Write -x^2 + 2x + 3 in vertex form

Step by step

The question

Question: write y = −x² + 2x + 3 in vertex form.

Vertex form is a times (x minus h) squared, plus k.

The vertex, the turning point, is (h, k).

1 Take −1 out of the x parts

The x² must have just 1 in front.

So we take −1 out of the two x parts.

The 3 stays outside.

So y is −1 times (x² − 2x), plus 3.

2 Complete the square

We add 1 to fill the corner.

And we take 1 away straight after, so nothing changes.

3 Write the square

The full square is x − 1 on each side.

So these three parts are (x − 1) squared.

We write the bracket squared.

The rest stays.

−1 times minus 1 is 1: the −1 multiplies both parts in the big bracket.

So y is −1 times the square, plus 1, plus 3.

4 Tidy up

1 plus 3 is 4.

So y = −(x − 1)² + 4.

This is vertex form.

5 Read the vertex

In vertex form we need x minus h.

x − 1 is x minus 1.

So h = 1, and k = 4.

So the vertex is (1, 4).

Careful!

x minus 1 means h is 1, not −1.

The sign in the bracket is the opposite of h.

6 Check

Let's check: open the brackets again.

(x − 1) squared is x² − 2x + 1.

We get −x² + 2x + 3 again.

It works!

Let's look back.

First: take −1 out of the x parts.

Then: complete the square.

Then: write the square.

Then: tidy up.

Then: read the vertex.

Last: check.