Quadratic Function: Vertex Form

Quadratic Function: Vertex Form

How to write a quadratic function in vertex form: formula, examples, and their solutions.

Formula

If the vertex of a quadratic function is (h, k), then the function is y = a(x - h)^2 + k.

The white graph shows the graph of a quadratic function.

Shape: parabola.
Blue dashed line: axis of symmetry
Red point: vertex

If the vertex of the quadratic function is (p, q),
then the quadratic function in vertex form is
y = (x - p)2 + q.

Example 1

Write the given quadratic function in the vertex form. y = x^2 - 4x + 5

To complete the square:

Change -4x to -2⋅x⋅2.

Write +22.

To undo +22, write -22.

Factor the left side trinomial:
x2 - 2⋅x⋅2 + 22 = (x - 2)2.

After arranging the constants,
the function is y = (x - 2)2 + 1.

To graph y = (x - 2)2 + 1:

Draw the vertex (2, 1).

Find the points near the vertex
by putting x = 0, 1, 3, 4 into the function.

Draw a parabola by connecting the vertex and the near points.

Example 2

Write the given quadratic function in the vertex form. y = x^2 + 2x - 2

To complete the square:

Change +2x to +2⋅x⋅1.

Write +12.

To undo +12, write -12.

Factor the left side trinomial:
x2 + 2⋅x⋅1 + 12 = (x + 1)2.

After arranging the constants,
the function is y = (x + 1)2 - 3.

You can either write the answer like this:
y = (x - (-1))2 - 3.

To graph y = (x + 1)2 - 3:

Draw the vertex (-1, -3).

Find the points near the vertex
by putting x = -3, -2, 0, 1 into the function.

Draw a parabola by connecting the vertex and the near points.

Example 3

Write the given quadratic function in the vertex form. y = -x^2 + 6x - 9

To complete the square:

Change -x2 + 6x to -(x2 - 6x).
Leave some space behind - 6x.

Change -6x to -2⋅x⋅3.

Write +32 behind -2⋅x⋅3.

To undo -(+32), write +32.

Factor the left side trinomial:
-(x2 - 2⋅x⋅3 + 32) = -(x - 3)2.

After arranging the constants,
the function is y = -(x - 3)2.

To graph y = -(x - 3)2:

Draw the vertex (3, 0).

Find the points near the vertex
by putting x = 1, 2, 4, 5 into the function.

Draw a parabola by connecting the vertex and the near points.

Because of the (-) in front of (x - 3)2,
the parabola is opened downward.