Math Academy

1.

Sequences and Series

18 topics

1.1. Arithmetic Series

	1.1.1.	Introduction to Sigma Notation
	1.1.2.	Properties of Finite Series
	1.1.3.	Expressing an Arithmetic Series in Sigma Notation
	1.1.4.	Finding the Sum of an Arithmetic Series
	1.1.5.	Finding the First Term of an Arithmetic Series
	1.1.6.	Calculating the Number of Terms in an Arithmetic Series
	1.1.7.	Modeling With Arithmetic Series

1.2. Finite Geometric Series

	1.2.1.	The Sum of a Finite Geometric Series
	1.2.2.	The Sum of the First N Terms of a Geometric Series
	1.2.3.	Writing Geometric Series in Sigma Notation
	1.2.4.	Finding the Sum of a Geometric Series Given in Sigma Notation
	1.2.5.	Solving Geometric Series Problems Using Exponential Equations and Inequalities
	1.2.6.	Modeling With Geometric Series
	1.2.7.	Modeling Financial Problems Using Geometric Series

1.3. The Binomial Theorem

	1.3.1.	Pascal's Triangle and the Binomial Coefficients
	1.3.2.	Expanding a Binomial Using Binomial Coefficients
	1.3.3.	The Special Case of the Binomial Theorem
	1.3.4.	Approximating Values Using the Binomial Theorem

2.

Rational Functions

16 topics

2.4. Rational Expressions and Equations

	2.4.1.	Closure Properties of Polynomials
	2.4.2.	Closure Properties of Rational Expressions
	2.4.3.	Rational Equations With Three Terms
	2.4.4.	Advanced Rational Equations
	2.4.5.	Further Advanced Rational Equations

2.5. Rational Functions

	2.5.1.	Finding Roots of Rational Functions
	2.5.2.	Vertical Asymptotes of Rational Functions
	2.5.3.	Locating Holes in Rational Functions
	2.5.4.	Horizontal Asymptotes of Rational Functions
	2.5.5.	End Behavior of Rational Functions
	2.5.6.	Infinite Limits of Rational Functions
	2.5.7.	Infinite Limits of Rational Functions: Advanced Cases
	2.5.8.	The Domain and Range of a Rational Function
	2.5.9.	Identifying a Rational Function From a Graph
	2.5.10.	Identifying a Rational Function From a Graph Containing Holes
	2.5.11.	Identifying the Graph of a Rational Function

3.

Inequalities

18 topics

3.6. Quadratic Inequalities

	3.6.1.	Solving Elementary Quadratic Inequalities
	3.6.2.	Solving Quadratic Inequalities From Graphs
	3.6.3.	Solving Quadratic Inequalities Using the Graphical Method
	3.6.4.	Solving Quadratic Inequalities Using the Sign Table Method
	3.6.5.	Solving Discriminant Problems Using Quadratic Inequalities

3.7. Polynomial Inequalities

	3.7.1.	Inequalities Involving Powers of Binomials
	3.7.2.	Solving Polynomial Inequalities Using a Graphical Method
	3.7.3.	Solving Polynomial Inequalities Using Special Factoring Techniques and the Graphical Method
	3.7.4.	Solving Polynomial Inequalities Using the Sign Table Method

3.8. Rational Inequalities

	3.8.1.	Solving Rational Inequalities
	3.8.2.	Further Solving of Rational Inequalities

3.9. Non-Polynomial Inequalities

	3.9.1.	Solving Inequalities Involving Radical Functions
	3.9.2.	Solving Inequalities Involving Exponential Functions
	3.9.3.	Solving Inequalities Involving Logarithmic Functions
	3.9.4.	Solving Inequalities Involving Exponential Functions and Polynomials
	3.9.5.	Solving Inequalities Involving Positive and Negative Factors
	3.9.6.	Solving Two-Variable Nonlinear Inequalities
	3.9.7.	Further Solving of Two-Variable Nonlinear Inequalities

4.

Trigonometry

42 topics

4.10. The Inverse Trigonometric Functions

	4.10.1.	Graphing the Inverse Sine Function
	4.10.2.	Graphing the Inverse Cosine Function
	4.10.3.	Graphing the Inverse Tangent Function
	4.10.4.	Evaluating Expressions Containing Inverse Trigonometric Functions
	4.10.5.	Further Evaluating Expressions Containing Inverse Trigonometric Functions

4.11. Trigonometric Identities

	4.11.1.	Simplifying Expressions Using Basic Trigonometric Identities
	4.11.2.	Simplifying Expressions Using the Pythagorean Identity
	4.11.3.	Alternate Forms of the Pythagorean Identity
	4.11.4.	Simplifying Expressions Using the Secant-Tangent Identity
	4.11.5.	Alternate Forms of the Secant-Tangent Identity
	4.11.6.	Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity
	4.11.7.	Simplifying Trigonometric Expressions Using Cofunction Identities

4.12. The Sum and Difference Formulas

	4.12.1.	The Sum and Difference Formulas for Sine
	4.12.2.	The Sum and Difference Formulas for Cosine
	4.12.3.	The Sum and Difference Formulas for Tangent
	4.12.4.	Calculating Trigonometric Ratios Using the Sum Formula for Sine
	4.12.5.	Calculating Trigonometric Ratios Using the Sum Formula for Cosine
	4.12.6.	Calculating Trigonometric Ratios Using the Sum Formula for Tangent
	4.12.7.	Writing Sums of Trigonometric Functions in Amplitude-Phase Form

4.13. The Double-Angle Formulas

	4.13.1.	The Double-Angle Formula for Sine
	4.13.2.	Verifying Trigonometric Identities Using the Double-Angle Formula for Sine
	4.13.3.	Using the Double-Angle Formula for Sine With the Pythagorean Theorem
	4.13.4.	The Double-Angle Formula for Cosine
	4.13.5.	Verifying Trigonometric Identities Using the Double-Angle Formulas for Cosine
	4.13.6.	Finding Exact Values of Trigonometric Expressions Using the Double-Angle Formulas for Cosine
	4.13.7.	Simplifying Expressions Using the Double-Angle Formula for Tangent
	4.13.8.	Verifying Trigonometric Identities Using the Double-Angle Formula for Tangent

4.14. Elementary Trigonometric Equations

	4.14.1.	Elementary Trigonometric Equations Containing Sine
	4.14.2.	Elementary Trigonometric Equations Containing Cosine
	4.14.3.	Elementary Trigonometric Equations Containing Tangent
	4.14.4.	Elementary Trigonometric Equations Containing Secant
	4.14.5.	Elementary Trigonometric Equations Containing Cosecant
	4.14.6.	Elementary Trigonometric Equations Containing Cotangent
	4.14.7.	General Solutions of Elementary Trigonometric Equations

4.15. Trigonometric Equations Containing Transformed Functions

	4.15.1.	General Solutions of Trigonometric Equations With Transformed Functions
	4.15.2.	Trigonometric Equations Containing Transformed Sine Functions
	4.15.3.	Trigonometric Equations Containing Transformed Cosine Functions
	4.15.4.	Trigonometric Equations Containing Transformed Tangent Functions

4.16. Advanced Trigonometric Equations

	4.16.1.	Solving Trigonometric Equations Using the Sin-Cos-Tan Identity
	4.16.2.	Solving Trigonometric Equations Using the Zero-Product Property
	4.16.3.	Quadratic Trigonometric Equations Containing Sine or Cosine
	4.16.4.	Quadratic Trigonometric Equations Containing Tangent or Cotangent

5.

Vectors

29 topics

5.17. Introduction to Vectors

	5.17.1.	Representing Given Information as a Vector
	5.17.2.	The Triangle Law for the Addition and Subtraction of Two Vectors
	5.17.3.	Calculating the Magnitude of a Vector From Given Information
	5.17.4.	Problem Solving Using Vector Diagrams
	5.17.5.	Parallel Vectors
	5.17.6.	Unit Vectors
	5.17.7.	Linear Combinations of Vectors and Their Properties
	5.17.8.	Describing the Position Vector of a Point Using Known Vectors

5.18. Vectors in 2D Cartesian Coordinates

	5.18.1.	Two-Dimensional Vectors Expressed in Component Form
	5.18.2.	Addition and Scalar Multiplication of Cartesian Vectors in 2D
	5.18.3.	Calculating the Magnitude of Cartesian Vectors in 2D
	5.18.4.	Calculating the Direction of Cartesian Vectors in 2D
	5.18.5.	Calculating the Components of Cartesian Vectors in 2D
	5.18.6.	Velocity and Acceleration for Plane Motion
	5.18.7.	Calculating Displacement for Plane Motion

5.19. Vectors in 3D Cartesian Coordinates

	5.19.1.	Three-Dimensional Vectors Expressed in Component Form
	5.19.2.	Addition and Scalar Multiplication of Cartesian Vectors in 3D
	5.19.3.	Calculating the Magnitude of Cartesian Vectors in 3D

5.20. The Dot Product

	5.20.1.	Calculating the Dot Product Using Angle and Magnitude
	5.20.2.	Calculating the Dot Product Using Components
	5.20.3.	The Angle Between Two Vectors
	5.20.4.	Calculating a Scalar Projection
	5.20.5.	Calculating a Vector Projection

5.21. The Cross Product

	5.21.1.	Calculating the Cross Product of Two Vectors Using the Definition
	5.21.2.	Calculating the Cross Product Using Determinants
	5.21.3.	Finding Areas Using the Cross Product
	5.21.4.	The Scalar Triple Product
	5.21.5.	Volumes of Parallelepipeds
	5.21.6.	Finding Volumes of Tetrahedrons and Pyramids Using Vector Products

6.

Matrices

36 topics

6.22. Introduction to Matrices

	6.22.1.	Introduction to Matrices
	6.22.2.	Index Notation for Matrices
	6.22.3.	Adding and Subtracting Matrices
	6.22.4.	Properties of Matrix Addition
	6.22.5.	Scalar Multiplication of Matrices
	6.22.6.	Zero, Square, Diagonal and Identity Matrices
	6.22.7.	The Transpose of a Matrix

6.23. Matrix Multiplication

	6.23.1.	Multiplying a Matrix by a Column Vector
	6.23.2.	Multiplying Square Matrices
	6.23.3.	Conformability for Matrix Multiplication
	6.23.4.	Multiplying Matrices
	6.23.5.	Powers of Matrices
	6.23.6.	Multiplying a Matrix by the Identity Matrix
	6.23.7.	Properties of Matrix Multiplication
	6.23.8.	Representing 2x2 Systems of Equations Using a Matrix Product
	6.23.9.	Representing 3x3 Systems of Equations Using a Matrix Product

6.24. Determinants

	6.24.1.	The Determinant of a 2x2 Matrix
	6.24.2.	The Geometric Interpretation of the 2x2 Determinant
	6.24.3.	The Minors of a 3x3 Matrix
	6.24.4.	The Determinant of a 3x3 Matrix

6.25. The Inverse of a Matrix

	6.25.1.	Introduction to the Inverse of a Matrix
	6.25.2.	Calculating the Inverse of a 2x2 Matrix
	6.25.3.	Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method
	6.25.4.	Solving 2x2 Systems of Equations Using Inverse Matrices
	6.25.5.	Solving 3x3 Systems of Equations Using Inverse Matrices

6.26. Linear Transformations

	6.26.1.	Introduction to Linear Transformations
	6.26.2.	The Standard Matrix of a Linear Transformation
	6.26.3.	Linear Transformations of Points and Lines in the Plane
	6.26.4.	Linear Transformations of Objects in the Plane
	6.26.5.	Dilations and Reflections as Linear Transformations
	6.26.6.	Shear and Stretch as Linear Transformations
	6.26.7.	Right-Angle Rotations as Linear Transformations
	6.26.8.	Rotations as Linear Transformations
	6.26.9.	Combining Linear Transformations Using 2x2 Matrices
	6.26.10.	Using Inverse Matrices to Reverse Linear Transformations
	6.26.11.	Determining the Area Scale Factor of a Linear Transformation

7.

Conic Sections

20 topics

7.27. Ellipses as Conic Sections

	7.27.1.	Introduction to Ellipses
	7.27.2.	Equations of Ellipses Centered at the Origin
	7.27.3.	Equations of Ellipses Centered at a General Point
	7.27.4.	Finding the Center and Axes of Ellipses by Completing the Square
	7.27.5.	Finding Intercepts of Ellipses
	7.27.6.	Finding Intersections of Ellipses and Lines
	7.27.7.	Foci of Ellipses
	7.27.8.	Vertices and Eccentricity of Ellipses
	7.27.9.	Directrices of Ellipses
	7.27.10.	The Area of an Ellipse

7.28. Hyperbolas as Conic Sections

	7.28.1.	Equations of Hyperbolas Centered at the Origin
	7.28.2.	Equations of Hyperbolas Centered at a General Point
	7.28.3.	Asymptotes of Hyperbolas Centered at the Origin
	7.28.4.	Asymptotes of Hyperbolas Centered at a General Point
	7.28.5.	Finding Intercepts and Intersections of Hyperbolas
	7.28.6.	Transverse Axes of Hyperbolas
	7.28.7.	Conjugate Axes of Hyperbolas
	7.28.8.	Foci of Hyperbolas
	7.28.9.	Eccentricity and Vertices of Hyperbolas
	7.28.10.	Directrices of Hyperbolas

8.

Parametric Equations

7 topics

8.29. Parametric Equations

	8.29.1.	Graphing Curves Defined Parametrically
	8.29.2.	Finding the Cartesian Equation of Curves Defined Parametrically
	8.29.3.	Finding Intercepts of Curves Defined Parametrically
	8.29.4.	Finding Intersections of Parametric Curves and Lines
	8.29.5.	Parametric Equations of Circles
	8.29.6.	Parametric Equations of Ellipses
	8.29.7.	Parametric Equations of Parabolas Centered at the Origin

9.

Polar Equations

6 topics

9.30. Polar Coordinates

	9.30.1.	Introduction to Polar Coordinates
	9.30.2.	Converting from Polar Coordinates to Cartesian Coordinates
	9.30.3.	Polar Equations of Circles Centered at the Origin
	9.30.4.	Polar Equations of Radial Lines
	9.30.5.	Polar Equations of Circles Centered on the Coordinate Axes
	9.30.6.	Finding Intersections of Polar Curves

10.

Complex Numbers

27 topics

10.31. The Complex Plane

	10.31.1.	The Complex Plane
	10.31.2.	The Magnitude of a Complex Number
	10.31.3.	The Argument of a Complex Number
	10.31.4.	Arithmetic in the Complex Plane
	10.31.5.	Geometry in the Complex Plane

10.32. Further Complex Numbers

	10.32.1.	The Complex Conjugate
	10.32.2.	Special Properties of the Complex Conjugate
	10.32.3.	The Complex Conjugate and the Roots of a Quadratic Equation
	10.32.4.	Dividing Complex Numbers
	10.32.5.	Solving Equations by Equating Real and Imaginary Parts
	10.32.6.	Extending Polynomial Identities to the Complex Numbers

10.33. Complex Numbers in Polar Form

	10.33.1.	The Polar Form of a Complex Number
	10.33.2.	Products of Complex Numbers Expressed in Polar Form
	10.33.3.	Quotients of Complex Numbers Expressed in Polar Form
	10.33.4.	The CIS Notation

10.34. De Moivre's Theorem

	10.34.1.	De Moivre's Theorem
	10.34.2.	Finding Powers of Complex Numbers Using De Moivre's Theorem
	10.34.3.	The Power-Reducing Formulas for Sine and Cosine
	10.34.4.	Euler's Formula
	10.34.5.	The Roots of Unity
	10.34.6.	Square Roots of Complex Numbers
	10.34.7.	Higher Roots of Complex Numbers

10.35. The Fundamental Theorem of Algebra

	10.35.1.	The Fundamental Theorem of Algebra for Quadratic Equations
	10.35.2.	The Fundamental Theorem of Algebra with Cubic Equations
	10.35.3.	Solving Cubic Equations With Complex Roots
	10.35.4.	The Fundamental Theorem of Algebra with Quartic Equations
	10.35.5.	Solving Quartic Equations With Complex Roots

11.

Probability & Combinatorics

42 topics

11.36. Introduction to Probability

	11.36.1.	Probability From Experimental Data
	11.36.2.	Sample Spaces and Events in Probability
	11.36.3.	Single Events in Probability
	11.36.4.	The Complement of an Event
	11.36.5.	Introduction to Sets
	11.36.6.	The Union of Sets
	11.36.7.	The Intersection of Sets
	11.36.8.	Venn Diagrams in Probability
	11.36.9.	Geometric Probability

11.37. Compound Events in Probability

	11.37.1.	Compound Events in Probability From Experimental Data
	11.37.2.	Computing Probabilities for Compound Events Using Venn Diagrams
	11.37.3.	Computing Probabilities for Three Events Using Venn Diagrams
	11.37.4.	The Addition Law of Probability
	11.37.5.	Mutually Exclusive Events

11.38. Conditional Probability

	11.38.1.	Conditional Probabilities From Venn Diagrams
	11.38.2.	Conditional Probabilities From Tables
	11.38.3.	The Multiplication Law for Conditional Probability
	11.38.4.	The Law of Total Probability
	11.38.5.	Tree Diagrams for Dependent Events
	11.38.6.	Tree Diagrams for Dependent Events: Applications
	11.38.7.	Independent Events
	11.38.8.	Tree Diagrams for Independent Events

11.39. Discrete Random Variables

	11.39.1.	Probability Mass Functions of Discrete Random Variables
	11.39.2.	Cumulative Distribution Functions for Discrete Random Variables
	11.39.3.	Expected Values of Discrete Random Variables
	11.39.4.	The Binomial Distribution
	11.39.5.	Modeling With the Binomial Distribution
	11.39.6.	The Geometric Distribution
	11.39.7.	Modeling With the Geometric Distribution

11.40. The Normal Distribution

	11.40.1.	The Standard Normal Distribution
	11.40.2.	The Normal Distribution
	11.40.3.	Mean and Variance of the Normal Distribution
	11.40.4.	Percentage Points of the Standard Normal Distribution
	11.40.5.	Modeling With the Normal Distribution
	11.40.6.	The Empirical Rule for the Normal Distribution

11.41. Combinatorics

	11.41.1.	The Rule of Sum and the Rule of Product
	11.41.2.	Factorials
	11.41.3.	Factorials in Variable Expressions
	11.41.4.	Ordering Objects
	11.41.5.	Permutations
	11.41.6.	Combinations
	11.41.7.	Computing Probabilities Using Combinatorics

12.

Statistics

18 topics

12.42. Analyzing Data

	12.42.1.	Sampling
	12.42.2.	The Mean of a Data Set
	12.42.3.	Variance and Standard Deviation
	12.42.4.	Covariance
	12.42.5.	The Z-Score

12.43. Correlation

	12.43.1.	Scatter Plots
	12.43.2.	Trend Lines
	12.43.3.	Making Predictions Using Trend Lines
	12.43.4.	Interpreting Coefficients of Trend Lines
	12.43.5.	Linear Correlation
	12.43.6.	Residuals and Residual Plots
	12.43.7.	The Linear Correlation Coefficient
	12.43.8.	Correlation vs. Causation

12.44. Regression

	12.44.1.	Selecting a Regression Model
	12.44.2.	Linear Regression
	12.44.3.	Quadratic Regression
	12.44.4.	Semi-Log Scatter Plots
	12.44.5.	Exponential Regression

Precalculus

Outcomes

Content

Rational Functions

Advanced Trigonometry

Conic Sections

Parametric and Polar Curves

Complex Numbers

Vectors and Matrices

Probability

Statistics