Factoring Patterns
Difference of Squares
a² − b² = (a − b)(a + b)
Perfect Square Trinomial
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
a² − 2ab + b² = (a − b)²
Sum of Cubes
a³ + b³ = (a + b)(a² − ab + b²)
Difference of Cubes
a³ − b³ = (a − b)(a² + ab + b²)
GCF Factoring
ab + ac = a(b + c)
Always try this first, before any other pattern.
Factoring by Grouping
ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
Function Transformations
| Form | Effect on the graph of f(x) |
|---|---|
| f(x) + k | Shift up k units |
| f(x) − k | Shift down k units |
| f(x + h) | Shift left h units |
| f(x − h) | Shift right h units |
| −f(x) | Reflect over the x-axis |
| f(−x) | Reflect over the y-axis |
| a · f(x), |a| > 1 | Vertical stretch by a factor of a |
| a · f(x), 0 < |a| < 1 | Vertical compression by a factor of a |
| f(bx), |b| > 1 | Horizontal compression by a factor of 1/b |
| f(bx), 0 < |b| < 1 | Horizontal stretch by a factor of 1/b |
Order of operations matters when stacking transformations: apply horizontal shifts/stretches
(inside the parentheses) before vertical ones (outside), reading the function definition from
the inside out.
Quadratics
Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
For ax² + bx + c = 0
Discriminant
Δ = b² − 4ac
Δ > 0: two real roots · Δ = 0: one real root · Δ < 0: two complex roots
Vertex Form
y = a(x − h)² + k
Vertex is at (h, k)
Vertex from Standard Form
h = −b / 2a, k = f(h)
For y = ax² + bx + c
Imaginary Unit
i = √−1, i² = −1
Radicals & Rational Exponents
Rational Exponent
am/n = (n√a)m = n√(am)
Product Rule for Radicals
n√a · n√b = n√(ab)
Quotient Rule for Radicals
n√a / n√b = n√(a/b)
Rationalizing the Denominator
1/√a × √a/√a = √a / a
Logarithms & Exponentials
Log – Exponential Relationship
logb(x) = y ⇔ by = x
Product Rule
logb(xy) = logb(x) + logb(y)
Quotient Rule
logb(x/y) = logb(x) − logb(y)
Power Rule
logb(xn) = n · logb(x)
Change of Base
logb(x) = ln(x) / ln(b)
Applications
Distance Formula
d = √((x₂−x₁)² + (y₂−y₁)²)
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Compound Interest
A = P(1 + r/n)nt
P = principal, r = annual rate, n = times compounded/year, t = years
Continuous Compounding
A = Pert
Direct Variation
y = kx
Inverse Variation
y = k/x
Joint Variation
y = kxz