Master the algebra of advanced functions including quadratics, logarithms, and trigonometry. Dive deep into the theory of polynomials, learn the basics of limits, derivatives, and integrals from calculus, and explore a variety of concepts from higher math including complex numbers, vectors, probability, and statistics.

- Understand the relationship between logarithms and exponents and use this understanding to evaluate logarithms.

- Apply trigonometric functions to solve for unknown sides and angles in right triangles.
- Leverage the unit circle as a conceptual framework for evaluating special trigonometric ratios.
- Graph and describe properties of trigonometric functions by relating them to the unit circle.
- Use the law of sines and the law of cosines to solve for unknown angles and sides in general triangles.

- Leverage factoring and the quadratic formula as complementary techniques for solving quadratic equations and graphing quadratic functions.
- Extend previous knowledge of algebraic techniques to solve nonlinear equations involving polynomials, radicals, exponents, and logarithms.
- Understand how transformations affect the shape of a function's graph and use this understanding to graph transformed quadratic, absolute value, exponential, radical, logarithmic, and trigonometric functions.

- Understand the relationship between zeros and factors of polynomials.
- Divide polynomials using synthetic and long division.
- Understand the relationship between the value of a polynomial at a given input and the remainder obtained when dividing the polynomial by the corresponding binomial.
- Leverage the rational roots theorem as a strategy to factor polynomials.

- Understand how the multiplicity of a root of a polynomial relates to the shape of the graph near the root.
- Sketch graphs of polynomial functions by identifying end behavior, roots, and behavior near roots.
- Sketch graphs of rational functions by identifying asymptotes.

- Estimate limits from graphs and compute limits using algebraic manipulation.
- Identify continuous functions from graphs and define continuity in terms of limits.
- Interpret the derivative as the instantaneous rate of change or the slope of the tangent line.
- Apply rules to compute derivatives of a variety of algebraic functions, including higher-order derivatives.
- Evaluate indefinite integrals by finding antiderivatives.

- Evaluate terms of a sequence and compute sums given in sigma notation.
- Determine the formula of an arithmetic or geometric sequence.

- Manipulate complex numbers algebraically and visualize them geometrically in the complex plane.
- Generalize prior intuitions about arithmetic to vectors.

- Define probability using the formal language of sets.
- Apply combinatorial techniques to compute probabilities in real-world modeling contexts.
- Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
- Compute statistical measures of the center and spread of a data set.
- Fit a linear regression to a data set and use it to make predictions.

1.

Quadratics
21 topics

1.1. Quadratic Equations

1.1.1. | Introduction to Quadratic Equations | |

1.1.2. | Solving Perfect Square Quadratic Equations | |

1.1.3. | Perfect Square Quadratic Equations with One or No Solutions | |

1.1.4. | The Zero Product Rule for Solving Quadratic Equations | |

1.1.5. | Solving Quadratic Equations Using a Difference of Squares | |

1.1.6. | Solving Quadratic Equations with No Constant Term | |

1.1.7. | Solving Quadratic Equations by Factoring | |

1.1.8. | Solving Quadratic Equations with Leading Coefficients by Factoring | |

1.1.9. | Completing the Square | |

1.1.10. | Completing the Square With Odd Linear Terms | |

1.1.11. | Completing the Square With Leading Coefficients | |

1.1.12. | Solving Quadratic Equations by Completing the Square | |

1.1.13. | Solving Quadratic Equations With Leading Coefficients by Completing the Square | |

1.1.14. | The Quadratic Formula | |

1.1.15. | The Discriminant of a Quadratic Equation |

1.2. Quadratic Functions

1.2.1. | Graphing Elementary Quadratic Functions | |

1.2.2. | Vertical Reflections of Quadratic Functions | |

1.2.3. | Graphs of General Quadratic Functions | |

1.2.4. | Roots of Quadratic Functions | |

1.2.5. | The Discriminant of a Quadratic Function | |

1.2.6. | Finding Intersections of Lines and Quadratic Functions |

2.

Polynomials
25 topics

2.3. Polynomials

2.3.1. | Closure Properties of Polynomials | |

2.3.2. | The Least Common Multiple of Two Monomials | |

2.3.3. | The Least Common Multiple of Two Polynomials | |

2.3.4. | Determining the Roots of Polynomials | |

2.3.5. | Graphing Elementary Cubic Functions | |

2.3.6. | End Behavior of Polynomials |

2.4. Factoring Polynomials

2.4.1. | Factoring Polynomials Using the Greatest Common Factor | |

2.4.2. | Factoring Higher-Order Polynomials as a Difference of Squares | |

2.4.3. | Factoring Cubic Expressions by Grouping | |

2.4.4. | Factoring Sums and Differences of Cubes | |

2.4.5. | Factoring Biquadratic Expressions |

2.5. Dividing Polynomials

2.5.1. | Dividing Polynomials Using Synthetic Division | |

2.5.2. | Dividing Polynomials by Linear Binomials Using Long Division | |

2.5.3. | Dividing Polynomials Using Long Division | |

2.5.4. | Dividing Polynomials by Manipulating Rational Expressions |

2.6. Polynomial Theorems

2.6.1. | The Factor Theorem | |

2.6.2. | Determining Polynomial Coefficients Using the Factor Theorem | |

2.6.3. | Factoring Cubic Polynomials Using the Factor Theorem | |

2.6.4. | Multiplicities of the Roots of Polynomials | |

2.6.5. | The Remainder Theorem | |

2.6.6. | The Rational Roots Theorem |

2.7. Graphs of Polynomials

2.7.1. | Graphing Cubic Curves Containing Three Distinct Real Roots | |

2.7.2. | Graphing Cubic Curves Containing a Double Root | |

2.7.3. | Graphing Cubic Curves Containing One Distinct Real Root | |

2.7.4. | Graphing General Polynomials |

3.

Number Systems
13 topics

3.8. The Real Number System

3.8.1. | Natural Numbers, Integers, and Rational Numbers | |

3.8.2. | The Real Number System | |

3.8.3. | Sums and Products of Rational and Irrational Numbers |

3.9. Introduction to Complex Numbers

3.9.1. | Imaginary Numbers | |

3.9.2. | Solving Quadratic Equations with Purely Imaginary Solutions | |

3.9.3. | Complex Numbers | |

3.9.4. | Adding and Subtracting Complex Numbers | |

3.9.5. | Multiplying Complex Numbers | |

3.9.6. | Solving Quadratic Equations With Complex Roots | |

3.9.7. | The Cyclic Property of the Imaginary Unit |

3.10. The Complex Plane

3.10.1. | The Complex Plane | |

3.10.2. | The Magnitude of a Complex Number | |

3.10.3. | The Argument of a Complex Number |

4.

Functions
24 topics

4.11. Functions

4.11.1. | The Arithmetic of Functions | |

4.11.2. | Composition of Functions | |

4.11.3. | Finding Algebraic Expressions of Composite Functions | |

4.11.4. | Local Extrema of Functions | |

4.11.5. | One-To-One Functions | |

4.11.6. | Introduction to Inverse Functions | |

4.11.7. | Calculating the Inverse of a Function | |

4.11.8. | Graphs of Inverse Functions | |

4.11.9. | Domain and Range of Inverse Functions | |

4.11.10. | Invertible Functions | |

4.11.11. | Plotting X as a Function of Y | |

4.11.12. | Periodic Functions | |

4.11.13. | Even and Odd Functions | |

4.11.14. | Unbounded Behavior of Functions Near a Point |

4.12. Graph Transformations of Functions

4.12.1. | Vertical Translations of Functions | |

4.12.2. | Horizontal Translations of Functions | |

4.12.3. | Vertical Stretches of Functions | |

4.12.4. | Horizontal Stretches of Functions | |

4.12.5. | Vertical Reflections of Functions | |

4.12.6. | Horizontal Reflections of Functions | |

4.12.7. | Combining Graph Transformations: Two Operations | |

4.12.8. | Combining Graph Transformations: Three or More Operations | |

4.12.9. | Constructing Functions Using Transformations | |

4.12.10. | Combining Reflections With Other Graph Transformations |

5.

Exponentials & Logarithms
29 topics

5.13. Introduction to Logarithms

5.13.1. | Converting From Exponential to Logarithmic Form | |

5.13.2. | Converting From Logarithmic to Exponential Form | |

5.13.3. | Evaluating Logarithms | |

5.13.4. | The Natural Logarithm | |

5.13.5. | The Common Logarithm | |

5.13.6. | Simplifying Logarithmic Expressions |

5.14. The Laws of Logarithms

5.14.1. | The Product Rule for Logarithms | |

5.14.2. | The Quotient Rule for Logarithms | |

5.14.3. | The Power Rule for Logarithms | |

5.14.4. | Combining the Laws of Logarithms | |

5.14.5. | The Change of Base Formula for Logarithms |

5.15. Exponential Equations

5.15.1. | Solving Exponential Equations Using Logarithms | |

5.15.2. | Solving Equations Containing the Exponential Function | |

5.15.3. | Solving Exponential Equations With Different Bases | |

5.15.4. | Solving Exponential Equations With Different Bases Using Logarithms | |

5.15.5. | Solving Exponential Equations Using the Zero-Product Property |

5.16. Logarithmic Equations

5.16.1. | Solving Logarithmic Equations | |

5.16.2. | Solving Logarithmic Equations Containing the Natural Logarithm | |

5.16.3. | Solving Logarithmic Equations Using the Laws of Logarithms | |

5.16.4. | Solving Logarithmic Equations by Combining the Laws of Logarithms | |

5.16.5. | Solving Logarithmic Equations With Logarithms on Both Sides |

5.17. Graphs of Exponential Functions

5.17.1. | Vertical Translations of Exponential Growth Functions | |

5.17.2. | Vertical Translations of Exponential Decay Functions | |

5.17.3. | Combining Graph Transformations of Exponential Functions | |

5.17.4. | Properties of Transformed Exponential Functions | |

5.17.5. | Inverses of Exponential and Logarithmic Functions |

5.18. Graphs of Logarithmic Functions

5.18.1. | Graphing Logarithmic Functions | |

5.18.2. | Combining Graph Transformations of Logarithmic Functions | |

5.18.3. | Properties of Transformed Logarithmic Functions |

6.

Radical & Rational Functions
25 topics

6.19. Rational Expressions

6.19.1. | Simplifying Rational Expressions Using Polynomial Factorization | |

6.19.2. | Adding and Subtracting Rational Expressions | |

6.19.3. | Adding Rational Expressions With No Common Factors in the Denominator | |

6.19.4. | Multiplying Rational Expressions | |

6.19.5. | Dividing Rational Expressions | |

6.19.6. | Expressing Rational Functions as Sums of Partial Fractions |

6.20. Rational Equations

6.20.1. | Rational Equations With Three Terms | |

6.20.2. | Advanced Rational Equations |

6.21. Rational Functions

6.21.1. | Graphing Reciprocal Functions | |

6.21.2. | Graph Transformations of Reciprocal Functions | |

6.21.3. | Combining Graph Transformations of Reciprocal Functions | |

6.21.4. | Domain and Range of Transformed Reciprocal Functions | |

6.21.5. | Inverses of Reciprocal Functions | |

6.21.6. | Finding Intersections of Lines and Reciprocal Functions | |

6.21.7. | Finding Roots of Rational Functions | |

6.21.8. | Horizontal Asymptotes of Rational Functions |

6.22. Radical Functions

6.22.1. | Solving Radical Equations | |

6.22.2. | Inverses of Radical Functions | |

6.22.3. | Graphing the Square Root Function | |

6.22.4. | Graph Transformations of Square Root Functions | |

6.22.5. | Graphing the Cube Root Function | |

6.22.6. | Domain, Range, and Roots of Transformed Square Root Functions | |

6.22.7. | The Domain of a Transformed Radical Function | |

6.22.8. | The Range of a Transformed Radical Function | |

6.22.9. | Roots of Transformed Radical Functions |

7.

Sequences
18 topics

7.23. Introduction to Sequences

7.23.1. | Introduction to Sequences | |

7.23.2. | Recursive Sequences | |

7.23.3. | Fibonacci Sequences | |

7.23.4. | Introduction to Sigma Notation | |

7.23.5. | Properties of Finite Series |

7.24. Arithmetic Sequences

7.24.1. | Arithmetic Sequences | |

7.24.2. | Recursive Formulas for Arithmetic Sequences | |

7.24.3. | The Nth Term of an Arithmetic Sequence | |

7.24.4. | Translating Between Explicit and Recursive Formulas for Arithmetic Sequences | |

7.24.5. | Finding the Common Difference of an Arithmetic Sequence Given Two Terms | |

7.24.6. | Finding the Nth Term of an Arithmetic Sequence Given Two Terms | |

7.24.7. | Determining Indexes of Terms in Arithmetic Sequences |

7.25. Geometric Sequences

7.25.1. | Introduction to Geometric Sequences | |

7.25.2. | The Recursive Formula for a Geometric Sequence | |

7.25.3. | Finding the Nth Term of a Geometric Sequence | |

7.25.4. | Translating Between Explicit and Recursive Formulas for Geometric Sequences | |

7.25.5. | Finding the Common Ratio of a Geometric Sequence Given Two Terms | |

7.25.6. | Finding the Index of a Term in a Geometric Sequence |

8.

Geometry
30 topics

8.26. Coordinate Geometry

8.26.1. | Parallel Lines in the Coordinate Plane | |

8.26.2. | Finding the Equation of a Parallel Line | |

8.26.3. | Perpendicular Lines in the Coordinate Plane | |

8.26.4. | Finding Equations of Perpendicular Lines | |

8.26.5. | Midpoints in the Coordinate Plane | |

8.26.6. | The Distance Formula | |

8.26.7. | The Distance Formula in Three Dimensions |

8.27. Geometric Transformations

8.27.1. | Translations of Geometric Figures | |

8.27.2. | Rotations of Geometric Figures | |

8.27.3. | Rotating Objects in the Coordinate Plane Using Functions | |

8.27.4. | Reflections of Geometric Figures in the Cartesian Plane | |

8.27.5. | Reflections of Figures Across Arbitrary Lines | |

8.27.6. | Dilations of Geometric Figures | |

8.27.7. | Dilations of Figures in the Coordinate Plane | |

8.27.8. | Stretches of Geometric Figures | |

8.27.9. | Combining Stretches of Geometric Figures | |

8.27.10. | Combining Geometric Transformations | |

8.27.11. | Rigid Motions and Congruence | |

8.27.12. | Reflective Symmetry | |

8.27.13. | Rotational Symmetry |

8.28. Solid Geometry

8.28.1. | Identifying Three-Dimensional Shapes | |

8.28.2. | Faces, Vertices, and Edges of Polyhedrons | |

8.28.3. | Nets of Polyhedrons | |

8.28.4. | Volumes of Cubes | |

8.28.5. | Volumes of Rectangular Solids | |

8.28.6. | Surface Areas of Cubes | |

8.28.7. | Volumes of Spheres | |

8.28.8. | Surface Areas of Spheres | |

8.28.9. | Euler's Formula for Polyhedra | |

8.28.10. | The Five Platonic Solids |

9.

Trigonometry
50 topics

9.29. Introduction to Trigonometry

9.29.1. | Angles and Sides in Right Triangles | |

9.29.2. | The Trigonometric Ratios | |

9.29.3. | Calculating Trigonometric Ratios Using the Pythagorean Theorem | |

9.29.4. | Calculating Side Lengths of Right Triangles Using Trigonometry | |

9.29.5. | Calculating Angles in Right Triangles Using Trigonometry | |

9.29.6. | The Reciprocal Trigonometric Ratios | |

9.29.7. | Special Trigonometric Ratios | |

9.29.8. | Calculating the Area of a Right Triangle Using Trigonometry | |

9.29.9. | Introduction to Radians | |

9.29.10. | Trigonometric Ratios With Radians |

9.30. Trigonometry with General Triangles

9.30.1. | The Law of Sines | |

9.30.2. | The Law of Cosines | |

9.30.3. | The Area of a General Triangle |

9.31. The Unit Circle

9.31.1. | Angles in the Coordinate Plane | |

9.31.2. | Negative Angles in the Coordinate Plane | |

9.31.3. | Coterminal Angles | |

9.31.4. | Calculating Reference Angles | |

9.31.5. | Properties of the Unit Circle in the First Quadrant | |

9.31.6. | Extending the Trigonometric Ratios Using the Unit Circle | |

9.31.7. | Extending the Trigonometric Ratios Using Angles in Radians | |

9.31.8. | Extending the Trigonometric Ratios to Negative Angles | |

9.31.9. | Extending the Trigonometric Ratios to Large Angles | |

9.31.10. | Using the Pythagorean Identity in the First Quadrant | |

9.31.11. | Extending the Pythagorean Identity to All Quadrants |

9.32. Special Trigonometric Ratios

9.32.1. | Finding Trigonometric Ratios of Quadrantal Angles | |

9.32.2. | Trigonometric Ratios of Quadrantal Angles Outside the Standard Range | |

9.32.3. | Finding Trigonometric Ratios of Special Angles Using the Unit Circle | |

9.32.4. | Evaluating Trigonometric Expressions | |

9.32.5. | Further Extensions of the Special Trigonometric Ratios |

9.33. Graphing Trigonometric Functions

9.33.1. | Graphing Sine and Cosine | |

9.33.2. | Graphing Tangent and Cotangent | |

9.33.3. | Graphing Secant and Cosecant |

9.34. Properties of Trigonometric Functions

9.34.1. | Describing Properties of the Sine Function | |

9.34.2. | Describing Properties of the Cosine Function | |

9.34.3. | Describing Properties of the Tangent Function | |

9.34.4. | Describing Properties of the Secant Function | |

9.34.5. | Describing Properties of the Cosecant Function | |

9.34.6. | Describing Properties of the Cotangent Function |

9.35. Graph Transformations of Trigonometric Functions

9.35.1. | Vertical Translations of Trigonometric Functions | |

9.35.2. | Vertical Stretches of Trigonometric Functions | |

9.35.3. | Horizontal Translations of Trigonometric Functions | |

9.35.4. | Horizontal Stretches of Trigonometric Functions | |

9.35.5. | Combining Graph Transformations of Sine and Cosine | |

9.35.6. | Graph Transformations of Tangent and Cotangent | |

9.35.7. | Combining Graph Transformations of Tangent and Cotangent | |

9.35.8. | Combining Graph Transformations of Secant and Cosecant | |

9.35.9. | Graphing Reflections of Trigonometric Functions |

9.36. Properties of Transformed Trigonometric Functions

9.36.1. | Properties of Transformed Sine and Cosine Functions | |

9.36.2. | Properties of Transformed Tangent and Cotangent Functions | |

9.36.3. | Properties of Transformed Secant and Cosecant Functions |

10.

Vectors
11 topics

10.37. Introduction to Vectors

10.37.1. | Representing Given Information as a Vector | |

10.37.2. | The Triangle Law for the Addition and Subtraction of Two Vectors | |

10.37.3. | Calculating the Magnitude of a Vector From Given Information | |

10.37.4. | Problem Solving Using Vector Diagrams | |

10.37.5. | Parallel Vectors | |

10.37.6. | Unit Vectors | |

10.37.7. | Linear Combinations of Vectors and Their Properties | |

10.37.8. | Describing the Position Vector of a Point Using Known Vectors |

10.38. Vectors in 2D Cartesian Coordinates

10.38.1. | Two-Dimensional Vectors Expressed in Component Form | |

10.38.2. | Addition and Scalar Multiplication of Cartesian Vectors in 2D | |

10.38.3. | Calculating the Magnitude of Cartesian Vectors in 2D |

11.

Limits & Continuity
22 topics

11.39. Estimating Limits from Graphs

11.39.1. | The Finite Limit of a Function | |

11.39.2. | The Left and Right-Sided Limits of a Function | |

11.39.3. | Finding the Existence of a Limit Using One-Sided Limits | |

11.39.4. | Limits at Infinity from Graphs | |

11.39.5. | Infinite Limits from Graphs |

11.40. The Algebra of Limits

11.40.1. | Limits of Power Functions, and the Constant Rule for Limits | |

11.40.2. | The Sum Rule for Limits | |

11.40.3. | The Product and Quotient Rules for Limits | |

11.40.4. | The Power and Root Rules for Limits |

11.41. Limits of Functions

11.41.1. | Limits at Infinity of Polynomials | |

11.41.2. | Limits of Reciprocal Functions | |

11.41.3. | Limits of Exponential Functions | |

11.41.4. | Limits of Logarithmic Functions | |

11.41.5. | Limits of Radical Functions | |

11.41.6. | Limits of Trigonometric Functions | |

11.41.7. | Limits of Reciprocal Trigonometric Functions | |

11.41.8. | Limits of Piecewise Functions | |

11.41.9. | Calculating Limits of Rational Functions by Factoring |

11.42. Continuity

11.42.1. | Determining Continuity from Graphs | |

11.42.2. | Defining Continuity at a Point | |

11.42.3. | Left and Right Continuity | |

11.42.4. | Continuity Over an Interval |

12.

Introduction to Calculus
28 topics

12.43. Introduction to Differentiation

12.43.1. | The Average Rate of Change of a Function | |

12.43.2. | The Average Rate of Change of a Function over a Varying Interval | |

12.43.3. | The Instantaneous Rate of Change of a Function at a Point | |

12.43.4. | Defining the Derivative Using Derivative Notation | |

12.43.5. | The Power Rule for Differentiation | |

12.43.6. | The Sum and Constant Multiple Rules for Differentiation | |

12.43.7. | Calculating the Slope of a Tangent Line Using Differentiation | |

12.43.8. | Calculating the Equation of a Tangent Line Using Differentiation | |

12.43.9. | Calculating the Equation of a Normal Line Using Differentiation | |

12.43.10. | Interpreting the Meaning of the Derivative in Context |

12.44. Derivatives of Functions and the Rules of Differentiation

12.44.1. | Differentiating Exponential Functions | |

12.44.2. | Differentiating Logarithmic Functions | |

12.44.3. | Differentiating Trigonometric Functions | |

12.44.4. | Second and Higher Order Derivatives | |

12.44.5. | The Product Rule for Differentiation | |

12.44.6. | The Quotient Rule for Differentiation | |

12.44.7. | Differentiating Reciprocal Trigonometric Functions |

12.45. Differentiating Composite Functions

12.45.1. | The Chain Rule for Differentiation | |

12.45.2. | The Chain Rule With Exponential Functions | |

12.45.3. | The Chain Rule With Logarithmic Functions | |

12.45.4. | The Chain Rule With Trigonometric Functions | |

12.45.5. | Selecting Procedures for Calculating Derivatives |

12.46. Indefinite Integrals

12.46.1. | The Antiderivative | |

12.46.2. | The Constant Multiple Rule for Indefinite Integrals | |

12.46.3. | The Sum Rule for Indefinite Integrals | |

12.46.4. | Integrating the Reciprocal Function | |

12.46.5. | Integrating Exponential Functions | |

12.46.6. | Integrating Trigonometric Functions |

13.

Statistics
12 topics

13.47. Analyzing Data

13.47.1. | The Mean of a Data Set | |

13.47.2. | Variance and Standard Deviation | |

13.47.3. | Covariance | |

13.47.4. | The Z-Score |

13.48. Correlation and Regression

13.48.1. | Scatter Plots | |

13.48.2. | Trend Lines | |

13.48.3. | Making Predictions Using Trend Lines | |

13.48.4. | Linear Correlation | |

13.48.5. | The Linear Correlation Coefficient | |

13.48.6. | Linear Regression | |

13.48.7. | Residuals and Residual Plots | |

13.48.8. | Selecting a Regression Model |

14.

Probability & Combinatorics
18 topics

14.49. Introduction to Probability

14.49.1. | Probability From Experimental Data | |

14.49.2. | Sample Spaces and Events in Probability | |

14.49.3. | Single Events in Probability | |

14.49.4. | The Complement of an Event | |

14.49.5. | Introduction to Sets | |

14.49.6. | The Union of Sets | |

14.49.7. | The Intersection of Sets | |

14.49.8. | Venn Diagrams in Probability |

14.50. Compound Events in Probability

14.50.1. | Compound Events in Probability From Experimental Data | |

14.50.2. | Computing Probabilities for Compound Events Using Venn Diagrams | |

14.50.3. | The Addition Law of Probability | |

14.50.4. | Mutually Exclusive Events |

14.51. Combinatorics

14.51.1. | The Rule of Sum and the Rule of Product | |

14.51.2. | Factorials | |

14.51.3. | Factorials in Variable Expressions | |

14.51.4. | Ordering Objects | |

14.51.5. | Permutations | |

14.51.6. | Combinations |