Graph y = -x^3 - 2 from y = x^3 Step by Step

Step by step

The question

Question: graph y = −x³ − 2 from y = x³.

y = −x³ − 2

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

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

So the graph moves down 2.

− 2 outside → down 2

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 down 2.

(−1, −1) stays where it is.

(0, 0) goes to (0, −2).

(1, 1) goes to (1, −3).

4 Draw the new graph

Join the new points with the same shape, flipped.

This is y = −x³ − 2.

5 Check

Let's check.

Put x = 0 in the new rule.

We get y = −2: the point (0, −2) we drew.

x = 0: y = −(0)³ − 2 = 0 − 2 = −2

And x = 1 gives y = −3: the point (1, −3).

x = 1: y = −(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