What you'll learn
The Advanced Math domain (~35% of SAT Math) tests nonlinear relationships: quadratic equations, polynomials, exponential functions, and function notation. It rewards fluency with algebraic manipulation.
Quadratics
A quadratic is ax² + bx + c = 0. Three ways to solve:
- Factoring:
x² − 5x + 6 = 0→(x − 2)(x − 3) = 0→ x = 2 or 3. - Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a.
- Completing the square (useful for vertex form).
The discriminant b² − 4ac tells you the number of real solutions: positive = two, zero = one (repeated), negative = none.
Vertex form
y = a(x − h)² + k has its vertex at (h, k). Useful for finding maximum/minimum points and the axis of symmetry (x = h).
Exponents and exponentials
Key rules: xᵃ · xᵇ = xᵃ⁺ᵇ, xᵃ / xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁰ = 1.
Exponential growth/decay: y = a·bᵗ. If a quantity doubles each period, b = 2; halves, b = ½. Distinguish from linear growth (adds a fixed amount each period).
Polynomials
- Multiply with FOIL:
(x + 3)(x − 2) = x² + x − 6. - Difference of squares:
x² − 9 = (x − 3)(x + 3). - The sum of the roots of
x² + bx + cis −b; the product is c.
Function notation
f(x) means "output for input x". To evaluate f(3), substitute 3 for x. Composite functions: f(g(x)) means apply g first, then f.
Exam strategy
- Recognise structure: a question hiding
(x + 3)²may simplify fast. - Use the calculator's graphing/solver for messy quadratics where allowed.
- For "how many solutions", reach for the discriminant.
Common mistakes
- Dropping the ± when taking a square root.
- Mis-applying exponent rules (adding bases instead of exponents).
- Confusing linear and exponential models.
Drill factoring and the quadratic formula until they're automatic — they unlock most of this domain.