Step by step
The question
Question: does a linear or an exponential model fit this scatterplot better?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Look at the points
Each dot is one point of the data: across is x, up is y.
x goes up by 1 each time, the same step.
So we can compare each point with the one before.
We write the points in a table.
2 Check the differences
Linear means: about the same number is added each step.
So we look at the change from one y to the next: plus 10, then plus 30, then plus 90, then plus 270.
The changes are getting bigger and bigger.
They are not about the same.
So it is not linear.
Careful!
Going up each step does not make it linear.
Linear needs about the same number added every step.
3 Check the ratios
Exponential means: we multiply by about the same number each step.
So we divide each y by the one before.
15/5 is 3.
45/15 is 3.
135/45 is 3.
405/135 is 3.
The same number.
y is multiplied by 3 each step.
So an exponential model fits better.
4 Check
Let's check on the picture.
A straight line from the first point to the last one misses the points in the middle.
A curve that multiplies by about 3 each step goes through the points, or very close.
So the exponential model is right!
5 Pick the choice
Our answer is exponential.
That is choice C.
6 Why not the others
A is linear.
Wrong: the steps are not equal: × 3 each step.
That is the Careful mistake from before.
B is quadratic.
Wrong: each y is 3 times the one before: that is exponential.
D is neither.
Wrong: each y is 3 times the one before.
Let's look back.
First: look at the points.
Then: check the differences.
Then: check the ratios.
Then: check.
Then: pick the choice.
Last: why not the others.