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 x2 - 4x
to make a perfect square trinomial.

x2 is x2.

-4x is
-2 times
x times,
(-4x)/(-2⋅x), 2.

Quadratic Equation: Completing the Square

Write +22.

To undo +22,
write -22.

Write +5.

So y = x2 - 4x + 5 becomes
y = x2 - 2⋅x⋅2 + 22 - 22 + 5.

x2 - 2⋅x⋅2 + 22
= (x - 2)2

Factor a Perfect Square Trinomial

-22 = -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 x2 + 6x
to make a perfect square trinomial.

x2 is x2.

+6x is
+2 times
x times,
(+6x)/(+2⋅x), 3.

Write +32.

To undo +32,
write -32.

Write -1.

So y = x2 + 6x - 1 becomes
y = x2 + 2⋅x⋅3 + 32 - 32 - 1.

x2 + 2⋅x⋅3 + 32
= (x + 3)2

-32 = -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 -x2 to -(x2.

+8x = -(-8x)

-8x is
-2 times
x times,
(-8x)/(-2⋅x), 4.

To make a perfect square trinomial
in the parentheses,
write +42).

To undo -(+42),
write +42.

Write -16.

So y = -x2 + 8x - 16 becomes
y = -(x2 - 2⋅x⋅4 + 42) + 42 - 16.

-(x2 - 2⋅x⋅4 + 42)
= -(x - 4)2

+42 = +16

+16 - 16 = 0

y = -(x - 4)2

Then the vertex of the quadratic function is
(4, 0).

So (4, 0) is the answer.