Worked examples
The question
Question: find the average rate of change of f(x) = x² from x = 1 to x = 4.
f(x) = x2
1 Find the two points
We put 1 in place of x.
f(1) = 1.
f(1) = 12 = 1
We put 4 in place of x.
f(4) = 16.
f(4) = 42 = 16
So the two points are (1, 1) and (4, 16).
(1, 1) and (4, 16)
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(1).
At the bottom: 4 − 1.
End minus start, on top and at the bottom.
rate = f(4) − f(1)4 − 1
Careful!
The change in y goes on top, not the change in x.
Careful: not 4 − 116 − 1 ✗
We put in the numbers: 16 minus 1, over 4 − 1.
= 16 − 14 − 1
16 minus 1 is 15.
4 − 1 is 3.
= 153
15 divided by 3 is 5.
= 5
So the average rate of change is 5.
On average, f(x) goes up 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 up.
Start at f(1) = 1.
Add the rate, 5, 3 times.
We land on 16, which is f(4).
So the answer is right!
1 + 3 × 5 = 16 = f(4)
Let's look back.
First: find the two points.
Then: change over change.
Last: check.
▶ Watch it solved
The question
Question: find the average rate of change of f(x) = −x² + 6 from x = −1 to x = 3.
f(x) = −x2 + 6
1 Find the two points
We put −1 in place of x.
f(−1) = 5.
f(−1) = −(−1)2 + 6 = −1 + 6 = 5
We put 3 in place of x.
f(3) = −3.
f(3) = −32 + 6 = −9 + 6 = −3
So the two points are (−1, 5) and (3, −3).
(−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.
rate = f(3) − f(−1)3 − (−1)
Careful!
The change in y goes on top, not the change in x.
Careful: not 3 − (−1)−3 − 5 ✗
We put in the numbers: −3 minus 5, over 3 − (−1).
= −3 − 53 − (−1)
−3 minus 5 is −8.
3 − (−1) is 4.
= −84
−8 divided by 4 is −2.
= −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!
5 + 4 × (−2) = −3 = f(3)
Let's look back.
First: find the two points.
Then: change over change.
Last: check.
▶ Watch it solved
The question
Question: find the average rate of change of f(x) = x³ − 2x from x = 0 to x = 2.
f(x) = x3 − 2x
1 Find the two points
We put 0 in place of x.
f(0) = 0.
f(0) = 03 − 2(0) = 0 + 0 = 0
We put 2 in place of x.
f(2) = 4.
f(2) = 23 − 2(2) = 8 − 4 = 4
So the two points are (0, 0) and (2, 4).
(0, 0) and (2, 4)
2 Change over change
Average rate of change means: how much y changes, divided by how much x changes.
On top: f(2) minus f(0).
At the bottom: 2 − 0.
End minus start, on top and at the bottom.
rate = f(2) − f(0)2 − 0
Careful!
The change in y goes on top, not the change in x.
Careful: not 2 − 04 − 0 ✗
We put in the numbers: 4 minus 0, over 2 − 0.
= 4 − 02 − 0
4 minus 0 is 4.
2 − 0 is 2.
= 42
4 divided by 2 is 2.
= 2
So the average rate of change is 2.
On average, f(x) goes up 2 for each 1 that x goes up.
3 Check
Let's check on a graph.
This is the graph of f(x).
Our two points.
The line through them.
Its slope is the average rate of change.
Run: 2 across.
Rise: 4 up.
Start at f(0) = 0.
Add the rate, 2, 2 times.
We land on 4, which is f(2).
So the answer is right!
0 + 2 × 2 = 4 = f(2)
Let's look back.
First: find the two points.
Then: change over change.
Last: check.
▶ Watch it solved
