Worked examples
The question
Question: write y = x² + 6x + 5 in vertex form.
y = x2 + 6x + 5
Vertex form is a times (x minus h) squared, plus k.
The vertex, the turning point, is (h, k).
vertex form: a(x − h)2 + k
1 Complete the square
We add 9 to fill the corner.
And we take 9 away straight after, so nothing changes.
= x2 + 6x + 9 − 9 + 5
2 Write the square
The full square is x + 3 on each side.
So these three parts are (x + 3) squared.
x2 + 6x + 9 = (x + 3)2
We write the bracket squared.
The rest stays.
= (x + 3)2 − 9 + 5
3 Tidy up
−9 plus 5 is −4.
−9 + 5 = −4
So y = (x + 3)² − 4.
This is vertex form.
= (x + 3)2 − 4
4 Read the vertex
In vertex form we need x minus h.
x + 3 is x minus (−3).
So h = −3, and k = −4.
x + 3 = x − (−3)
So the vertex is (−3, −4).
vertex = (−3, −4)
Careful!
x plus 3 means h is −3, not 3.
The sign in the bracket is the opposite of h.
Careful: not vertex (3, −4) ✗
5 Check
Let's check: open the brackets again.
(x + 3) squared is x² + 6x + 9.
Check: (x + 3)2 − 4 = x2 + 6x + 9 − 4
= x2 + 6x + 5
We get x² + 6x + 5 again.
It works!
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Last: check.
▶ Watch it solved
The question
Question: write f(x) = x² + 8x + 12 in vertex form.
f(x) = x2 + 8x + 12
Vertex form is a times (x minus h) squared, plus k.
The vertex, the turning point, is (h, k).
vertex form: a(x − h)2 + k
1 Complete the square
We add 16 to fill the corner.
And we take 16 away straight after, so nothing changes.
= x2 + 8x + 16 − 16 + 12
2 Write the square
The full square is x + 4 on each side.
So these three parts are (x + 4) squared.
x2 + 8x + 16 = (x + 4)2
We write the bracket squared.
The rest stays.
= (x + 4)2 − 16 + 12
3 Tidy up
−16 plus 12 is −4.
−16 + 12 = −4
So f(x) = (x + 4)² − 4.
This is vertex form.
= (x + 4)2 − 4
4 Read the vertex
In vertex form we need x minus h.
x + 4 is x minus (−4).
So h = −4, and k = −4.
x + 4 = x − (−4)
So the vertex is (−4, −4).
vertex = (−4, −4)
Careful!
x plus 4 means h is −4, not 4.
The sign in the bracket is the opposite of h.
Careful: not vertex (4, −4) ✗
5 Check
Let's check: open the brackets again.
(x + 4) squared is x² + 8x + 16.
Check: (x + 4)2 − 4 = x2 + 8x + 16 − 4
= x2 + 8x + 12
We get x² + 8x + 12 again.
It works!
Let's look back.
First: complete the square.
Then: write the square.
Then: tidy up.
Then: read the vertex.
Last: check.
▶ Watch it solved
The question
Question: write y = −x² − 6x + 4 in vertex form.
y = −x2 − 6x + 4
Vertex form is a times (x minus h) squared, plus k.
The vertex, the turning point, is (h, k).
vertex form: a(x − h)2 + 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 4 stays outside.
−x² ÷ (−1) = x², −6x ÷ (−1) = 6x
So y is −1 times (x² + 6x), plus 4.
= −(x2 + 6x) + 4
2 Complete the square
We add 9 to fill the corner.
And we take 9 away straight after, so nothing changes.
= −(x2 + 6x + 9 − 9) + 4
3 Write the square
The full square is x + 3 on each side.
So these three parts are (x + 3) squared.
x2 + 6x + 9 = (x + 3)2
We write the bracket squared.
The rest stays.
= −[(x + 3)2 − 9] + 4
−1 times minus 9 is 9: the −1 multiplies both parts in the big bracket.
−1 × (−9) = 9
So y is −1 times the square, plus 9, plus 4.
= −(x + 3)2 + 9 + 4
4 Tidy up
9 plus 4 is 13.
9 + 4 = 13
So y = −(x + 3)² + 13.
This is vertex form.
= −(x + 3)2 + 13
5 Read the vertex
In vertex form we need x minus h.
x + 3 is x minus (−3).
So h = −3, and k = 13.
x + 3 = x − (−3)
So the vertex is (−3, 13).
vertex = (−3, 13)
Careful!
x plus 3 means h is −3, not 3.
The sign in the bracket is the opposite of h.
Careful: not vertex (3, 13) ✗
6 Check
Let's check: open the brackets again.
(x + 3) squared is x² + 6x + 9.
Check: −(x + 3)2 + 13 = −(x2 + 6x + 9) + 13
= −x2 − 6x − 9 + 13 = −x2 − 6x + 4
We get −x² − 6x + 4 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.
▶ Watch it solved
