AP Precalculus: Change in Tandem and Rates of Change

Flashcards covering key definitions and concepts from AP Precalculus Topics 1.1 through 1.3, including function definitions, domain/range, concavity, rates of change, and secant lines.

Function

A mathematical relation 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 of a function.

Range

The set of output values of a function.

Independent Variable

The variable representing input values of a function.

Dependent Variable

The variable representing output values of a function.

Increasing Function

A function where output values always increase as input values increase, satisfying if $a < b$, then $f(a) < f(b)$.

Decreasing Function

A function where output values always decrease as input values increase, satisfying if $a < b$, then $f(a) > f(b)$.

Concave Up

A condition describing a function or graph whose rate of change is increasing.

Concave Down

A condition describing a function or graph whose rate of change is decreasing.

Positive Function

A function $f$ whose outputs ($y$-values) are greater than zero, meaning its graph lies above the $x$-axis.

Negative Function

A function $f$ whose outputs ($y$-values) are less than zero, meaning its graph lies below the $x$-axis.

Rate of Change

A term synonymous with slope, where a positive value indicates an increasing function and a negative value indicates a decreasing function.

Rate of Change at a Point

The rate at which the output values would change if the input values were to change at a specific point.

Average Rate of Change

The constant rate of change yielding the same change in output values as the function on an interval, given by the ratio $\frac{f(b) - f(a)}{b - a}$.

Secant Line

A line passing through points $(a, f(a))$ and $(b, f(b))$ on a function's graph, whose slope equals the average rate of change over the closed interval $[a, b]$.