AP Precalculus Notes: Unit 1 Topic 1.2 Rates of Change

Vocabulary flashcards covering average rate of change, rate of change, secant line, slope, and related concepts from the notes.

Average Rate of Change

The constant rate of change that yields the same change in output over an interval; equals (f(b) − f(a)) / (b − a).

Rate of Change

The rate at which the output changes with respect to the input; at a point it can be approximated by average rates over small intervals.

Secant Line

The straight line through (a, f(a)) and (b, f(b)) on a function's graph; its slope equals the average rate of change.

Slope

The rate of change of a line; for y = mx + b, the slope is m, the constant rate of change.

Constant Function

A function f(x) = c where the output does not depend on x; its rate of change is zero.

Linear Function

A function f(x) = mx + b with constant rate of change equal to the slope m.

Quadratic Function

A function f(x) = ax^2 + bx + c whose rate of change is not constant and varies with x.

Delta (Δ)

The Greek letter delta; denotes change, as in Δy = change in y and Δx = change in x.

Δy and Δx

Change in the output (Δy) and change in the input (Δx) used in rate-of-change calculations.

Average Rate of Change Formula

Average rate of change from a to b is (f(b) − f(a)) / (b − a), the slope of the secant line.

Positive/Negative Rate of Change

A positive rate means the output increases with input; a negative rate means the output decreases as the input increases.

Slope of a Secant Line (Theorem)

The average rate of change of f from a to b equals the slope of the secant line through (a, f(a)) and (b, f(b)).

Secant Line Equation (Point-Slope)

Equation of the secant line through (a, f(a)) with slope msec: y − f(a) = msec (x − a).

Average Rate of Change on an Interval

The slope of the line joining the endpoints of the graph over [a, b], i.e., (f(b) − f(a)) / (b − a).