Step by step
The question
Question: find the average rate of change of f(x) = x² − 4 from x = 2 to x = 5.
1 Find the two points
We put 2 in place of x.
f(2) = 0.
We put 5 in place of x.
f(5) = 21.
So the two points are (2, 0) and (5, 21).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(5) minus f(2).
At the bottom: 5 − 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: 21 minus 0, over 5 − 2.
21 minus 0 is 21.
5 − 2 is 3.
21 divided by 3 is 7.
So the average rate of change is 7.
On average, f(x) goes up 7 for each 1 that x goes up.
3 Check
Let's check on a graph.
Each square up is 5.
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: 21 up.
Start at f(2) = 0.
Add the rate, 7, 3 times.
We land on 21, which is f(5).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.