Average Rate of Change for Quadratic Functions
The average rate of change of a quadratic function f(x) = ax² over an interval [p, q] is computed as [f(q) - f(p)] / (q - p), a concept developed in enVision Algebra 1 Chapter 8 for Grade 11. For f(x) = x² over [1, 3], the average rate is (9 - 1)/(3 - 1) = 4. For f(x) = 2x² over [-1, 2], it is (8 - 2)/(2 - (-1)) = 2. Unlike linear functions, the average rate of change of a quadratic is not constant — it varies with the interval chosen, reflecting the curve's changing steepness. This is a key distinction between linear and quadratic behavior.
Key Concepts
The average rate of change of a quadratic function $f(x) = ax^2$ over an interval $[p, q]$ is:.
$$\text{Average rate of change} = \frac{f(q) f(p)}{q p}$$.
Common Questions
What is the formula for average rate of change?
Average rate of change = [f(q) - f(p)] / (q - p), where [p, q] is the interval. This is also called the slope of the secant line connecting the two points on the graph.
For f(x) = x² over [1, 3], what is the average rate of change?
f(1) = 1 and f(3) = 9, so (9 - 1)/(3 - 1) = 8/2 = 4.
For f(x) = -x² over [0, 4], what is the average rate of change?
f(0) = 0 and f(4) = -16, so (-16 - 0)/(4 - 0) = -4.
Why is the average rate of change of a quadratic not constant?
A quadratic is a curve, not a straight line. The steepness changes at every point, so the ratio (f(q)-f(p))/(q-p) depends on which interval you choose, unlike linear functions where slope is always the same.
How does the interval choice affect the average rate of change for f(x) = 2x²?
Over [-1, 2] the average rate is 2, but over [0, 2] it is (8-0)/(2-0) = 4. Different intervals give different average rates, showing the non-constant nature of quadratic change.