AP Precalculus Unit 1: Rates of Change and Polynomial Functions Vocabulary

Vocabulary flashcards defining fundamental terms, properties, formulas, and concepts related to function behavior, rates of change, concavity, and polynomial functions.

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 or independent variables (typically $x$) for a function.

Range

The set of output values or dependent variables (typically $y$) for a function.

Function Notation

A mathematical notation such as $h(t)$ that immediately identifies which variable is the input (such as $t$) and which is the output.

Increasing Function

A function where output values increase as input values increase; analytically, for all $a$ and $b$ in an interval, if $a < b$, then $f(a) < f(b)$.

Decreasing Function

A function where output values decrease as input values increase; analytically, for all $a$ and $b$ in an interval, if $a < b$, then $f(a) > f(b)$.

Zero of a Function

An input value where the graph of a function intersects the x-axis, corresponding to an output value of zero.

y-intercept

The output value where the graph of a function intersects the y-axis, corresponding to an input value of zero.

Concave Up

A graphical shape facing upward like a cup, where the slope or rate of change of the function is increasing.

Concave Down

A graphical shape facing downward like a frown, where the slope or rate of change of the function is decreasing.

Point of Inflection

A point on a graph where the rate of change changes from increasing to decreasing (or vice versa), marked by a change in concavity from concave up to concave down (or vice versa).

Average Rate of Change

The ratio of the change in output values to the change in input values over an interval, given by $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$.

Secant Line

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

Rate of Change at a Point

The rate at which output values would change relative to input values at a single point, approximated using average rates of change over small intervals containing that point.

Positive Rate of Change

A relationship where two quantities change in the same direction (as one increases or decreases, the other does the same).

Negative Rate of Change

A relationship where two quantities change in opposite directions (as one quantity increases, the other decreases).

Linear Function Average Rate of Change

A constant average rate of change over any input-value interval length, resulting in a rate of change of average rates of change equal to $0$.

Quadratic Function Average Rate of Change

An average rate of change that varies across intervals, but whose rate of change of average rates of change is constant over consecutive equal-length input intervals.

Polynomial Function

A nonconstant function with multiple terms in the form $p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$, where $n$ is a positive integer and each $a_i$ is a real number.

Leading Term

The term in a polynomial function that contains the variable with the largest exponent.

Degree of a Polynomial

The largest exponent $n$ among all variable terms in a polynomial function.

Leading Coefficient

The coefficient $a_n$ of the leading term in a polynomial function.

Local (Relative) Maximum

An output value where a polynomial switches from increasing to decreasing, or at an included endpoint of a restricted domain.

Local (Relative) Minimum

An output value where a polynomial switches from decreasing to increasing, or at an included endpoint of a restricted domain.

Global (Absolute) Maximum

The greatest output value among all local maxima of a function.

Global (Absolute) Minimum

The least output value among all local minima of a function.