Step by step
The question
Question: find the average rate of change of f(x) = x² + 2x from x = −3 to x = 1.
1 Find the two points
We put −3 in place of x.
f(−3) = 3.
We put 1 in place of x.
f(1) = 3.
So the two points are (−3, 3) and (1, 3).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(1) minus f(−3).
At the bottom: 1 − (−3).
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: 3 minus 3, over 1 − (−3).
3 minus 3 is 0.
1 − (−3) 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(−3) = 3.
Add the rate, 0, 4 times.
We land on 3, which is f(1).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.