All Flashcards
General form of an exponential function?
f(x) = a ⋅ bˣ, where a = initial value, b = base (growth factor)
What is exponential growth?
When b > 1 in f(x) = a ⋅ bˣ, the function increases rapidly.
What is exponential decay?
When 0 < b < 1 in f(x) = a ⋅ bˣ, the function approaches zero asymptotically.
Formula for continuously compounded interest?
A(t) = P ⋅ eʳᵗ, where P = principal, r = interest rate, t = time.
General form of a quadratic function?
f(x) = ax² + bx + c, where a ≠ 0
What is the axis of symmetry for a parabola?
The vertical line x = -b / 2a that passes through the vertex.
Quadratic Formula?
x = (-b ± √(b² - 4ac)) / 2a
What is the discriminant?
Δ = b² - 4ac. It determines the number and type of solutions.
General form of a polynomial function?
f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
What is multiplicity of a zero?
The number of times a root appears as a solution of a polynomial equation.
What is a rational function?
f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0
What are vertical asymptotes?
Vertical lines where the denominator of a rational function equals zero.
What are horizontal asymptotes?
Horizontal lines that the graph of a rational function approaches as x tends to +∞ or -∞.
How to model population growth?
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.
How to find the vertex of a parabola?
The x-coordinate is h = -b / 2a, and the y-coordinate is k = f(h).
How does the discriminant determine the number of real solutions?
Δ > 0: Two distinct real solutions; Δ = 0: One repeated real solution; Δ < 0: No real solutions.
How to determine end behavior of a polynomial with even degree and positive leading coefficient?
The graph rises on both ends (as x approaches positive or negative infinity, f(x) approaches positive infinity).
How to determine end behavior of a polynomial with odd degree and negative leading coefficient?
The graph rises to the left and falls to the right.
How does multiplicity affect the graph at x-intercepts?
Odd multiplicity: graph crosses the x-axis. Even multiplicity: graph touches the x-axis.
How to find vertical asymptotes of a rational function?
Set the denominator equal to zero and solve for x. These x-values are the vertical asymptotes.
How to find horizontal asymptotes when the degree of the numerator is less than the degree of the denominator?
The horizontal asymptote is y = 0.
How to find horizontal asymptotes when the degree of the numerator is equal to the degree of the denominator?
The horizontal asymptote is y = ratio of leading coefficients.
How to solve rational equations?
Multiply by the least common denominator (LCD), solve the resulting polynomial equation, and check for extraneous solutions.