Graph y = (x - 4)^2 - 2 from y = x^2 Step by Step

Step by step

The question

Question: graph y = (x − 4)² − 2 from y = x².

y = (x − 4)² − 2

We know the graph of y = x².

It is a U, with its turning point at (0, 0).

we know: y = x²

1 Read the changes

Inside, next to x, we have − 4.

x = 4 makes the inside 0.

So the turning point moves to x = 4: right 4.

x − 4 inside → right 4

Careful!

Inside, it goes the opposite way to the sign.

Careful: − 4 inside goes right, not left

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

So the graph moves down 2.

− 2 outside → down 2

2 Start from y = x²

Draw the grid.

The x-axis goes across.

The y-axis goes up and down.

They meet at 0.

That point is called the origin.

Draw y = x² first.

Its turning point is at (0, 0).

3 Move the turning point

Move it right 4.

Then down 2.

The new turning point is (4, −2).

4 Draw the new graph

The shape stays the same.

From the turning point, 1 right, 1 up.

And 1 left, 1 up.

Join them with the same U.

This is y = (x − 4)² − 2.

5 Check

Let's check.

Put x = 4 in the new rule.

We get y = −2: the turning point (4, −2).

x = 4: y = (4 − 4)² − 2 = (0)² − 2 = 0 − 2

= −2

And x = 5 gives y = −1: the point (5, −1) we drew.

x = 5: y = (5 − 4)² − 2 = (1)² − 2 = 1 − 2

= −1

Let's look back.

First: read the changes.

Then: start from y = x².

Then: move the turning point.

Then: draw the new graph.

Last: check.

Answer