Step by step
The question
Question: find the average rate of change of f(x) = x³ − x from x = −1 to x = 2.
1 Find the two points
We put −1 in place of x.
f(−1) = 0.
We put 2 in place of x.
f(2) = 6.
So the two points are (−1, 0) and (2, 6).
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(−1).
At the bottom: 2 − (−1).
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: 6 minus 0, over 2 − (−1).
6 minus 0 is 6.
2 − (−1) is 3.
6 divided by 3 is 2.
So the average rate of change is 2.
On average, f(x) goes up 2 for each 1 that x goes up.
3 Check
Let's check on a graph.
Each square up is 2.
This is the graph of f(x).
Our two points.
The line through them.
Its slope is the average rate of change.
Run: 3 across.
Rise: 6 up.
Start at f(−1) = 0.
Add the rate, 2, 3 times.
We land on 6, which is f(2).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.