Step by step
The question
Question: find the average rate of change of f(x) = 3x + 2 from x = 1 to x = 5.
1 Find the two points
We put 1 in place of x.
f(1) = 5.
We put 5 in place of x.
f(5) = 17.
So the two points are (1, 5) and (5, 17).
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(5) minus f(1).
At the bottom: 5 − 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: 17 minus 5, over 5 − 1.
17 minus 5 is 12.
5 − 1 is 4.
12 divided by 4 is 3.
So the average rate of change is 3.
On average, f(x) goes up 3 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: 4 across.
Rise: 12 up.
Start at f(1) = 5.
Add the rate, 3, 4 times.
We land on 17, which is f(5).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.