All Flashcards
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.
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 -∞.