AP Precalculus Ultimate Guide

Function

A mathematical relationship that maps a set of input values to output values, with each input resulting in exactly one output.

Input Values

Also known as domain; these are the independent variable values (x) that a function takes.

Output Values

Also known as range; these are the dependent variable values (y) derived from the input values of a function.

Increasing Function

A function is increasing on an interval if output values increase as input values increase; for all a and b in the interval, if a < b, then f(a) < f(b).

Decreasing Function

A function is decreasing on an interval if output values decrease as input values increase; for all a and b in the interval, if a < b, then f(a) > f(b).

Graph of a Function

A visual representation of input-output pairs that illustrates how the output values vary with the input values.

Concave Up

A portion of a graph where the rate of change is increasing.

Concave Down

A portion of a graph where the rate of change is decreasing.

X-Intercepts

Also known as zeros of the function; points where the function intersects the x-axis.

Average Rate of Change

The average change of a function over the interval [a, b], calculated as the slope of the secant line from (a, f(a)) to (b, f(b)).

Positive Rate of Change

When one quantity increases, and the other quantity also increases.

Negative Rate of Change

When one quantity increases, while the other quantity decreases.

Local Maxima/Minima

Points where a polynomial function changes between increasing and decreasing on a restricted domain.

Global Maxima/Minima

The highest point (global maximum) or lowest point (global minimum) across the entire domain of a function.

Endpoints for Inequality Intervals

Real zeros of a polynomial function that serve as boundaries when solving inequalities.

Multiplicity of a Zero

The number of times a zero (x-a) appears as a factor in a polynomial function; affects the graph's behavior at the x-intercept.

Even Polynomial

A polynomial with an even degree, which can have a global maximum or minimum.

Non-Real Zeros

Zerothat occur in complex conjugate pairs, meaning if p(a + bi) = 0, then p(a - bi) = 0.

End Behavior of a Polynomial

Determined by the degree and sign of the leading term, indicating in which direction the function approaches as input values increase or decrease.

Rational Function

A function expressed as the ratio of two polynomials where the polynomial in the denominator is not zero.

Vertical Asymptote

Zeros of the polynomial in the denominator indicating where a function approaches infinity.

Standard Form

A way of writing polynomial and rational functions to analyze their end behavior.

Factor Form

Representation of polynomial functions useful for finding x-intercepts, asymptotes, and domain/range.

Polynomial Long Division

A method used to find equations of slant asymptotes in rational functions.

Binomial Theorem

Used to expand expressions of the form (a + b)^n, with applications in polynomial functions.

Transformations of Functions

Changes made to a function that alter the graph vertically or horizontally with respect to the parent function.

Linear Functions

Functions used for scenarios with roughly constant rates of change.

Quadratic Functions

Used for data sets with roughly linear rates of change, often exhibiting a unique minimum or maximum value.

Piecewise Functions

Functions defined by different expressions on non-overlapping intervals of their domain.

Exponential Function

A mathematical function f(x) = b^x where the rate of growth is proportional to its current value.

Logarithmic Functions

The inverse of exponential functions; they're used to model situations involving proportional growth or repeated multiplication.

Identity Function

A function defined as f(x) = x, serving as the multiplicative and additive identity.

Inverse Functions

Functions that reverse the mapping of original functions, satisfying f(f^-1(x)) = x.

Unit Circle

A circle with radius 1, used in defining trigonometric functions and their values.

Coterminal Angles

Angles that share the same initial and terminal side but differ in degree or radian measure.

Period of a Function

The length of one complete cycle of a periodic function.

Amplitude of a Function

The distance from the midline to the maximum or minimum value of the function.

Pythagorean Identity

The fundamental relation among the sine and cosine functions, expressed as sin²θ + cos²θ = 1.

Transition Matrix

A matrix that represents the rates of transition between different states in a system.

Position Vectors

Describes a particle's position using a vector originating from the origin to the particle's location.

Matrix Determinant

A scalar value obtained from a square matrix that can indicate its invertibility and the area defined by its vectors.

Linear Transformation

A function that maps input vectors to output vectors while preserving the operations of vector addition and scalar multiplication.

Scalar

A single number used to multiply a vector, determining its magnitude and potentially its direction.

Matrix Multiplication

An operation where the elements of two matrices are combined according to specific rules to produce a third matrix.

Matrix Inverse

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

Implicitly Defined Functions

Functions defined by equations in which the dependent variable is not isolated on one side of the equation.

Conic Sections

The curves obtained by intersecting a plane with a cone, including parabolas, ellipses, and hyperbolas.

Unit Vector

A vector with a magnitude of 1, indicating direction only.

Magnitude of a Vector

The length or size of a vector, calculated using the Pythagorean theorem.

Velocity Vector

The derivative of the position vector, indicating the direction and speed of a moving object.

Radian

A unit of angle measure defined as the angle subtended by an arc that is equal in length to the radius of the circle.

Sine Function

A primary trigonometric function, representing the ratio of the opposite side to the hypotenuse in a right triangle.

Cosine Function

A primary trigonometric function, representing the ratio of the adjacent side to the hypotenuse in a right triangle.

Tangent Function

A primary trigonometric function, representing the ratio of the opposite side to the adjacent side in a right triangle.

Secant Function

The reciprocal of the cosine function, defined as 1/cos(θ).

Cosecant Function

The reciprocal of the sine function, defined as 1/sin(θ).

Cotangent Function

The reciprocal of the tangent function, defined as cos(θ)/sin(θ).

Graphing the Sine Function

Plotting the sine values, typically resulting in a smooth wave-like shape.

Graphing the Cosine Function

Plotting the cosine values, similar in shape to the sine function but phase-shifted.

Complete Cycle of a Periodic Function

A segment of graph that shows complete repetitions of its values.

Phase Shift

The horizontal translation of a periodic function, altering when a function starts its cycle.

State Change in Matrices

A representation of how a system evolves from one state to another using matrices.

Circular Motion

The motion of an object that moves along a circular path.

Roots of a Polynomial

The values of x for which the polynomial equals zero; also called solutions or zeros.