Step by step
The question
Question: find the average rate of change of f(x) = x² from x = 1 to x = 4.
1 Find the two points
We put 1 in place of x.
f(1) = 1.
We put 4 in place of x.
f(4) = 16.
So the two points are (1, 1) and (4, 16).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(4) minus f(1).
At the bottom: 4 − 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: 16 minus 1, over 4 − 1.
16 minus 1 is 15.
4 − 1 is 3.
15 divided by 3 is 5.
So the average rate of change is 5.
On average, f(x) goes up 5 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: 15 up.
Start at f(1) = 1.
Add the rate, 5, 3 times.
We land on 16, which is f(4).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.