Step by step
The question
Question: find the average rate of change of f(x) = 3x² − 5 from x = 1 to x = 3.
1 Find the two points
We put 1 in place of x.
f(1) = −2.
We put 3 in place of x.
f(3) = 22.
So the two points are (1, −2) and (3, 22).
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(1).
At the bottom: 3 − 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: 22 minus negative 2, over 3 − 1.
22 minus negative 2 is 24.
3 − 1 is 2.
24 divided by 2 is 12.
So the average rate of change is 12.
On average, f(x) goes up 12 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: 2 across.
Rise: 24 up.
Start at f(1) = −2.
Add the rate, 12, 2 times.
We land on 22, which is f(3).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.