Worked examples
The question
Question: is this table linear or exponential: x is 0, 1, 2, 3, and y is 2, 5, 8, 11?
x: 0, 1, 2, 3; y: 2, 5, 8, 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: plus 3, then plus 3, then plus 3.
We add 3 every time.
The same number.
So the table is linear.
= + 3, + 3, + 3: the same → linear
2 Write the rule
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.
y = mx + b
m = the change for 1, b = start
Careful!
Not just 3x.
At x = 0, y is 2, not 0.
The start value goes in too.
Careful: not y = 3x ✗
So the rule is y = 3x + 2.
= y = 3x + 2
3 Check
Let's check with the last pair: x = 3.
The rule gives 11.
The table says 11.
So the rule is right!
x = 3: 3(3) + 2 = 11
Let's look back.
First: check the differences.
Then: write the rule.
Last: check.
▶ Watch it solved
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 × bx
a = start, b = the ratio
So the rule is y = 80 times 0.5 to the power x.
= y = 80 × 0.5x
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.
▶ Watch it solved
The question
Question: is this table linear or exponential: x is 0, 2, 4, 6, and y is 1, 7, 13, 19?
x: 0, 2, 4, 6; y: 1, 7, 13, 19
We write it as a table.
x goes up by 2 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: plus 6, then plus 6, then plus 6.
We add 6 every time.
The same number.
So the table is linear.
= + 6, + 6, + 6: the same → linear
2 Write the rule
y changes 6 when x goes up 2.
For each 1 of x, that is 3.
+6 for every 2: m = 6 ÷ 2 = 3
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.
y = mx + b
m = the change for 1, b = start
Careful!
Not just 3x.
At x = 0, y is 1, not 0.
The start value goes in too.
Careful: not y = 3x ✗
So the rule is y = 3x + 1.
= y = 3x + 1
3 Check
Let's check with the last pair: x = 6.
The rule gives 19.
The table says 19.
So the rule is right!
x = 6: 3(6) + 1 = 19
Let's look back.
First: check the differences.
Then: write the rule.
Last: check.
▶ Watch it solved
