Algebra 2 Review
What You’ll Learn
Section titled “What You’ll Learn”This review lesson brings together the major concepts from the entire Algebra 2 course. Use it to solidify your understanding before moving on.
Key Concepts Review
Section titled “Key Concepts Review”1. Functions and Their Properties
Section titled “1. Functions and Their Properties”- Function notation: f(x), evaluation, domain, and range
- Inverse functions: f(f⁻¹(x)) = x
- Piecewise functions: different rules for different parts of the domain
2. Quadratic Functions
Section titled “2. Quadratic Functions”- Graphing in standard and vertex form
- Vertex form: f(x) = a(x − h)² + k
- Solving quadratic equations by factoring, completing the square, and quadratic formula
- The discriminant b² − 4ac tells you the nature of the solutions:
3. Complex Numbers
Section titled “3. Complex Numbers”- Imaginary unit: i = √(−1), i² = −1
- Standard form: a + bi
- Operations: add, subtract, multiply, divide (multiply by conjugate)
- Solving quadratics with negative discriminant
4. Polynomial Functions
Section titled “4. Polynomial Functions”- Degree and leading coefficient
- Adding, subtracting, and multiplying polynomials
- Factoring higher-degree polynomials using grouping and Rational Root Theorem
5. Rational Expressions and Equations
Section titled “5. Rational Expressions and Equations”- Simplifying by factoring and canceling
- Multiplying and dividing (flip and multiply for division)
- Solving rational equations and checking for extraneous solutions
6. Radical Expressions and Equations
Section titled “6. Radical Expressions and Equations”- Simplifying radicals and rational exponents
- Operations with radicals (add/subtract like terms, multiply/divide using rules)
- Solving radical equations and checking solutions
7. Conic Sections
Section titled “7. Conic Sections”- Identifying from equation: circle, parabola, ellipse, hyperbola
- Standard forms and key features (center, radius, vertex, foci, etc.)
8. Exponential and Logarithmic Functions
Section titled “8. Exponential and Logarithmic Functions”- Exponential growth: y = a(1 + r)ᵗ
- Exponential decay and half-life
- Logarithms as inverses of exponentials
- Properties of logarithms (product, quotient, power)
9. Systems of Equations
Section titled “9. Systems of Equations”- Linear systems with three variables
- Nonlinear systems (especially linear + quadratic)
- Basic matrix representation and solving small systems
10. Sequences
Section titled “10. Sequences”- Arithmetic sequences: constant difference, explicit formula aₙ = a₁ + (n − 1)d
- Geometric sequences: constant ratio, explicit formula aₙ = a₁ · r^(n − 1)
Mixed Worked Examples
Section titled “Mixed Worked Examples”1. Solve 2x² − 5x − 3 = 0 using the quadratic formula
Discriminant = 25 + 24 = 49.
So x = 3 or x = −0.5.
2. Simplify (x² − 9) / (x² − 6x + 9)
3. Find the 10th term of the arithmetic sequence with a₁ = 8 and d = −3
4. Solve the nonlinear system: y = x + 2 and y = x² − 4x + 3
Set equal: x + 2 = x² − 4x + 3, which gives x² − 5x + 1 = 0. After solving and substituting back:
Solutions: (1, 3) and (5, 7).
Real-World Connections
Section titled “Real-World Connections”- Quadratics: profit maximization, projectile motion
- Rational expressions: work rates, average cost
- Exponentials and logs: compound interest, half-life, pH scale
- Sequences: savings plans, population growth
- Conics: satellite dishes, planetary orbits, bridge design
Final Tips for Success
Section titled “Final Tips for Success”- Always check solutions in the original equation (especially rational and radical equations)
- Factor completely before simplifying rational expressions
- Use the discriminant to predict the nature of quadratic solutions
- Practice converting between exponential and logarithmic forms
- Sketch graphs when possible. Visual understanding helps a lot