Step by step
The question
Question: find the average rate of change of f(x) = x² from x = −2 to x = 2.
1 Find the two points
We put −2 in place of x.
f(−2) = 4.
We put 2 in place of x.
f(2) = 4.
So the two points are (−2, 4) and (2, 4).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(2) minus f(−2).
At the bottom: 2 − (−2).
End minus start, on top and at the bottom.
Careful!
The change in y goes on top, not the change in x.
We put in the numbers: 4 minus 4, over 2 − (−2).
4 minus 4 is 0.
2 − (−2) is 4.
0 divided by 4 is 0.
So the average rate of change is 0.
On average, f(x) does not go up or down: it ends where it started.
3 Check
Let's check on a graph.
This is the graph of f(x).
Our two points.
The line through them.
Its slope is the average rate of change.
Run: 4 across.
Rise: 0 up.
Start at f(−2) = 4.
Add the rate, 0, 4 times.
We land on 4, which is f(2).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.