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

Step by step

The question

Question: graph y = −(x + 1)² + 2 from y = x².

y = −(x + 1)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 + 1.

x = −1 makes the inside 0.

So the turning point moves to x = −1: left 1.

x + 1 inside → left 1

Careful!

Inside, it goes the opposite way to the sign.

Careful: + 1 inside goes left, not right

Outside, at the end, + 2 adds 2 to every y.

So the graph moves up 2.

+ 2 outside → up 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 left 1.

Then up 2.

The new turning point is (−1, 2).

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 + 1)² + 2.

5 Check

Let's check.

Put x = −1 in the new rule.

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

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

= 0 + 2 = 2

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

x = 0: y = −(0 + 1)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