Graph y = (x - 1)^3 + 2 from y = x^3 Step by Step

Step by step

The question

Question: graph y = (x − 1)³ + 2 from y = x³.

y = (x − 1)³ + 2

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 − 1.

x = 1 makes the inside 0.

So the middle point moves to x = 1: right 1.

x − 1 inside → right 1

Outside, at the end, + 2 adds 2 to every y.

So the graph moves up 2.

+ 2 outside → up 2

Careful!

Inside, it goes the opposite way to the sign.

Careful: − 1 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 1.

Then up 2.

(−1, −1) goes to (0, 1).

(0, 0) goes to (1, 2).

(1, 1) goes to (2, 3).

4 Draw the new graph

Join the new points with the same shape.

This is y = (x − 1)³ + 2.

5 Check

Let's check.

Put x = 1 in the new rule.

We get y = 2: the point (1, 2) we drew.

x = 1: y = (1 − 1)³ + 2 = (0)³ + 2 = 0 + 2

= 2

And x = 2 gives y = 3: the point (2, 3).

x = 2: y = (2 − 1)³ + 2 = (1)³ + 2 = 1 + 2

= 3

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