Master the algebra of advanced functions including quadratics, logarithms, trigonometry, and more. Dive deep into the theory of polynomials.

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

- Understand trigonometric functions as representing ratios of side lengths in right triangles.
- 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.
- Apply trigonometric identities to simplify trigonometric expressions.

- 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, 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.

- Perform arithmetic operations with complex numbers.

1.

Number Systems
11 topics

1.1. The Real Number System

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

1.1.2. | The Real Number System | |

1.1.3. | Writing Repeating Decimals as Fractions | |

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

1.2. Introduction to Complex Numbers

1.2.1. | Imaginary Numbers | |

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

1.2.3. | Complex Numbers | |

1.2.4. | Adding and Subtracting Complex Numbers | |

1.2.5. | Multiplying Complex Numbers | |

1.2.6. | Solving Quadratic Equations With Complex Roots | |

1.2.7. | The Cyclic Property of the Imaginary Unit |

2.

Polynomials
30 topics

2.3. Polynomials

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

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

2.3.3. | Describing Numerical Relationships Using Polynomial Identities |

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 Equations

2.6.1. | Determining the Roots of Polynomials | |

2.6.2. | Solving Polynomial Equations Using the Greatest Common Factor | |

2.6.3. | Solving Cubic Equations by Grouping | |

2.6.4. | Solving Biquadratic Equations |

2.7. Polynomial Theorems

2.7.1. | The Factor Theorem | |

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

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

2.7.4. | Factoring Quartic Polynomials Using the Factor Theorem | |

2.7.5. | Multiplicities of the Roots of Polynomials | |

2.7.6. | Finding Multiplicities of the Roots of Quartic Polynomials by Factoring | |

2.7.7. | The Remainder Theorem | |

2.7.8. | The Rational Roots Theorem |

2.8. Graphs of Polynomials

2.8.1. | Graphing Elementary Cubic Functions | |

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

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

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

2.8.5. | End Behavior of Polynomials | |

2.8.6. | Graphing General Polynomials |

3.

Functions
25 topics

3.9. Functions

3.9.1. | Local Extrema of Functions | |

3.9.2. | One-To-One Functions | |

3.9.3. | Graphs of Inverse Functions | |

3.9.4. | The Domain and Range of an Inverse Function | |

3.9.5. | Invertible Functions | |

3.9.6. | Calculating the Inverse of a Function | |

3.9.7. | The Inverse of a Quadratic Function | |

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

3.9.9. | Periodic Functions | |

3.9.10. | Even and Odd Functions | |

3.9.11. | Unbounded Behavior of Functions Near a Point | |

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

3.10. Graph Transformations of Functions

3.10.1. | Vertical Translations of Functions | |

3.10.2. | Horizontal Translations of Functions | |

3.10.3. | Vertical Stretches of Functions | |

3.10.4. | Horizontal Stretches of Functions | |

3.10.5. | Combining Graph Transformations: Two Operations | |

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

3.10.7. | Constructing Functions Using Transformations | |

3.10.8. | Vertical Reflections of Functions | |

3.10.9. | Horizontal Reflections of Functions | |

3.10.10. | Combining Reflections With Other Graph Transformations | |

3.10.11. | Finding Points on Transformed Curves | |

3.10.12. | The Domain and Range of Transformed Functions | |

3.10.13. | Absolute Value Graph Transformations |

4.

Exponentials & Logarithms
34 topics

4.11. Introduction to Logarithms

4.11.1. | Converting From Exponential to Logarithmic Form | |

4.11.2. | Converting From Logarithmic to Exponential Form | |

4.11.3. | Evaluating Logarithms | |

4.11.4. | The Natural Logarithm | |

4.11.5. | The Common Logarithm | |

4.11.6. | Simplifying Logarithmic Expressions |

4.12. The Laws of Logarithms

4.12.1. | The Product Rule for Logarithms | |

4.12.2. | The Quotient Rule for Logarithms | |

4.12.3. | The Power Rule for Logarithms | |

4.12.4. | Combining the Laws of Logarithms | |

4.12.5. | The Change of Base Formula for Logarithms |

4.13. Exponential Equations

4.13.1. | Solving Exponential Equations Using Logarithms | |

4.13.2. | Solving Equations Containing the Exponential Function | |

4.13.3. | Solving Exponential Equations With Different Bases | |

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

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

4.14. Logarithmic Equations

4.14.1. | Solving Logarithmic Equations | |

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

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

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

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

4.14.6. | Solving Logarithmic Equations Using the Zero-Product Property |

4.15. Graphs of Exponential Functions

4.15.1. | Vertical Translations of Exponential Growth Functions | |

4.15.2. | Vertical Translations of Exponential Decay Functions | |

4.15.3. | Interpreting Graphs of Exponential Functions | |

4.15.4. | Combining Graph Transformations of Exponential Functions | |

4.15.5. | Properties of Transformed Exponential Functions |

4.16. Graphs of Logarithmic Functions

4.16.1. | Graphing Logarithmic Functions | |

4.16.2. | Combining Graph Transformations of Logarithmic Functions | |

4.16.3. | Properties of Transformed Logarithmic Functions | |

4.16.4. | Finding the Inverse of a Transformed Logarithmic Function |

4.17. Modeling with Exponential Functions

4.17.1. | Modeling With Compound Interest | |

4.17.2. | Continuously Compounded Interest | |

4.17.3. | Converting Between Exponents |

5.

Rational & Radical Functions
11 topics

5.18. Rational Expressions

5.18.1. | Simplifying Rational Expressions Using Polynomial Factorization | |

5.18.2. | Splitting Rational Expressions Into Separate Terms | |

5.18.3. | Adding and Subtracting Rational Expressions | |

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

5.18.5. | Multiplying Rational Expressions | |

5.18.6. | Dividing Rational Expressions |

5.19. Reciprocal Functions

5.19.1. | Graphing Reciprocal Functions | |

5.19.2. | Graph Transformations of Reciprocal Functions | |

5.19.3. | Combining Graph Transformations of Reciprocal Functions | |

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

5.19.5. | Finding Intersections of Lines and Reciprocal Functions |

6.

Radical Expressions & Functions
10 topics

6.20. Radical Expressions

6.20.1. | Simplifying Square Root Expressions Using Polynomial Factorization | |

6.20.2. | Solving Advanced Radical Equations |

6.21. Graphs of Radical Functions

6.21.1. | Graphing the Square Root Function | |

6.21.2. | Graph Transformations of Square Root Functions | |

6.21.3. | Graphing the Cube Root Function | |

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

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

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

6.21.7. | Roots of Transformed Radical Functions | |

6.21.8. | Finding Intersections of Lines and Radical Functions |

7.

Conic Sections
15 topics

7.22. Circles as Conic Sections

7.22.1. | The Center and Radius of a Circle in the Coordinate Plane | |

7.22.2. | Equations of Circles Centered at the Origin | |

7.22.3. | Equations of Circles Centered at a General Point | |

7.22.4. | Finding the Center and Radius of a Circle by Completing the Square | |

7.22.5. | Calculating Intercepts of Circles | |

7.22.6. | Intersections of Circles with Lines |

7.23. Parabolas as Conic Sections

7.23.1. | Upward and Downward Opening Parabolas | |

7.23.2. | Left and Right Opening Parabolas | |

7.23.3. | The Vertex of a Parabola | |

7.23.4. | Calculating the Vertex of a Parabola by Completing the Square | |

7.23.5. | The Focus-Directrix Property of a Parabola | |

7.23.6. | Calculating the Focus of a Parabola | |

7.23.7. | Calculating the Directrix of a Parabola | |

7.23.8. | Calculating Intercepts of Parabolas | |

7.23.9. | Intersections of Parabolas With Lines |

8.

The Unit Circle
20 topics

8.24. The Unit Circle

8.24.1. | Angles in the Coordinate Plane | |

8.24.2. | Negative Angles in the Coordinate Plane | |

8.24.3. | Coterminal Angles | |

8.24.4. | Calculating Reference Angles | |

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

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

8.24.7. | Expressing Negative Angles in Terms of Reference Angles | |

8.24.8. | Expressing Large Angles in Terms of Reference Angles | |

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

8.24.10. | Extending the Pythagorean Identity to All Quadrants |

8.25. Special Trigonometric Ratios

8.25.1. | Finding Trigonometric Ratios of Quadrantal Angles | |

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

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

8.25.4. | Evaluating Trigonometric Expressions | |

8.25.5. | Further Extensions of the Special Trigonometric Ratios |

8.26. Trigonometry with General Triangles

8.26.1. | The Law of Sines | |

8.26.2. | The Law of Cosines | |

8.26.3. | The Area of a General Triangle | |

8.26.4. | Modeling Using the Law of Sines | |

8.26.5. | Modeling Using the Law of Cosines |

9.

Trigonometric Functions
23 topics

9.27. Graphing Trigonometric Functions

9.27.1. | Graphing Sine and Cosine | |

9.27.2. | Graphing Tangent and Cotangent | |

9.27.3. | Graphing Secant and Cosecant |

9.28. Properties of Trigonometric Functions

9.28.1. | Describing Properties of the Sine Function | |

9.28.2. | Describing Properties of the Cosine Function | |

9.28.3. | Describing Properties of the Tangent Function | |

9.28.4. | Describing Properties of the Secant Function | |

9.28.5. | Describing Properties of the Cosecant Function | |

9.28.6. | Describing Properties of the Cotangent Function |

9.29. Graph Transformations of Trigonometric Functions

9.29.1. | Vertical Translations of Trigonometric Functions | |

9.29.2. | Vertical Stretches of Trigonometric Functions | |

9.29.3. | Horizontal Translations of Trigonometric Functions | |

9.29.4. | Horizontal Stretches of Trigonometric Functions | |

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

9.29.6. | Graph Transformations of Tangent and Cotangent | |

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

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

9.29.9. | Graphing Reflections of Trigonometric Functions | |

9.29.10. | Graphing Reflections of Trigonometric Functions: Three or More Transformations |

9.30. Properties of Transformed Trigonometric Functions

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

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

9.30.3. | Properties of Transformed Secant and Cosecant Functions | |

9.30.4. | Modeling With Trigonometric Functions |