Step by step
The question
Question: graph y = 81 × (1/3)ˣ.
1 Read a and b
This is the rule y = a times b to the power x.
a = 81 is the start: the y when x is 0.
b = 1/3 is the decay factor: each time x goes up 1, y is multiplied by 1/3.
1/3 is between 0 and 1, so y gets smaller.
That is called decay.
Careful!
Power first, then times.
Only 1/3 is squared, not 81 times 1/3.
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 81 times 1, that is 81: the start.
x = 1: (1/3) to the power 1 is just 1/3.
81 times 1/3 is 27.
x = 2: (1/3) to the power 2 is 1/9.
Times 81: 9.
x = 3: (1/3) to the power 3 is 1/27.
Times 81: 3.
x = 4: (1/3) to the power 4 is 1/81.
Times 81: 1.
Look at the y's: each one is the one before, times 1/3.
That is what exponential means.
3 Plot the points
Draw the grid.
The y numbers go up in 10s, because y gets small fast.
Put the points from the table on the grid.
4 Draw the curve
Join the points with a smooth curve.
It goes down, slower and slower.
It gets close to 0, but never reaches it.
5 Check
Let's check with one more x.
The rule with x = 5 gives 0.333333.
One more step of the table: 1 times 1/3 is 0.333333 too.
It works!
Let's look back.
First: read a and b.
Then: make a table.
Then: plot the points.
Then: draw the curve.
Last: check.