# Parabola: Directrix

How to find the directrix of the given parabola: definition, formulas, examples, and their solutions.

## Formula: Focus of *y*^{2} = 4*px*

For the parabola [*y*^{2} = 4*px*],

the directrix is [*x* = -*p*].

Parabola - Proof of the formula *y*^{2} = 4*px*

## Example 1: Directrix of *y*^{2} = 8*x*

Change the right side into 4*px*.

Then 8*x* = 4⋅2⋅*x*.

*y*^{2} = 4⋅2⋅*x*

So the directrix is *x* = -2.

This is the graph of *y*^{2} = 4⋅2⋅*x*.

Its directrix is *x* = -2.

## Example 2: Directrix of *y*^{2} = -4*x*

Change the right side into 4*px*.

Then -4*x* = 4⋅(-1)⋅*x*.

*y*^{2} = 4⋅(-1)⋅*x*

So the directrix is *x* = -(-1).

-(-1) = +1

So *x* = 1 is the answer.

This is the graph of *y*^{2} = 4⋅(-1)⋅*x*.

Its directrix is *x* = -(-1).

## Formula: Focus of *x*^{2} = 4*py*

For the parabola [*x*^{2} = 4*py*],

the directrix is [*y* = -*p*].

Parabola - Proof of the formula *x*^{2} = 4*py*

## Example 3: Directrix of *y* = *x*^{2}

Switch both sides:*x*^{2} = *y*.

Then change the right side into 4*py*.

So *y* = 4⋅(1/4)⋅*y*.

*x*^{2} = 4⋅(1/4)⋅*y*

So the directrix is *y* = -1/4.

This is the graph of *x*^{2} = 4⋅(1/4)⋅*y*.

Its directrix is *y* = -1/4.