Step by step
The question
Question: graph y = (x − 2)³ from y = x³.
y = (x − 2)3
We know the graph of y = x³.
It is an S shape.
we know: y = x³
1 Read the changes
Inside, next to x, we have − 2.
x = 2 makes the inside 0.
So the middle point moves to x = 2: right 2.
x − 2 inside → right 2
Careful!
Inside, it goes the opposite way to the sign.
Careful: − 2 inside goes right, not left
2 Start from y = x³
Draw the grid.
Draw y = x³ first.
It goes through (−1, −1), (0, 0) and (1, 1).
3 Move the points
The middle point (0, 0) moves right 2.
(−1, −1) goes to (1, −1).
(0, 0) goes to (2, 0).
(1, 1) goes to (3, 1).
4 Draw the new graph
Join the new points with the same shape.
This is y = (x − 2)³.
5 Check
Let's check.
Put x = 2 in the new rule.
We get y = 0: the point (2, 0) we drew.
x = 2: y = (2 − 2)3 = (0)3 = 0
And x = 3 gives y = 1: the point (3, 1).
x = 3: y = (3 − 2)3 = (1)3 = 1
Let's look back.
First: read the changes.
Then: start from y = x³.
Then: move the points.
Then: draw the new graph.
Last: check.
Answer
x − 2 inside → right 2