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

Step by step

The question

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

y = −(x + 2)³ + 1

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 + 2.

x = −2 makes the inside 0.

So the middle point moves to x = −2: left 2.

x + 2 inside → left 2

Outside, at the end, + 1 adds 1 to every y.

So the graph moves up 1.

+ 1 outside → up 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 left 2.

Then up 1.

(−1, −1) goes to (−3, 2).

(0, 0) goes to (−2, 1).

(1, 1) goes to (−1, 0).

4 Draw the new graph

Join the new points with the same shape, flipped.

This is y = −(x + 2)³ + 1.

5 Check

Let's check.

Put x = −2 in the new rule.

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

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

= 0 + 1 = 1

And x = −1 gives y = 0: the point (−1, 0).

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

= −1 + 1 = 0

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