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

Step by step

The question

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

y = −(x − 1)³

We know the graph of y = x³.

It is an S shape.

we know: y = x³

1 Read the changes

The minus in front changes the sign of every y.

Up becomes down.

So the graph flips over the x-axis.

− in front → flip over the x-axis

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

Careful!

A minus inside flips left and right.

A minus outside flips up and down.

Careful: inside → left-right, outside → up-down

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.

(−1, −1) goes to (0, 1).

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

(1, 1) goes to (2, −1).

4 Draw the new graph

Join the new points with the same shape, flipped.

This is y = −(x − 1)³.

5 Check

Let's check.

Put x = 1 in the new rule.

We get y = 0: the point (1, 0) we drew.

x = 1: y = −(1 − 1)³ = −(0)³ = 0

And x = 2 gives y = −1: the point (2, −1).

x = 2: y = −(2 − 1)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