# Quadratic Function: Vertex Form

How to find the vertex of a quadratic function (by changing it to vertex form): formula, 3 examples, and their solutions.

## Formula

### Formula

y = a(x - h)^{2} + k

is a quadratic function in vertex form.

The vertex of the quadratic function is

(h, k).

## Example 1

### Example

### Solution

To find the vertex of the quadratic function,

change the quadratic function to vertex form.

Use x^{2} - 4x

to make a perfect square trinomial.

x^{2} is x^{2}.

-4x is

-2 times

x times,

(-4x)/(-2⋅x), 2.

Quadratic Equation: Completing the Square

Write +2^{2}.

To undo +2^{2},

write -2^{2}.

Write +5.

So y = x^{2} - 4x + 5 becomes

y = x^{2} - 2⋅x⋅2 + 2^{2} - 2^{2} + 5.

x^{2} - 2⋅x⋅2 + 2^{2}

= (x - 2)^{2}

Factor a Perfect Square Trinomial

-2^{2} = -4

-4 + 5 = +1

y = (x - 2)^{2} + 1

Then the vertex of the quadratic function is

(2, 1).

So (2, 1) is the answer.

## Example 2

### Example

### Solution

To find the vertex,

change the quadratic function to vertex form.

Use x^{2} + 6x

to make a perfect square trinomial.

x^{2} is x^{2}.

+6x is

+2 times

x times,

(+6x)/(+2⋅x), 3.

Write +3^{2}.

To undo +3^{2},

write -3^{2}.

Write -1.

So y = x^{2} + 6x - 1 becomes

y = x^{2} + 2⋅x⋅3 + 3^{2} - 3^{2} - 1.

x^{2} + 2⋅x⋅3 + 3^{2}

= (x + 3)^{2}

-3^{2} = -9

To see the x value of the vertex easily,

change (x + 3)^{2} to (x - (-3))^{2}.

-9 - 1 = -10

y = (x - (-3))^{2} - 10

Then the vertex of the quadratic function is

(-3, -10).

So (-3, -10) is the answer.

## Example 3

### Example

### Solution

To find the vertex,

change the quadratic function to vertex form.

First change -x^{2} to -(x^{2}.

+8x = -(-8x)

-8x is

-2 times

x times,

(-8x)/(-2⋅x), 4.

To make a perfect square trinomial

in the parentheses,

write +4^{2}).

To undo -(+4^{2}),

write +4^{2}.

Write -16.

So y = -x^{2} + 8x - 16 becomes

y = -(x^{2} - 2⋅x⋅4 + 4^{2}) + 4^{2} - 16.

-(x^{2} - 2⋅x⋅4 + 4^{2})

= -(x - 4)^{2}

+4^{2} = +16

+16 - 16 = 0

y = -(x - 4)^{2}

Then the vertex of the quadratic function is

(4, 0).

So (4, 0) is the answer.