SAT Advanced Math: Quadratics, Exponents, and Curved Graphs

Advanced Math is Algebra's less-linear sibling: instead of straight lines, you're working with quadratics, exponentials, rational expressions, and curved graphs. It's just as heavily weighted as Algebra, so it deserves the same amount of practice time.

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Did you know

Advanced Math and Algebra together make up roughly 70% of the entire Math section, about 35% each. Between the two, they're worth more than the other two math domains combined.

What's tested

Equivalent expressions

Rewrite an expression (factor it, expand it, simplify a rational expression) into an equivalent form.

Learn more: factoring fast and checking your work

Always check for a common factor first before trying anything else, pulling out a shared term often turns a hard-looking expression into a simple one.
For quadratics in the form x² + bx + c, look for two numbers that multiply to c and add to b. If the leading coefficient isn't 1, check the answer choices by expanding them instead of factoring from scratch, it's often faster.
Plug in a real number for the variable as a check. Compute the original expression and your simplified version with the same number, if they don't match, you made an error somewhere.
For rational expressions, factor the top and bottom fully before canceling anything. Canceling too early, or canceling terms that are added instead of multiplied, is the single most common mistake here.

Nonlinear equations and systems

Solve quadratic, exponential, or radical equations, including a linear equation paired with a nonlinear one.

Learn more: choosing the fastest solving method

For a quadratic equal to zero, factor first if it looks clean. If it doesn't factor nicely, the quadratic formula always works, but plugging numbers into the answer choices can be faster on a multiple-choice question.
Quadratics usually have two solutions. If a question asks for "the sum of all solutions" or "the greater solution," make sure you find both roots before answering, picking just the first one you find is a common way to lose an easy point.
For a system with one linear and one nonlinear equation, solve the linear one for a variable and substitute it into the nonlinear equation, this almost always the fastest path.
Watch for extraneous solutions on equations involving square roots or with variables in a denominator, always check candidate answers back in the original equation, since squaring both sides can introduce a solution that doesn't actually work.

Nonlinear functions

Reading and interpreting graphs of quadratics, exponentials, and other curves, vertex, intercepts, growth and decay.

Learn more: reading a parabola or exponential graph

For a parabola in vertex form, y = a(x − h)² + k, the vertex is at (h, k) directly, no calculation needed. Watch the sign inside the parentheses; it flips h's sign.
The sign of "a" tells you the direction: positive opens upward (a minimum), negative opens downward (a maximum). This alone can eliminate half the answer choices on some questions.
For exponential growth or decay, y = a(b)ˣ: a is the starting value, b is the growth factor. b greater than 1 means growth; b between 0 and 1 means decay.
Word problems about "doubling" or "halving" are exponential functions in disguise, if a quantity multiplies by the same factor over equal time periods, it's exponential, not linear.

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Gotcha

A question asks for "the value of x" in a quadratic equation, and you correctly find x = 3 or x = −5. If the answer choices only list one of them, or the question asks for something like "the greatest possible value of x," picking the first root you calculated instead of checking both is one of the most common avoidable errors on this domain. Quadratics almost always have two answers, find both before you decide which one the question actually wants.

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Quick tip

When a question describes a graph but doesn't show one, use the built-in calculator. Type in the equation, look at the actual shape, and read off whatever the question is asking for, it's often faster and safer than working it out algebraically.

Reading about the traps is step one.

Ember has original Advanced Math questions, each with the reasoning explained the same way, including exactly which trap answer it's testing.

Practice Advanced Math now