# Quadratic Function: Vertex Form

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

## Formula

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

To complete the square:

Change -4*x* to -2⋅*x*⋅2.

Write +2^{2}.

To undo +2^{2}, write -2^{2}.

Factor the left side trinomial:*x*^{2} - 2⋅*x*⋅2 + 2^{2} = (*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

To complete the square:

Change +2*x* to +2⋅*x*⋅1.

Write +1^{2}.

To undo +1^{2}, write -1^{2}.

Factor the left side trinomial:*x*^{2} + 2⋅*x*⋅1 + 1^{2} = (*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

To complete the square:

Change -*x*^{2} + 6*x* to -(*x*^{2} - 6*x*).

Leave some space behind - 6*x*.

Change -6*x* to -2⋅*x*⋅3.

Write +3^{2} behind -2⋅*x*⋅3.

To undo -(+3^{2}), write +3^{2}.

Factor the left side trinomial:

-(*x*^{2} - 2⋅*x*⋅3 + 3^{2}) = -(*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.