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SAT Math — Advanced Math Quick Reference
Core Concepts
Quadratics — Three Forms
| 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 |
Key quadratic tools:
| 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 |
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
| 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
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