SAT Math Mastery

Slope-intercept: y = mx + b

Point-slope: y−y₁ = m(x−x₁)

Parallel lines: equal slopes, different y-intercepts

No solution ⟺ same slope, different intercept

|x| < a ⟺ −a < x < a

|x| > a ⟺ x < −a or x > a

Example Solve: |2x − 3| ≤ 7
Step 1: −7 ≤ 2x − 3 ≤ 7 → Step 2: −4 ≤ 2x ≤ 10 → Step 3: −2 ≤ x ≤ 5

Quadratic: x = (−b ± √(b²−4ac)) / 2a

Discriminant: Δ = b²−4ac
Δ>0: 2 real; Δ=0: 1 real; Δ<0: no real

Composition: (f∘g)(x) = f(g(x))

Remainder Theorem: p(a) = remainder of p(x)÷(x−a)

Rational exponents: x^(m/n) = (ⁿ√x)^m

Complex: i² = −1, i³ = −i, i⁴ = 1

Vertex form: y = a(x−h)²+k
Vertex at (h, k)

LCD for rational expressions: factor each denominator

Example — Remainder Theorem Find remainder when p(x) = x³ − 4x + 6 is divided by (x − 2).
p(2) = 8 − 8 + 6 = 6

Distance: d = r × t

Work rate: combined rate = 1/t₁ + 1/t₂

Conditional probability: P(A|B) = P(A∩B)/P(B)

Geometric sequence: aₙ = a₁ · r^(n−1)

Mean = Σx / n; Median = middle value

Std. deviation measures spread, not center

Example — Geometric Sequence a₁ = 3, r = 2. Find a₅.
a₅ = 3 · 2^(5−1) = 3 · 16 = 48

Circle: (x−h)²+(y−k)² = r² center (h,k)

Complete the square: add (b/2)² to both sides

sin θ = opp/hyp cos θ = adj/hyp

tan θ = opp/adj = sin θ / cos θ

Exponential growth: A = P(1+r)^t

sin²θ + cos²θ = 1 (Pythagorean identity)

Example — Circle Equation x² + y² − 6x + 4y = 12
(x−3)² + (y+2)² = 12 + 9 + 4 = 25 → Center (3, −2), r = 5

Core Concepts & Key Formulas