Algebra 2 Review: Key Topics, Formulas, and Exam Checklist
An Algebra 2 final centers on functions and their graphs, quadratics and complex numbers, polynomial behavior, rational and radical equations, and exponential and logarithmic functions. For each topic, recall the rule, work one example by hand, and substitute the answer back into the original equation. Use the sections below to locate a weak unit, then open the dedicated guide for that topic.
What This Review Covers
This page reviews the topics and formulas an Algebra 2 final tests, with one short refresher problem per topic. The Algebra 2 study guide lays out the course as units and lessons. Vocabulary definitions and a separate practice set sit on their own pages, so this review stays on what to memorize and redo.
Function Foundations and Transformations
A function assigns exactly one output to each input. The domain is the set of allowed inputs, and the range is the set of resulting outputs; on a graph, the vertical line test confirms that no vertical line crosses the curve more than once.
A parent function changes through shifts, reflections, and stretches. For a transformed graph written as g(x) = a·f(x - h) + k, h moves the graph horizontally, k moves it vertically, and a stretches or reflects it. The functions guide develops notation, domain, and these transformations in full.
Quadratic Functions and Complex Numbers
A quadratic has the form ax^2 + bx + c = 0 with a ≠ 0. Two questions matter here: how to solve it, and what the roots look like.
The Quadratic Formula and the Discriminant
When factoring is not quick, use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)The expression under the square root, b^2 - 4ac, is the discriminant. A positive discriminant gives two real roots, zero gives one repeated real root, and a negative discriminant gives no real root.
Refresher: solve x^2 - 4x + 1 = 0. Here a = 1, b = -4, c = 1, so the discriminant is 16 - 4 = 12. The roots are x = (4 ± √12)/2 = 2 ± √3, two distinct real roots. The quadratic equations guide compares factoring, completing the square, and the formula.
Complex Numbers and Complex Roots
Complex numbers extend the real numbers with i, where i^2 = -1. When the discriminant is negative, the roots form a conjugate pair a + bi and a - bi.
Refresher: solve x^2 + 2x + 5 = 0. The discriminant is 4 - 20 = -16, so there are no real roots. Applying the formula gives x = (-2 ± √(-16))/2 = (-2 ± 4i)/2 = -1 ± 2i, a conjugate pair.
Polynomial Functions: Roots, Factors, and Graphs
A polynomial in factored form reveals its roots: each linear factor (x - r) gives a zero at x = r. The degree and leading coefficient set the end behavior of the graph.
Refresher: factor x^3 - 4x. Pull out x first: x(x^2 - 4) = x(x - 2)(x + 2), so the zeros are -2, 0, and 2. With a positive leading coefficient, the cubic falls on the left and rises on the right. The polynomials guide covers division, the Factor Theorem, and graphing in depth.
Rational Functions and Equations
A rational function has a variable in the denominator. Before simplifying or solving, write the excluded values where the denominator equals zero; those values stay out of the domain even if a factor cancels. A vertical asymptote sits where a non-cancelled factor makes the denominator zero.
Refresher: for f(x) = 1/(x - 3), x = 3 is excluded and the vertical asymptote is x = 3. When you clear denominators to solve a rational equation, reject any candidate that lands on an excluded value. The rational expressions guide walks through simplifying, asymptotes, and equations.
Radical Functions and Equations
A radical equation contains a variable under a root, most often a square root. Squaring both sides can produce an extraneous root, so every candidate must be substituted back into the original equation.
Refresher: solve √(2x + 7) = x + 1. Squaring gives 2x + 7 = (x + 1)^2 = x^2 + 2x + 1, so x^2 - 6 = 0 and x = ±√6. Substitute back: √6 works because both sides equal 1 + √6; -√6 fails because the right side is negative while the square root is positive, so it is rejected. The radicals guide covers rational exponents and radical equations in full.
Exponential and Logarithmic Functions
A logarithm is the inverse of an exponent: log_b(x) = y means b^y = x. Logarithm properties turn products into sums, quotients into differences, and powers into multipliers, and the change-of-base rule writes a logarithm in any base:
log_b(x) = ln(x) / ln(b)Refresher: solve log_2(x + 3) = 4. In exponential form, x + 3 = 2^4 = 16, so x = 13. Check that the logarithm's argument stays positive, here 13 + 3 = 16.
Systems, Matrices, and Sequences
Many courses close the year with solving systems, basic matrix operations, and arithmetic or geometric sequences. A sequence gives an explicit or recursive rule for its terms; a geometric sequence multiplies by a constant ratio at each step. The course catalog covers these extensions in full, so this section lists them for scoping only.
The Algebra 2 Formula Sheet
Topic | Formula to remember |
Quadratic formula | x = (-b ± √(b^2 - 4ac)) / (2a) |
Discriminant | b^2 - 4ac: positive two roots, zero one root, negative complex pair |
Complex unit | i^2 = -1 |
Logarithm inverse | log_b(x) = y means b^y = x |
Change of base | log_b(x) = ln(x) / ln(b) |
Rational exponent | x^(m/n) = ⁿ√(x^m) |
Final Review Self-Check Checklist
Mark every topic where a refresher problem still trips you up, then return to that section before the exam.
- Functions: Read domain and range and run the vertical line test.
- Quadratics: Solve with the formula and read the discriminant.
- Complex numbers: Write a negative-discriminant root as a conjugate pair.
- Polynomials: Factor and name the zeros and end behavior.
- Rational: Write excluded values before clearing denominators.
- Radicals: Square to solve, then substitute candidates back.
- Exponential/log: Convert between log form and exponential form.
- Systems and sequences: Solve a system and name a sequence rule.
Common Mistakes and How to Check Your Work
- Treating a negative discriminant as "no answer" instead of a complex conjugate pair.
- Forgetting to write excluded values before clearing a rational denominator.
- Keeping a squared candidate that the original equation rejects.
- Applying a logarithm rule to a sum instead of a product.
- Stopping after the algebra instead of substituting back into the original equation.
- Reading the end behavior from the constant term rather than the degree and leading coefficient.
Where This Review Fits
This page consolidates topics and formulas; each section points to the guide that explains a method in depth. If Algebra 1 skills such as factoring, solving equations, or exponent rules feel shaky, start with the Algebra 1 review and return here. How Algebra 2 compares with Geometry is covered in the course hub rather than repeated here. College students working toward a different course can compare scope with the college algebra guide.
Practice Algebra 2 with Gauth
For unit practice, open the verified Algebra 2 course and work the matching lesson after each weak section. If one topic still trips you up, go to Gauth Math Courses, choose Create Course, enter the topic and your level, and generate a focused interactive course. Redo a fresh problem on paper before comparing answers.
Sources and Review
OpenStax: Algebra and Trigonometry 2e, accessed October 19, 2026.HCPSS: Summer Review For Students Entering Algebra 2, accessed October 19, 2026.HCPSS: Algebra II GT Essential Curriculum, accessed October 19, 2026.