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: left 1.

x + 1 inside → left 1

Outside, at the end, − 2 takes 2 away from every y.

So the graph moves down 2.

− 2 outside → down 2

Careful!

Inside, it goes the opposite way to the sign.

Careful: + 1 inside goes left, not right

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 left 1.

Then down 2.

(−1, −1) goes to (−2, −3).

(0, 0) goes to (−1, −2).

(1, 1) goes to (0, −1).

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 = 0 gives y = −1: the point (0, −1).

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

= −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