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M3 Nonlinear functions

One of the SAT's published Math skills, inside Advanced Math. 149 questions here, across 56 distinct question types.

Advanced Math covers nonlinear content: quadratic equations, functions and their graphs, exponential expressions, polynomials, and rational and radical equations. Success here means being comfortable moving between an equation, a table, and a graph, and knowing the go-to methods for solving quadratics. One concrete tip: learn to read a parabola's key features, like its vertex and where it crosses the x-axis, straight from the equation's form, because the form often hands you the answer. On this page you'll find a short lesson on functions, quadratics, and exponential relationships with worked examples, plus free interactive practice with instant explanations so you can strengthen the nonlinear skills that separate mid and high math scores.

Advanced Math covers quadratics, exponents, polynomials, and functions — the nonlinear content. This is where 1500+ scores are decided, because the hardest module-2 questions cluster here.

Factored form tells you the zeros. If y = a(x − r₁)(x − r₂), the graph crosses the x-axis at r₁ and r₂. Reading zeros straight from factors saves huge time.

Must-know facts

  • Function notation: f(3) means substitute 3 for every x. Don't overthink it.
  • Exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ; (xᵃ)ᵇ = xᵃᵇ; x⁰ = 1.
  • Vertex of a parabola y = ax² + bx + c is at x = −b/(2a); plug back in for the y-value (the min if a > 0, max if a < 0).
  • Sum of the zeros of a quadratic = −b/a; product = c/a. Handy shortcuts.
Desmos tip: A calculator is allowed on the entire Math section — there is no no-calculator portion. You can graph any equation and read zeros, intersections, and vertices directly. Use it to check — but know the algebra for the no-calculator instincts it builds.

Worked examples

Find the zeros of y = x² − 5x + 6.

  1. Factor: we need two numbers that multiply to 6 and add to −5. Those are −2 and −3.
  2. So y = (x − 2)(x − 3).
  3. Set each factor to zero: x − 2 = 0 → x = 2; x − 3 = 0 → x = 3.
  4. The zeros are x = 2 and x = 3 (the points where the graph crosses the x-axis).

Find the minimum value of y = x² − 6x + 5.

  1. The graph is a parabola opening upward (a = 1 > 0), so it has a minimum at its vertex.
  2. Vertex x-coordinate: x = −b / (2a) = −(−6) / (2 × 1) = 3.
  3. Substitute back: y = 3² − 6(3) + 5 = 9 − 18 + 5 = −4.
  4. The minimum value is −4, at the point (3, −4).

If f(x) = 2x² − 3x + 1, find f(−2).

  1. Replace every x with −2: f(−2) = 2(−2)² − 3(−2) + 1.
  2. Handle the exponent first: (−2)² = 4, so 2(4) = 8.
  3. Then −3(−2) = +6 (negative times negative is positive).
  4. Add it up: 8 + 6 + 1 = 15.

Common mistakes to avoid

  • Sign errors when substituting a negative number. Remember (−2)² = 4, but −2² = −4.
  • Using x = −b/(2a) and stopping there. That gives the x-coordinate; you still have to plug it back in to get the minimum or maximum VALUE.
  • Forgetting that the zeros of a(x − r₁)(x − r₂) are r₁ and r₂ — with the opposite sign of what's inside the parentheses.
  • Adding exponents when you should multiply: (x³)² = x⁶, but x³ · x² = x⁵.

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Frequently asked questions

What topics are in SAT Advanced Math?

Advanced Math focuses on nonlinear relationships: quadratic equations and their graphs, functions in equation, table, and graph form, exponential growth and decay, polynomials, and rational and radical equations. You'll factor, use the quadratic formula, interpret function notation, and connect an equation to its graph. It's the step beyond linear algebra, testing whether you can handle curves and more complex expressions.

How do I solve quadratic equations on the SAT?

Know several tools and pick the fastest for each problem. Factoring is quickest when the numbers cooperate. The quadratic formula always works when factoring is hard. Completing the square helps when a question involves the vertex. You can also read solutions from a graph as x-intercepts. Recognizing which method a problem invites saves time, so practice all of them until you can switch easily.

How do I understand function notation like f(x)?

Think of f(x) as a machine: you put in an x value and get out a result. To find f(3), substitute 3 for every x and simplify. If a question gives f(x) equals a value and asks for x, set the expression equal to that value and solve. On a graph, f(x) is the y-value at a given x. Getting comfortable with substitution unlocks most function questions.

How do I graph or interpret a parabola?

A parabola is the U-shaped graph of a quadratic. Its vertex is the highest or lowest point, and it's symmetric around a vertical line through that vertex. The x-intercepts are the solutions to the equation set equal to zero. Different forms of a quadratic reveal different features: vertex form shows the vertex, and factored form shows the x-intercepts. Match the form to what the question asks.