AP Precalculus flashcards

31 cards. Read the term and the definition here. The deck at the top of this page is optional.

Vertical line test

A relation is a function exactly when no vertical line intersects its graph more than once, meaning each input value has only one output value.

Average rate of change

For f on the interval from a to b, it equals (f(b) − f(a))/(b − a), the slope of the secant line joining the two graph points.

Even function

A function satisfying f(−x) = f(x) for every x in its domain; its graph is symmetric about the y-axis, as with f(x) = x².

Odd function

A function satisfying f(−x) = −f(x) for every x in its domain; its graph is symmetric about the origin, as with f(x) = x³.

Composite function domain

For (f ∘ g)(x) = f(g(x)), x must lie in g's domain and g(x) must lie in f's domain; both conditions must hold.

Inverse function reflection

If f is one-to-one with inverse f⁻¹, the graph of f⁻¹ is the reflection of f's graph across the line y = x.

One-to-one function

A function where distinct inputs always give distinct outputs; only such functions possess an inverse function on their full domain.

Horizontal shift direction

In g(x) = f(x − h), a positive h shifts the graph right by h units, while a negative h shifts it left.

Polynomial end behavior

Determined by the leading term aₙxⁿ: even n gives both tails the same direction, odd n gives opposite directions, and the sign of aₙ sets the right tail.

Multiplicity of a zero

If (x − c)^m is a factor, the zero c has multiplicity m; odd m crosses the axis, even m touches it and turns around.

Remainder theorem

When a polynomial p(x) is divided by x − c, the remainder equals p(c); a zero remainder means x − c is a factor.

Rational zeros theorem

For a polynomial with integer coefficients, any rational zero p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

Vertical asymptote

A line x = c where a rational function's denominator is zero without canceling; function values grow without bound as x approaches c.

Horizontal asymptote rule

For N(x)/D(x): smaller numerator degree gives y = 0, equal degrees give the ratio of leading coefficients, larger numerator degree gives no horizontal asymptote.

Removable discontinuity (hole)

A point where a common factor cancels from numerator and denominator; the function is undefined there but the limit exists.

Exponential growth factor

In f(t) = a·bᵗ, the base b > 1 produces growth with multiplier b per unit time, while 0 < b < 1 produces decay.

Continuous exponential model

f(t) = a·e^{kt} models growth when k > 0 and decay when k < 0, with a equal to the value at t = 0.

Logarithm definition

log_b(x) = y means bʸ = x, valid for base b > 0 with b ≠ 1 and for argument x > 0.

Product rule for logarithms

For positive m and n, log_b(mn) = log_b m + log_b n, turning a product inside one logarithm into a sum of logarithms.

Power rule for logarithms

log_b(mᵖ) = p·log_b m, allowing an exponent inside a logarithm to be moved to the front as a multiplier.

Change of base formula

log_b(x) = ln(x)/ln(b); any base may be used, letting you evaluate logs using only natural or common log keys.

Doubling time

For growth modeled by a·e^{kt} with k > 0, the time required for the quantity to double equals ln(2)/k.

Half-life

For decay modeled by a·e^{kt} with k < 0, the time for the quantity to fall to half its value equals ln(2)/|k|.

Unit circle coordinates

A point on the unit circle at angle θ has coordinates (cos θ, sin θ), which yields the identity sin²θ + cos²θ = 1.

Amplitude and period

For y = a·sin(bθ), the amplitude is |a| and the period is 2π/|b|; cosine follows the same rules.

Phase shift

In y = a·sin(bθ + c) + d, the graph shifts horizontally by −c/b and vertically by d, with midline y = d.

Tangent period

The tangent function repeats every π radians rather than 2π, and it is undefined wherever cos θ equals zero.

Reciprocal identities

csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ; each is undefined wherever its base function equals zero.

Inverse sine range

arcsin x is defined on [−1, 1] with range [−π/2, π/2], chosen so the restricted sine is one-to-one and invertible.

Polar to rectangular

A polar point (r, θ) converts to rectangular coordinates by x = r·cos θ and y = r·sin θ.

Rectangular to polar

Given (x, y), the polar radius satisfies r² = x² + y² and the angle satisfies tan θ = y/x, adjusted for the quadrant.