Step by step
The question
Question: find the average rate of change of f(x) = −x³ + 4x from x = 0 to x = 3.
1 Find the two points
We put 0 in place of x.
f(0) = 0.
We put 3 in place of x.
f(3) = −15.
So the two points are (0, 0) and (3, −15).
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(0).
At the bottom: 3 − 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: −15 minus 0, over 3 − 0.
−15 minus 0 is −15.
3 − 0 is 3.
−15 divided by 3 is −5.
So the average rate of change is −5.
On average, f(x) goes down 5 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: 3 across.
Rise: −15, so down.
Start at f(0) = 0.
Add the rate, −5, 3 times.
We land on −15, which is f(3).
So the answer is right!
Let's look back.
First: find the two points.
Then: change over change.
Last: check.