1.3 Rates of Change in Linear and Quadratic Functions

Average Rate of Change Recap

The average rate of change of a function over an interval [a,b] is (f(b)-f(a))/(b-a) — the slope of the secant line. For linear functions, this rate is constant everywhere. For quadratic functions, the average rate of change varies depending on the interval chosen.

Rate of Change of Quadratics

Unlike lines, quadratics have a rate of change that itself changes linearly. If you compute average rates of change over consecutive equal-width intervals, those rates form an arithmetic sequence — they increase or decrease by a constant amount, revealing the parabola's increasing/decreasing behavior and concavity.

Second Differences Confirm Quadratics

For evenly spaced x-values, linear functions have constant first differences. Quadratic functions have constant second differences (the differences of the differences). This pattern helps identify whether data represents a linear or quadratic model without graphing.

Key terms

Average rate of change formula

(f(b) - f(a)) / (b - a), the slope of the secant line between two points on the graph.

Rate of change for linear functions

Constant for every interval; equals the slope of the line.

Rate of change for quadratic functions

Not constant; it changes linearly as x increases, forming an arithmetic sequence over equal-width intervals.

First differences

Differences between consecutive y-values for equally spaced x-values; constant for linear functions.

Second differences

Differences of the first differences; constant for quadratic functions with equally spaced x-values.

Try an AP-style question

Which kind of function best fits this data?

1. Linear, since the first differences are constant
2. Cubic, since the third differences are constant
3. Exponential, since the ratios are constant
4. Quadratic, since the second differences are constant