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