Step by step
The question
Question: find the average rate of change of f(x) = x² − 6x + 8 from x = 1 to x = 6.
1 Find the two points
We put 1 in place of x.
f(1) = 3.
We put 6 in place of x.
f(6) = 8.
So the two points are (1, 3) and (6, 8).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(6) minus f(1).
At the bottom: 6 − 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: 8 minus 3, over 6 − 1.
8 minus 3 is 5.
6 − 1 is 5.
5 divided by 5 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.
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: 5 across.
Rise: 5 up.
Start at f(1) = 3.
Add the rate, 1, 5 times.
We land on 8, which is f(6).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.