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

Step by step

The question

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

y = −(x − 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

The minus in front turns every y into minus y.

So the U flips upside down.

− in front → flip

Inside, next to x, we have − 2.

x = 2 makes the inside 0.

So the turning point moves to x = 2: right 2.

x − 2 inside → right 2

Careful!

Inside, it goes the opposite way to the sign.

Careful: − 2 inside goes right, not left

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 2.

The new turning point is (2, 0).

4 Draw the new graph

The shape is flipped.

From the turning point, 1 right, 1 down.

And 1 left, 1 down.

Join them with an upside-down U.

This is y = −(x − 2)².

5 Check

Let's check.

Put x = 2 in the new rule.

We get y = 0: the turning point (2, 0).

x = 2: y = −(2 − 2)² = −(0)² = 0

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

x = 3: y = −(3 − 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