Step by step
The question
Question: find the average rate of change of f(x) = −2x + 7 from x = 0 to x = 4.
1 Find the two points
We put 0 in place of x.
f(0) = 7.
We put 4 in place of x.
f(4) = −1.
So the two points are (0, 7) and (4, −1).
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(0).
At the bottom: 4 − 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: −1 minus 7, over 4 − 0.
−1 minus 7 is −8.
4 − 0 is 4.
−8 divided by 4 is −2.
So the average rate of change is −2.
On average, f(x) goes down 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: 4 across.
Rise: −8, so down.
Start at f(0) = 7.
Add the rate, −2, 4 times.
We land on −1, which is f(4).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.