Linear or exponential: x 0, 1, 2, 3; y 80, 40, 20, 10 Step by Step

Step by step

The question

Question: is this table linear or exponential: x is 0, 1, 2, 3, and y is 80, 40, 20, 10?

x: 0, 1, 2, 3; y: 80, 40, 20, 10

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 40, then minus 20, then minus 10.

The changes are not the same.

So it is not linear.

= − 40, − 20, − 10: not the same

Careful!

Going down each time does not make it linear.

Linear needs the same number added every time.

Careful: it goes down each time, but that is not linear

2 Check the ratios

Exponential means: we multiply by the same number each time.

So we divide each y by the one before.

40/80 is 0.5.

20/40 is 0.5.

10/20 is 0.5.

= 4080 = 0.5, 2040 = 0.5, 1020 = 0.5

The same number.

So the table is exponential.

= × 0.5 every time → 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.

y = a × b^x

a = start, b = the ratio

So the rule is y = 80 times 0.5 to the power x.

= y = 80 × 0.5^x

4 Check

Let's check with the last pair: x = 3.

The rule gives 10.

The table says 10.

So the rule is right!

x = 3: 80 × 0.5³ = 80 × 0.125 = 10

Let's look back.

First: check the differences.

Then: check the ratios.

Then: write the rule.

Last: check.

Answer