Step by step
The question
Question: find the average rate of change of f(x) = x² from x = 1 to x = 5.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find the two points
We put 1 in place of x.
f(1) = 1.
We put 5 in place of x.
f(5) = 25.
So the two points are (1, 1) and (5, 25).
2 Change over change
f(x) is the y.
A rate tells how much y changes for each 1 that x moves.
Like speed: kilometres, divided by hours.
So we divide the change in y by the change in x.
On top: f(5) minus f(1), the change in y.
At the bottom: 5 − 1, the change in x.
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: 25 minus 1, over 5 − 1.
25 minus 1 is 24.
5 − 1 is 4.
24 divided by 4 is 6.
So the average rate of change is 6.
On average, f(x) goes up 6 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: 24 up.
Start at f(1) = 1.
x moves 4, so we add the rate, 6, 4 times.
We land on 25, which is f(5).
So the answer is right!
4 Pick the choice
Our answer is 6.
That is choice B.
5 Why not the others
A is 4.
Wrong: 5 − 1: that is the change in x.
C is 24.
Wrong: 25 − 1: forgot to divide by 5 − 1.
D is 26.
Wrong: 25 + 1: added the values.
Let's look back.
First: find the two points.
Then: change over change.
Then: check.
Then: pick the choice.
Last: why not the others.