Write x^2 - 4x + 1 in vertex form

Step by step

The question

Question: write y = x² − 4x + 1 in vertex form.

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

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

1 Complete the square

We add 4 to fill the corner.

And we take 4 away straight after, so nothing changes.

2 Write the square

The full square is x − 2 on each side.

So these three parts are (x − 2) squared.

We write the bracket squared.

The rest stays.

3 Tidy up

−4 plus 1 is −3.

So y = (x − 2)² − 3.

This is vertex form.

4 Read the vertex

In vertex form we need x minus h.

x − 2 is x minus 2.

So h = 2, and k = −3.

So the vertex is (2, −3).

Careful!

x minus 2 means h is 2, not −2.

The sign in the bracket is the opposite of h.

5 Check

Let's check: open the brackets again.

(x − 2) squared is x² − 4x + 4.

We get x² − 4x + 1 again.

It works!

Let's look back.

First: complete the square.

Then: write the square.

Then: tidy up.

Then: read the vertex.

Last: check.