AP Precalculus Ultimate Guide (copy)

Function

A mathematical relationship that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.

Domain

The set of input values for a function, also referred to as independent variable (x).

Range

The set of output values for a function, also referred to as dependent variable (y).

Increasing Function

A function is increasing over an interval if, for all a and b in the interval, if a < b, then f(a) < f(b).

Decreasing Function

A function is decreasing over an interval if, for all a and b in the interval, if a < b, then f(a) > f(b).

Graph

A visual display of input-output pairs that shows how values vary.

Concave Up

A curve is concave up when the rate of change is increasing.

Concave Down

A curve is concave down when the rate of change is decreasing.

Average Rate of Change

The average rate of change over the closed interval [a, b] is the slope of the secant line from the point (a, f(a)) to (b, f(b)).

Positive Rate of Change

Occurs when one quantity increases and another quantity increases as well.

Negative Rate of Change

Occurs when one quantity increases while another quantity decreases.

Local Maximum

The highest point in a local vicinity of a function's graph where the function changes from increasing to decreasing.

Global Maximum

The overall highest point on the graph of a function across its entire domain.

Polynomial Function

A function represented by a polynomial expression, defined generally as p(x) = anx^n + a{n-1}x^{n-1} + … + a1x + a0.

Complex Numbers

Numbers that include real numbers and non-real numbers, expressed in the form a + bi.

Real Zeros

Values of x for which the polynomial function p(x) = 0.

Vertical Asymptote

A vertical line x = a where the function approaches infinity.

Hole

Occurs when a zero appears more times in the numerator than in the denominator of a rational function.

Standard Form of Functions

The representation of polynomial and rational functions used to analyze their properties and behaviors.

Exponential Function

A function of the form f(x) = ab^x, where a is a non-zero constant and b > 0.

Logarithmic Function

A function defined as the inverse of an exponential function, expressed as log_b(c), representing the exponent required to obtain c from b.

Unit Circle

A circle with a radius of one centered at the origin of the coordinate plane, useful for defining trigonometric functions.

Sine Function

A periodic function that assigns a value to an angle that corresponds to the y-coordinate of a point on the unit circle.

Cosine Function

A periodic function that assigns a value to an angle that corresponds to the x-coordinate of a point on the unit circle.

Tangent Function

A periodic function defined as the ratio of the sine and cosine functions, tan(θ) = sin(θ)/cos(θ).

Parametric Function

A function defined by one or more parameters, expressed typically as (x(t), y(t)).

Matrix

An ordered array of numbers arranged in rows and columns, used for organizing data and performing linear transformations.

Determinant

A scalar value calculated from a square matrix that helps determine the matrix's invertibility.

Inverse Matrix

A matrix that, when multiplied with the original matrix, results in the identity matrix.

Linear Transformation

A mapping of vector spaces that preserves the operations of vector addition and scalar multiplication.

Vector

A mathematical entity that has both magnitude and direction, often represented as an arrow.