Rates of Change in Linear and Quadratic Functions

The average rate of change for linear functions is constant, while for quadratic functions, it varies linearly over equal-length intervals.

Functions30–40% of exam

Understand It

Ace It

Context

What this topic is and why it exists

Rates of change in linear and quadratic functions are about how outputs shift as inputs vary.

For linear functions, the average rate of change is constant.

This means the slope of the secant line between any two points on the function is the same.

The cognitive trap is assuming this applies to all functions.

It doesn't.

Quadratic functions have a changing average rate of change.

Over equal-length input intervals, this rate can be described by a linear function.

The misconception: thinking quadratics behave like lines.

The constant rate for linear functions is zero.

For quadratics, the rate changes linearly.

Recognizing this difference is crucial when analyzing function behavior.

The slope of the secant line for a quadratic function over an interval [a, b] is not fixed.

It reflects the quadratic's curvature.

Misidentifying these characteristics leads to incorrect conclusions about function behavior.

Master the distinction between constant and changing rates of change to accurately interpret linear versus quadratic functions.

1 / 9