SAT Advanced Math
Practice Questions
Advanced Math covers quadratics, polynomials, exponentials, nonlinear functions, and equivalent expressions. The domain rewards algebraic fluency plus careful interpretation of how an expression or graph behaves.
About Advanced Math
The Advanced Math domain builds directly on Algebra skills and introduces nonlinear relationships: quadratic and higher-degree polynomials, rational expressions, radical equations, and exponential and linear growth models. Questions from this domain can appear in either module; the overall Module 1 performance determines the Module 2 difficulty mix.
One defining feature of Advanced Math questions is the emphasis on equivalent expressions and strategic manipulation. Rather than always solving for a numerical answer, many questions ask you to simplify an expression, factor a polynomial, or identify an equivalent form. Recognizing patterns such as a difference of squares, a perfect-square trinomial, or a common factor helps you choose a valid rewrite instead of a merely plausible one.
Nonlinear equation and function questions can combine conceptual understanding with computation. For example, a quadratic question might ask for the vertex of a parabola and then ask what the vertex represents in a context about projectile motion. An exponential question might give you a growth formula and ask how the base relates to the percentage growth rate. These contextual layers require both an algebraic procedure and an interpretation of its result.
What You'll Practice
- Factoring quadratic and polynomial expressions using GCF, grouping, and standard patterns
- Solving quadratic equations by factoring, completing the square, and the quadratic formula
- Identifying and interpreting key features of parabolas (vertex, axis of symmetry, roots)
- Simplifying and manipulating rational and radical expressions
- Modeling exponential growth and decay and interpreting growth factors
- Recognizing equivalent algebraic forms and matching them to contextual descriptions
Why Advanced Math Matters for Your Score
Advanced Math is one of the two largest Math domains, accounting for about 35% of questions. Understanding quadratics, exponentials, and polynomial manipulation conceptually—not just procedurally—helps across the domain's easier and harder items; your scaled result still depends on performance across both Math modules and the test form's scoring model.
Advanced Math Subtopics
Each subtopic page combines its available practice questions with concept explanations, common mistakes, and strategy tips tailored to that specific skill.
Equivalent Expressions
Rewriting algebraic expressions into equivalent forms through factoring, expanding, and applying algebraic identities.
Nonlinear Equations in One Variable
Solving quadratic, radical, and rational equations, and understanding the conditions under which extraneous solutions arise.
Nonlinear Functions
Understanding polynomial, radical, and rational functions—evaluating them, identifying their key features, and interpreting them in context.
Quadratic Equations and Parabolas
Solving quadratics, identifying the vertex and intercepts of parabolas, and interpreting these features in applied contexts.
Exponential Functions
Modeling growth and decay with exponential functions, interpreting the base and exponent, and comparing exponential to linear growth.
Adv. Math Sample Questions
More questionsPick an answer and hit Check Answer to see the detailed explanation. Questions are from easy, medium, and hard difficulty levels.
Which of the following is equivalent to 3x^2 + 6x?
Show explanation
Correct answer: B. 3x(x + 2)
Explanation
Factor out the GCF of 3 and x: 3x^2 + 6x = 3x(x + 2). Choice A would equal 3x^2 + 18x. Choice C equals 6x^2 + 6x. Choice D is not valid.
Using the quadratic formula, what are the solutions to 2x^2 - 7x + 3 = 0?
Show explanation
Correct answer: D. x = 1/2 or x = 3
Explanation
Using the quadratic formula with a=2, b=-7, c=3: discriminant = 49 - 24 = 25. x = (7 ± 5)/4. So x = 12/4 = 3 or x = 2/4 = 1/2.
The function f(x) = (x^2 - 1)/(x - 1) has which of the following key features?
Show explanation
Correct answer: C. A hole at x = 1 and no asymptote
Explanation
Factor: (x^2-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1 for x ≠ 1. The (x-1) factor cancels, creating a hole at x = 1 (the point (1, 2) is missing from the graph), not an asymptote.
What is the vertex of the parabola y = (x - 3)^2 + 5?
Show explanation
Correct answer: A. (3, 5)
Explanation
The equation is in vertex form y = (x - h)^2 + k, where (h, k) is the vertex. Here h = 3 and k = 5, so the vertex is (3, 5).
The value of an investment after t years is V(t) = 2500 * (1.06)^t dollars. What does 1.06 represent in this context?
Show explanation
Correct answer: B. The investment earns 6% interest per year
Explanation
The base 1.06 = 1 + 0.06 = 1 + 6%. This means the investment grows by 6% per year (the annual interest rate is 6%).
Which of the following is equivalent to 8x^3 - 27?
Show explanation
Correct answer: C. (2x - 3)(4x^2 + 6x + 9)
Explanation
This is a difference of cubes: a^3 - b^3 = (a-b)(a^2+ab+b^2). Here a = 2x, b = 3: (2x)^3 - 3^3 = (2x-3)((2x)^2 + (2x)(3) + 3^2) = (2x-3)(4x^2 + 6x + 9).
What are the solutions to x^2 - 9 = 0?
Show explanation
Correct answer: A. x = 3 or x = -3
Explanation
Add 9: x^2 = 9. Take the square root of both sides: x = ±3. Both 3 and -3 are solutions.
The function f(x) = -x^2 + 6x - 5. At which values of x does f(x) cross the x-axis?
Show explanation
Correct answer: C. x = 1 and x = 5
Explanation
Set f(x) = 0: -x^2 + 6x - 5 = 0 → x^2 - 6x + 5 = 0 → (x-1)(x-5) = 0 → x = 1 or x = 5.
Want all Advanced Math questions?
Continue with more advanced math practice filtered by difficulty, score band, and skill.
Strategy Tips for Adv. Math
Factor before using the quadratic formula
Before reaching for the quadratic formula, inspect the coefficients for a common factor or an integer factor pair. If factoring is not clear, the quadratic formula gives a general route; write the signs and denominator carefully.
Memorize the key factoring patterns
Know the difference of squares, perfect-square trinomials, and sum or difference of cubes. Recognizing one of these structures lets you rewrite the expression directly, then verify the result by expanding.
Use vertex form to read off the vertex
If a parabola question asks for the vertex or maximum/minimum, convert to vertex form f(x) = a(x - h)^2 + k by completing the square, or use h = -b/(2a). Once you have the vertex (h, k), answer any related contextual question directly from the coordinates.
Understand exponential base meaning
For f(x) = a * b^x, the base b is the growth or decay factor per unit of x. If b = 1.05, the quantity grows by 5% per period; if b = 0.80, it decays by 20%. Keep the initial value a separate from that per-period factor.
Frequently Asked Questions — Adv. Math
What is the difference between Algebra and Advanced Math on the test?
Algebra focuses on linear relationships (equations and functions where the highest power of the variable is 1). Advanced Math includes nonlinear relationships—quadratic, polynomial, rational, radical, and exponential functions. In practice, Algebra questions tend to have more straightforward setups while Advanced Math questions require multi-step manipulation and a deeper understanding of function behavior.
Do I need to know the quadratic formula for the test?
Yes. A quadratic may not factor cleanly over the integers or may have irrational roots, so know x = (-b ± sqrt(b^2 - 4ac)) / (2a). The formula is not on the test's reference sheet. The discriminant also tells you the number of real solutions: positive gives two, zero gives one repeated real solution, and negative gives none.
How do exponential function questions appear on the test?
Exponential questions usually involve interpreting a given model (e.g., P(t) = 5000 * 1.03^t for a population growing 3% per year) and answering contextual questions: initial value, growth rate, value at a specific time, or what a change to the formula would do. A smaller number of questions require you to write the equation from given conditions (two data points, or initial value and growth rate).
Can I use the graphing calculator for Advanced Math questions?
Yes. The built-in graphing calculator is available throughout Math. A graph can help estimate a parabola's vertex or intercepts and inspect an exponential model, but use the equation when the question requires an exact value or parameter relationship.
What does 'equivalent expressions' mean on the test?
An 'equivalent expressions' question gives you an expression and asks which answer choice is equal to it for all values of the variable. These are identity questions, not equation questions. Strategies include expanding and simplifying both the original and each answer choice, substituting a specific number (like x = 2) into the original and each choice, or recognizing standard algebraic patterns that let you rewrite the expression in one step.
Other Math Topics
Ready for targeted Adv. Math practice?
Continue in the question bank for filtered practice, then use the timed Math test to check pacing across a full section.