# Algebra II

Our comprehensive, fully accredited, Common Core-aligned Algebra II course is designed to build upon the strong foundations acquired in Algebra I and Geometry, delving deeper into algebra, functions, geometry, and trigonometry. This course will further develop students' mathematical understanding and problem-solving skills, preparing them for success in our Precalculus course, the final stepping stone before calculus.

## Content

In this course, students will expand their understanding of quadratic functions to cubic and higher-order polynomials. They will gain detailed experience in dealing with fundamental concepts associated with these polynomials, including least common factors and multiples, polynomial factorization, division, solving polynomial equations, and applying key polynomial theorems in various contexts.

Building on the solid foundations laid out in Algebra I, this course enables students to deepen their understanding of functions. They will explore global and local extrema, periodicity, even and odd functions, and invertible functions. They will be carefully guided through the process of performing graph transformations of functions and will learn to compute inverse functions seamlessly.

As the course progresses, students will enrich their knowledge of exponential functions. This includes a deep dive into logarithms, examining their graphs and properties. Students will learn to solve both exponential and logarithmic equations and further develop their aptitude for modeling scenarios with exponential functions, such as calculating compound interest.

Building upon the knowledge acquired in Geometry, this course facilitates deep explorations of trigonometric functions. Students will dive into the intricacies of the unit circle and discover how it elegantly extends the trigonometric ratios. Students will learn to apply the Law of Sines and Cosines effectively, graph trigonometric functions with precision, describe their distinct properties and master the techniques for transforming the graphs of trigonometric functions.

Upon successful completion of this course, students will have mastered the following:
• Understand the hierarchy and properties of the real number system, including classifying natural, integer, rational, and irrational numbers.
• Convert repeating decimals to fractions and describe general properties of operations with rational and irrational numbers.
• Understand the fundamentals of complex numbers and perform arithmetic operations with them.
• Solve quadratic equations with imaginary and complex solutions, and understand the cyclic property of the imaginary unit.
• Identify and find the least common factors and multiples of monomials and polynomials.
• Factor polynomials using various techniques, including using the greatest common factor, difference of squares, grouping, sum/difference of cubes, and factoring biquadratic expressions.
• Divide polynomials using synthetic division, long division, and manipulation of rational expressions.
• Determine the roots of polynomial equations using the factor, remainder, and rational roots theorems.
• Graph polynomial functions, describing key features such as end behavior and roots.
• Identify one-to-one and invertible functions, restricting the range of a function to make it invertible, and the relationship between the domain and range of a function and that of its inverse.
• Analyze local extrema of functions, periodic functions, even and odd functions, and unbounded behavior of functions.
• Calculate the average rate of change of a function.
• Apply graph transformations, including translations, stretches, reflections, and combinations of these. Apply absolute value graph transformations.
• Determine the domain and range of transformed functions and locate points on the transformed graphs.
• Convert between exponential and logarithmic forms of expressions, evaluate logarithms, and use the properties of logarithms to simplify expressions.
• Solve exponential and logarithmic equations, including those with different bases.
• Graph exponential and logarithmic functions and understand the properties of these graphs.
• Model real-world scenarios with exponential functions, including financial applications such as compound and continuously compounded interest.
• Simplify, add, subtract, multiply, and divide rational expressions.
• Graph and analyze reciprocal and rational functions, including identifying asymptotes, roots, intercepts, and end behavior.
• Combine transformations of reciprocal functions and analyze their domains, ranges, and inverses.
• Simplify radical expressions, and solve radical equations, including those with extraneous solutions.
• Graph and analyze radical functions, including identifying key features of the graph such as domain, range, and intercepts.
• Write equations of circles, and calculate the center and radius of a circle from its equation, including completing the square.
• Calculate intercepts of circles and parabolas, and find intersections of these conics with lines.
• Understand and apply the focus-directrix property of a parabola.
• Develop a strong foundation in trigonometry, including understanding and applying trigonometric ratios, working with radians, and utilizing the unit circle.
• Master graphing and analyzing trigonometric functions, including applying graph transformations and describing their resulting properties. Use transformed trigonometric functions to model real-world scenarios.
• Develop and apply the Law of Sines and the Law of Cosines, using them to model and solve problems in real-world situations.
1.
Number Systems
10 topics
1.1. The Real Number System
 1.1.1. The Real Number System 1.1.2. Writing Repeating Decimals as Fractions 1.1.3. Sums and Products of Rational and Irrational Numbers
1.2. Introduction to Complex Numbers
 1.2.1. Imaginary Numbers 1.2.2. 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. Properties 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