f(x) = a ⋅ bˣ, where a = initial value, b = base (growth factor)

When b > 1 in f(x) = a ⋅ bˣ, the function increases rapidly.

When 0 < b < 1 in f(x) = a ⋅ bˣ, the function approaches zero asymptotically.

A(t) = P ⋅ eʳᵗ, where P = principal, r = interest rate, t = time.

f(x) = ax² + bx + c, where a ≠ 0

The vertical line x = -b / 2a that passes through the vertex.

x = (-b ± √(b² - 4ac)) / 2a

Δ = b² - 4ac. It determines the number and type of solutions.

f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ

The number of times a root appears as a solution of a polynomial equation.

f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0

Vertical lines where the denominator of a rational function equals zero.

Horizontal lines that the graph of a rational function approaches as x tends to +∞ or -∞.

Use A(t) = A₀ ⋅ (1 + r)ᵗ or A(t) = A₀ ⋅ eʳᵗ, where A₀ is the initial amount, r is the growth rate, and t is time.

The x-coordinate is h = -b / 2a, and the y-coordinate is k = f(h).

Δ > 0: Two distinct real solutions; Δ = 0: One repeated real solution; Δ < 0: No real solutions.

The graph rises on both ends (as x approaches positive or negative infinity, f(x) approaches positive infinity).

The graph rises to the left and falls to the right.

Odd multiplicity: graph crosses the x-axis. Even multiplicity: graph touches the x-axis.

Set the denominator equal to zero and solve for x. These x-values are the vertical asymptotes.

The horizontal asymptote is y = 0.

The horizontal asymptote is y = ratio of leading coefficients.

Multiply by the least common denominator (LCD), solve the resulting polynomial equation, and check for extraneous solutions.