AP Precalculus key terms

A rule that assigns to each input value in its domain exactly one output value. The set of all resulting outputs is the range.

The set of all input values for which a function is defined. For a real-valued function, these are the x-values that produce a real output.

The set of all output values a function actually produces as its input varies over the domain.

A graph represents a function exactly when no vertical line meets the graph more than once. This confirms each input has only one output.

A function is one-to-one exactly when no horizontal line meets its graph more than once. One-to-one functions have inverse functions that are also functions.

The operation forming f(g(x)), in which the output of the inner function g becomes the input of the outer function f. Its domain requires g(x) to lie in f's domain.

A function that reverses another function's action, so f(f⁻¹(x)) = x and f⁻¹(f(x)) = x wherever both are defined. Its graph is the original reflected across y = x.

A function satisfying f(−x) = f(x) for every x in its domain. Its graph is symmetric about the y-axis.

A function satisfying f(−x) = −f(x) for every x in its domain. Its graph has 180-degree rotational symmetry about the origin.

A shift, stretch, compression, or reflection applied to a parent graph. For example, af(b(x − h)) + k shifts, scales, and reflects the graph of f.

For a function on an interval [a, b], the quotient (f(b) − f(a))/(b − a). It is the slope of the secant line through the two corresponding graph points.

A description of how a graph bends. A graph is concave up where its slopes increase and concave down where its slopes decrease across an interval.

The behavior of a function's output values as the input grows without bound in the positive or negative direction, often described with limit notation.

A function defined by a finite sum of terms of the form axⁿ where n is a nonnegative integer and a is a real coefficient. Its domain is all real numbers.

The highest power of the variable appearing with a nonzero coefficient in a polynomial. It bounds the number of real zeros and turning points.

The term of a polynomial with the highest power of the variable. It alone determines the polynomial's end behavior.

An input value x for which f(x) = 0. Zeros correspond to x-intercepts of the graph and to factors of a polynomial.

The number of times a factor (x − c) appears in a polynomial's factorization. Even multiplicity makes the graph touch the axis; odd multiplicity makes it cross.

A function that is a ratio of two polynomials, written p(x)/q(x) with q not the zero polynomial. It is undefined wherever the denominator equals zero.

A vertical line x = a that a graph approaches as x approaches a, with function values growing without bound. For rational functions it typically arises where the denominator is zero but not canceled.

A horizontal line y = L that a graph approaches as x grows without bound in either direction. It describes end behavior rather than a value the function must reach.

A single missing point in a graph, also called a hole, where a common factor cancels in a rational expression. The limit exists there even though the function is undefined.

A polynomial p has factor (x − c) exactly when p(c) = 0. It links zeros, factors, and the remainder when dividing by x − c.

A function of the form f(x) = a·bˣ with b positive and not equal to 1. It grows or decays by a constant multiplicative factor over equal input intervals.

The constant multiplier b in an exponential function a·bˣ. Values above 1 give growth, values between 0 and 1 give decay, and the percent change is b − 1.

The time required for an exponentially decaying quantity to fall to half its initial amount. It corresponds to a decay factor of one-half over that interval.

The exponent to which a base must be raised to produce a given number: log_b(x) = y means bʸ = x. It is the inverse relationship to exponentiation.

The logarithm with base e, written ln(x), where e is approximately 2.718. It is the inverse function of the natural exponential eˣ.

The identity log_b(x) = log_c(x)/log_c(b), which rewrites a logarithm in a new base. It allows evaluation using common or natural logarithms.

A bounded growth model of the form L/(1 + a·e^(−kx)) that begins near zero, grows quickly, then levels off at a carrying capacity L.

A unit of angle measure defined so that a full rotation is 2π radians. One radian equals 180/π degrees, roughly 57.3 degrees.

The circle of radius 1 centered at the origin. A point on it at angle θ has coordinates (cos θ, sin θ), which defines the trigonometric functions for all angles.

The smallest positive input interval after which a periodic function repeats its values. Sine and cosine have period 2π; tangent has period π.

Half the vertical distance between a sinusoidal graph's maximum and minimum values. For y = a·sin(bx) + d it equals |a|.

The horizontal line halfway between a sinusoidal graph's maximum and minimum values, given by y = d in y = a·sin(b(x − h)) + d.

The function assigning to each angle the y-coordinate of the corresponding point on the unit circle. Its graph is a wave with period 2π and range from −1 to 1.

A way to locate a point using a distance r from the origin and an angle θ from the positive x-axis, written (r, θ). The same point has infinitely many polar representations.

An equation relating r and θ that describes a curve in the polar plane, such as r = 2cos θ. Converting to rectangular form often reveals the curve's shape.