Step by step
The question
Question: is this table linear or exponential: x is 1, 2, 3, 4, and y is 64, 32, 16, 8?
We write it as a table.
x goes up by 1 each time.
1 Check the differences
Linear means: the same number is added each time.
So we look at the change from one y to the next: minus 32, then minus 16, then minus 8.
The changes are not the same.
So it is not linear.
Careful!
Going down each time does not make it linear.
Linear needs the same number added every time.
2 Check the ratios
Exponential means: we multiply by the same number each time.
So we divide each y by the one before.
32/64 is 0.5.
16/32 is 0.5.
8/16 is 0.5.
The same number.
So the table is exponential.
3 Write the rule
An exponential rule is y = a times b to the power x.
a is the start value, at x = 0.
b is the number we multiply by.
The start value is y at x = 0.
The table starts at x = 1, so we go back: divide by 0.5.
y at 0 is 128.
So the rule is y = 128 times 0.5 to the power x.
4 Check
Let's check with the last pair: x = 4.
The rule gives 8.
The table says 8.
So the rule is right!
Let's look back.
First: check the differences.
Then: check the ratios.
Then: write the rule.
Last: check.