SAT Math — Advanced Math Quick Reference

Core Concepts

Quadratics — Three Forms

|---|---|---| Key quadratic tools: |---|---|---| Converting to vertex form: Complete the square: y = ax² + bx + c → y = a(x − h)² + k where h = −b/(2a) and k = f(h)

Exponential Functions

|---|---|

Form	Equation	What You Get Directly
Standard	ax² + bx + c	Y-intercept (c); direction (a sign)
Factored	a(x − r)(x − s)	Roots r and s; x-intercepts
Vertex	a(x − h)² + k	Vertex (h, k); axis of symmetry x = h
Tool	Formula	Use
Factoring	Find numbers: multiply to ac, add to b	When it factors cleanly
Quadratic formula	x = (−b ± √(b²−4ac)) / 2a	Always works
Vertex x-coordinate	x = −b/(2a)	From standard form
Discriminant	b² − 4ac	Positive → 2 solutions; zero → 1; negative → 0 real
Rule	Formula
General form	y = a · bˣ
a = initial value	y-value when x = 0
Growth (b > 1)	b = 1 + growth rate
Decay (0 < b < 1)	b = 1 − decay rate

Quick percent ↔ growth factor conversions:

• Grows 20% each year → b = 1.20

• Decays 15% each year → b = 0.85

• Doubles → b = 2

• Halves → b = 0.5

Linear vs. exponential test:

• Add same amount each step → linear

• Multiply by same factor each step → exponential

Polynomials and Function Notation

Operations:

• Add/subtract → combine like terms

• Multiply → distribute (FOIL for binomials)

FOIL: (a + b)(c + d) = ac + ad + bc + bd Function notation:

• f(3) → substitute x = 3 wherever you see x

• f(g(x)) → apply g first, then f to the result

Zeros/roots: Set factored polynomial = 0; solve each factor Degree = number of roots (counting multiplicity)

Common Exam Traps

• Vertex form sign: y = a(x − 2)² + 5 has vertex at (2, 5), NOT (−2, 5)

• Discriminant negative: No REAL solutions (there are complex solutions, but SAT cares about real)

• Factored form zeros: (x − r) → zero is r (positive); (x + s) → zero is −s (negative)

• Exponential growth factor: A 30% increase means b = 1.30, not 0.30

• f(g(x)) order: Apply g FIRST, then f — not the other way around

• Parallel lines in quadratics: Two quadratics with the same a and b but different c → same vertex x-coord, different vertex height

• Finding b from two exponential points: Use point (0, a) to find a, then substitute the second point to find b