Worked examples
The question
Question: graph y = 2ˣ.
y = 2x
1 Read a and b
This is the rule y = a times b to the power x.
a = 1 is the start: the y when x is 0.
b = 2 is the growth factor: each time x goes up 1, y is multiplied by 2.
2 is more than 1, so y grows.
y = a × bx
a = 1: the start
b = 2: times this each step
2 Make a table
We make a table: x from 0 to 4.
x = 0: anything to the power 0 is 1.
So y is 1: the start.
1
x = 1: 2 to the power 1 is just 2.
So y is 2.
2
x = 2: 2 to the power 2 is 4: 4.
x = 2: y = 22 = 4
4
x = 3: 2 to the power 3 is 8: 8.
x = 3: y = 23 = 8
8
x = 4: 2 to the power 4 is 16: 16.
x = 4: y = 24 = 16
16
Look at the y's: each one is the one before, times 2.
That is what exponential means.
3 Plot the points
Draw the grid.
The y numbers go up in 2s, because y gets big fast.
Put the points from the table on the grid.
4 Draw the curve
Join the points with a smooth curve.
It goes up faster and faster.
5 Check
Let's check with one more x.
The rule with x = 5 gives 32.
rule: x = 5, y = 25 = 32
One more step of the table: 16 times 2 is 32 too.
It works!
table: 16 × 2 = 32
Let's look back.
First: read a and b.
Then: make a table.
Then: plot the points.
Then: draw the curve.
Last: check.
▶ Watch it solved
The question
Question: $1000 grows by 5% a year.
How much is it after 3 years?
1 Find the start and the factor
The start is $1000.
Each year it keeps all of it, 100%, and gets 5% more.
So it becomes 105% of what it was.
105% is 1.05.
That is the factor.
start: a = $1000
100% + 5% = 105% = 1.05
factor: b = 1.05
2 Write the rule
x is the number of years.
Each year we multiply by 1.05 again.
So y = 1000 × 1.05ˣ.
y = 1000 × 1.05x
Careful!
The 5% is taken of the NEW amount every year, not always of $1000.
Careful: not $1000 + 3 × $50 = $1150 ✗
3 Put in x = 3
After 3 years, x = 3.
y = 1000 × 1.053
1.05 to the power 3 is 1.157625 (with a calculator).
1.053 = 1.157625
So we multiply 1000 by 1.157625.
= 1000 × 1.157625
That is 1157.625.
Money is rounded to cents: about $1157.63.
= ≈ $1157.63
4 Check
Let's check year by year.
We start at $1000, and multiply by 1.05 each year.
After 3 years we get 1157.625 again.
It works!
after 3 years: 1157.625
Let's look back.
First: find the start and the factor.
Then: write the rule.
Then: put in x = 3.
Last: check.
▶ Watch it solved
The question
Question: A lake has 6000 fish.
The number falls by 20% each year.
How many fish are there after 3 years?
1 Find the start and the factor
The start is 6000.
Each year it loses 20%.
So 80% is left.
80% is 0.8.
That is the factor.
start: a = 6000
100% − 20% = 80% = 0.8
factor: b = 0.8
2 Write the rule
x is the number of years.
Each year we multiply by 0.8 again.
So y = 6000 × 0.8ˣ.
y = 6000 × 0.8x
Careful!
0.2 is the part that is lost.
We multiply by the part that is left: 0.8.
Careful: not b = 0.2 ✗
3 Put in x = 3
After 3 years, x = 3.
y = 6000 × 0.83
0.8 to the power 3 is 0.512.
0.83 = 0.512
So we multiply 6000 by 0.512.
= 6000 × 0.512
So it is 3072.
= 3072
4 Check
Let's check year by year.
We start at 6000, and multiply by 0.8 each year.
After 3 years we get 3072 again.
It works!
after 3 years: 3072
Let's look back.
First: find the start and the factor.
Then: write the rule.
Then: put in x = 3.
Last: check.
▶ Watch it solved
