Step by step
The question
Question: find the average rate of change of f(x) = x² − 3x from x = 1 to x = 4.
1 Find the two points
We put 1 in place of x.
f(1) = −2.
We put 4 in place of x.
f(4) = 4.
So the two points are (1, −2) and (4, 4).
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: 4 minus negative 2, over 4 − 1.
4 minus negative 2 is 6.
4 − 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) = −2.
Add the rate, 2, 3 times.
We land on 4, which is f(4).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.