Step by step
The question
Question: find the average rate of change of f(x) = −x² + 6 from x = −1 to x = 3.
1 Find the two points
We put −1 in place of x.
f(−1) = 5.
We put 3 in place of x.
f(3) = −3.
So the two points are (−1, 5) and (3, −3).
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: −3 minus 5, over 3 − (−1).
−3 minus 5 is −8.
3 − (−1) 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(−1) = 5.
Add the rate, −2, 4 times.
We land on −3, which is f(3).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.