Step by step
The question
Question: is this table linear or exponential: x is 1, 2, 3, 4, and y is 20, 17, 14, 11?
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 3, then minus 3, then minus 3.
We add -3 every time.
The same number.
So the table is linear.
2 Write the rule
The start value is y at x = 0.
The table starts at x = 1, so we go back 1 step.
y at 0 is 23.
A linear rule is y = m x plus b.
m is the change when x goes up 1.
b is the start value, at x = 0.
Careful!
Not just −3x.
At x = 0, y is 23, not 0.
The start value goes in too.
So the rule is y = −3x + 23.
3 Check
Let's check with the last pair: x = 4.
The rule gives 11.
The table says 11.
So the rule is right!
Let's look back.
First: check the differences.
Then: write the rule.
Last: check.