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# Algebra II

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

## Content

### Logarithms

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

### Trigonometry

• 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.

### Nonlinear Equations and Graphing

• 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.

### Polynomial Division and Factoring

• 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.

### Graphing 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.

### Complex Numbers

• 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. Domain and Range of Inverse Functions 3.9.5. Invertible Functions 3.9.6. Calculating the Inverse of a Function 3.9.7. Inverses of Quadratic Functions 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. Inverses of Exponential and Logarithmic Functions
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 Expressions & Functions
12 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. Inverses of Reciprocal Functions 5.19.6. Finding Intersections of Lines and Reciprocal Functions
6.
11 topics
 6.20.1. Simplifying Square Root Expressions Using Polynomial Factorization 6.20.2. Solving Advanced Radical Equations
 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. Inverses of Radical Functions 6.21.9. 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
21 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. Extending the Trigonometric Ratios Using Angles in Radians 8.24.8. Extending the Trigonometric Ratios to Negative Angles 8.24.9. Extending the Trigonometric Ratios to Large Angles 8.24.10. Using the Pythagorean Identity in the First Quadrant 8.24.11. 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