SAT Advanced Math Study Guide
Prepare for SAT Math. Study equivalent expressions, quadratics, nonlinear functions and systems. Choose a lesson below, or try mixed practice to check your skills.
Alongside the core guide, the topic lessons include additional instruction, 3 worked examples and 6 targeted practice questions. Find these sets under Practice options.
All lessons
7 lessons
About this guide
Seven complete lessons, 105 worked examples, 35 topic-practice exercises and 42 mixed-practice questions. Written explanation tasks include model answers for self-review.
How to use this guide
Advanced Math is not simply “Algebra, but with more steps.” Its central challenge is choosing a representation that makes the question easier. A product exposes zeros. A completed square exposes an extreme value. A power with a carefully chosen exponent exposes a growth period. A denominator exposes a restriction that no amount of simplification can erase.
This volume follows the seven Advanced Math sections in the larger SAT Math study-guide outline. Every section contains a concept lesson, exactly 15 worked examples, and a five-question exit check. The final mixed practice contains 42 additional questions. All 105 worked examples, 35 exit-check questions, and 42 mixed questions have complete solutions.
A study cycle that builds independent skill
Learn the condition with the rule. Do not memorize “cancel” without “common nonzero factor,” or “use the discriminant” without “the equation is genuinely quadratic.” Conditions are part of the mathematics, not fine print.
Attempt before reading. Leave the worked solution closed while attempting each question. Write the target, restrictions, and first useful step. When stuck, open the worked solution, read the recognition cue, then close it and try again. The titles are teaching supports; the mixed set removes them.
Explain the method choice. A correct answer with an unjustified operation is not secure understanding. Explain why a factorization, substitution, graph, or coefficient comparison works. An alternative method is useful only when you can identify what it saves.
Retrieve and transfer. Re-solve missed problems after a gap, then use the exit checks and mixed practice without notes. Record the cause of an error, not merely the problem number. The review tools at the back make that process reusable.
The difficulty progression
Foundation examples isolate a key relationship. Standard examples connect representations or add context. Challenge examples emphasize parameters, hidden structure, solution counts, and restrictions. These labels do not predict an official difficulty rating or an SAT score.
Two routes through the book
Build the domain: study Sections 1–7 in order. Sections 1 and 2 supply the rewriting tools used everywhere else.
Repair a weakness: use the linked skills map in Review and reference, work the linked examples, and then attempt unfamiliar questions without the chapter’s hints.
Navigation. Use the guide contents for lessons and the skills map for links to worked examples.
Scope, priorities, and testing context
College Board organizes Advanced Math around equivalent expressions, nonlinear equations and systems, and nonlinear functions. The named families include quadratic, exponential, polynomial, rational, radical, and absolute-value relationships. This guide divides that material into seven teaching sections.[1]
Advanced Math accounts for approximately 35% of SAT Math. This is a domain-level weighting, not a promised frequency for every individual topic.[2]
Section | Central question |
|---|---|
1. Equivalent expressions and polynomials | Which form reveals the structure or the requested quantity? |
2. Exponents and radicals | How can powers and roots express the same quantity? |
3. Quadratic equations and functions | What do roots, a vertex, and coefficients tell us? |
4. Rational, radical, and absolute-value relationships | Which candidates survive the original restrictions? |
5. Exponential equations and functions | What changes by a constant factor, and over which interval? |
6. Nonlinear function features and transformations | How do inputs, outputs, and graph features move together? |
7. Nonlinear systems | Which values satisfy every relationship at once? |
The boundary of this volume
Linear equations and inequalities are assumed prerequisites and belong in the Algebra volume. Geometry appears only when it supplies a nonlinear model. This is not a general precalculus course: it does not devote chapters to logarithmic identities, complex-number arithmetic, trigonometric identities, or calculus. The coverage audit uses the official detailed testing points.[3]
Digital testing and calculator context
The standard Math section has 44 questions in two 35-minute modules. First-module performance influences the second module’s difficulty. The section includes multiple-choice and student-produced response questions.[4]
Bluebook includes Desmos graphing and scientific options, and calculators are allowed throughout Math. Current policy prohibits handheld calculators with computer algebra system (CAS) functionality. Recheck the official policy before test day; these lessons do not depend on symbolic solving software.[5]
A decimal display is evidence, not an exact proof
A graph can locate a root or suggest a tangent. It does not by itself establish an exact parameter condition, preserve a canceled domain restriction, or prove that no intersection exists beyond the current window. The calculator workflow in Review and reference keeps the mathematics in control.
Before you begin: the five habits that matter
1. Distinguish an expression, an equation, and a function
An expression such as x2 − 5x + 6 is a quantity to rewrite or evaluate. An equation such as x2 − 5x + 6
2. Write the target before doing algebra
“Find x,” “find the larger root,” “find the sum of the roots,” and “find the minimum value” are different requests. For f(x)
3. Record restrictions before changing form
A denominator must be nonzero. An even root must have a nonnegative radicand in real-number work. A context can require a positive length, an integer count, or nonnegative time. A simplified expression is equivalent only on the original domain.
4. Ask whether an operation is reversible
Adding the same quantity to both sides is reversible. Squaring can introduce extraneous solutions because opposite numbers have the same square. Dividing by an expression can remove valid solutions if that expression can be zero. Multiplying by a denominator requires preserving the restriction that made the denominator legal.
5. Keep exact form until a decimal is useful
A value such as or is exact. A rounded display is not. Use exact form for comparisons, symbolic answer choices, and intermediate steps. When a numeric answer is requested, follow the question’s rounding instruction and the test’s entry directions.
A compact decision sequence
Target ⟶ domain ⟶ structure ⟶ method ⟶ check. Before expanding, ask: Can I factor? Can I isolate a square or root? Can I rewrite a base? Can I substitute a repeated expression? Can I compare coefficients? Can I find the target without finding every unknown?
Prerequisite check. You should be comfortable distributing a negative sign, combining like terms, solving a linear equation, adding numerical fractions, and reading coordinate axes. When one of those skills repeatedly breaks down, repair it in the Algebra volume rather than memorizing a workaround here.