4.1 Solving Trigonometric Equations

3.4.2 Solving Trigonometric Equations by Factoring

3.4.3 Solving Trigonometric Equations with Coefficients in the Argument

3.4.4 Solving Trigonometric Equations Using the Quadratic Formula

3.4.5 Solving Word Problems Involving Trigonometric Equations




3.5 The Sum and Difference Identities

3.5.1 Identities for Sums and Differences of Angles

3.5.2 Using Sum and Difference Identities

3.5.3 Using Sum and Difference Identities to Simplify an Expression




3.6 Double-Angle Identities

3.6.1 Confirming a Double-Angle Identity

3.6.2 Using Double-Angle Identities

3.6.3 Solving Word Problems Involving Multiple-Angle Identities




3.7 Other Advanced Identities

3.7.1 Using a Cofunction Identity

3.7.2 Using a Power-Reducing Identity

3.7.3 Using Half-Angle Identities to Solve a Trigonometric Equation

Applications of Trigonometry



4.1 The Law of Sines

4.1.1 The Law of Sines

4.1.2 Solving a Triangle Given Two Sides and One Angle

4.1.3 Solving a Triangle (SAS): Another Example

4.1.4 The Law of Sines: An Application




4.2 The Law of Cosines

4.2.1 The Law of Cosines

4.2.2 The Law of Cosines (SSS)

4.2.3 The Law of Cosines (SAS): An Application

4.2.4 Heron's Formula




4.3 Vector Basics

4.3.1 An Introduction to Vectors

4.3.2 Finding the Magnitude and Direction of a Vector

4.3.3 Vector Addition and Scalar Multiplication




4.4 Components of Vectors and Unit Vectors

4.4.1 Finding the Components of a Vector

4.4.2 Finding a Unit Vector

4.4.3 Solving Word Problems Involving Velocity or Forces

Complex Numbers and Polar Coordinates



5.1 Complex Numbers

5.1.1 Introducing and Writing Complex Numbers

5.1.2 Rewriting Powers of i

5.1.3 Adding and Subtracting Complex Numbers

5.1.4 Multiplying Complex Numbers

5.1.5 Dividing Complex Numbers




5.2 Complex Numbers in Trigonometric Form

5.2.1 Graphing a Complex Number and Finding Its Absolute Value

5.2.2 Expressing a Complex Number in Trigonometric or Polar Form

5.2.3 Multiplying and Dividing Complex Numbers in Trigonometric or Polar Form




5.3 Powers and Roots of Complex Numbers

5.3.1 Using DeMoivre's Theorem to Raise a Complex Number to a Power

5.3.2 Roots of Complex Numbers

5.3.3 More Roots of Complex Numbers

5.3.4 Roots of Unity




5.4 Polar Coordinates

5.4.1 An Introduction to Polar Coordinates

5.4.2 Converting between Polar and Rectangular Coordinates

5.4.3 Graphing Simple Polar Equations

Exponential and Logarithmic Functions



6.1 Exponential Functions

6.1.1 An Introduction to Exponential Functions

6.1.2 Graphing Exponential Functions: Useful Patterns

6.1.3 Graphing Exponential Functions: More Examples




6.2 Applying Exponential Functions

6.2.1 Using Properties of Exponents to Solve Exponential Equations

6.2.2 Finding Present Value and Future Value

6.2.3 Finding an Interest Rate to Match Given Goals




6.3 The Number e

6.3.1 e

6.3.2 Applying Exponential Functions




6.4 Logarithmic Functions

6.4.1 An Introduction to Logarithmic Functions

6.4.2 Converting between Exponential and Logarithmic Functions




6.5 Solving Logarithmic Functions

6.5.1 Finding the Value of a Logarithmic Function

6.5.2 Solving for x in Logarithmic Equations

6.5.3 Graphing Logarithmic Functions

6.5.4 Matching Logarithmic Functions with Their Graphs




6.6 Properties of Logarithms

6.6.1 Properties of Logarithms

6.6.2 Expanding a Logarithmic Expression Using Properties

6.6.3 Combining Logarithmic Expressions




6.7 Evaluating Logarithms

6.7.1 Evaluating Logarithmic Functions Using a Calculator

6.7.2 Using the Change of Base Formula




6.8 Applying Logarithmic Functions

6.8.1 The Richter Scale

6.8.2 The Distance Modulus Formula




6.9 Solving Exponential and Logarithmic Equations

6.9.1 Solving Exponential Equations

6.9.2 Solving Logarithmic Equations

6.9.3 Solving Equations with Logarithmic Exponents




6.10 Applying Exponents and Logarithms

6.10.1 Compound Interest

6.10.2 Predicting Change




6.11 Word Problems Involving Exponential Growth and Decay

6.11.1 An Introduction to Exponential Growth and Decay

6.11.2 Half-Life

6.11.3 Newton's Law of Cooling

6.11.4 Continuously Compounded Interest

Conic Sections



7.1 Conic Sections: Parabolas

7.1.1 An Introduction to Conic Sections

7.1.2 An Introduction to Parabolas

7.1.3 Determining Information about a Parabola from Its Equation

7.1.4 Writing an Equation for a Parabola




7.2 Conic Sections: Ellipses

7.2.1 An Introduction to Ellipses

7.2.2 Finding the Equation for an Ellipse

7.2.3 Applying Ellipses: Satellites




7.3 Conic Sections: Hyperbolas

7.3.1 An Introduction to Hyperbolas

7.3.2 Finding the Equation for a Hyperbola

7.3.3 Applying Hyperbolas: Navigation




7.4 Conic Sections

7.4.1 Identifying a Conic

7.4.2 Name That Conic

Edward Burger
Williams College

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.