# Parabola: Formula

How to write the equation of a parabola by using its focus and directrix: formulas, examples, and their solutions.

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

For a parabola

whose focus is (*p*, 0)

and whose directrix is *x* = -*p*,

the equation of the parabola is*y*^{2} = 4*px*.

Parabola - Focus

Parabola - Directrix

## Example 1: Focus (3, 0), Directrix *x* = -3, Parabola?

The focus is (3, 0).

And the directrix is *x* = -3.

So *p* = 3.

*p* = 3

And the directrix is [*x* = ...].

Then the equation of the parabola is*y*^{2} = 4⋅3⋅*x*.

4⋅3 = 12

So *y*^{2} = 12*x*.

This is the graph of *y*^{2} = 12*x*.

The points on the parabola are equidistant

from the focus (3, 0) and the directrix *x* = -3.

## Example 2: Focus (3, 4), Directrix *x* = -1, Parabola?

Lightly draw the given conditions.

It says the focus is (3, 4).

And the directrix is *x* = -1.

So draw the parabola like this.

Draw a segment

between the focus (3, 4) and the directrix *x* = -1.

The distance

between the vertex of the parabola and (3, 4)

is *p*.

And the distance

between the vertex of the parabola and *x* = -1

is also *p*.

So, the length of the whole segment, 2*p*

is equal to,

the distance between (3, 4) and *x* = -1, 3 - (-1).

3 - (-1)

= 3 + 1

= 4

Divide both sides by 2.

Then *p* = 2.

*p* = 2

And the directrix is [*x* = ...].

So the original focus is (2, 0).

But the given focus is (3, 4).

So the focus is under a translation.

Use (2, 0) to find the translation:

(3, 4) = (2 + 1, 0 + 4).

So the translation is

(*x*, *y*) → (*x* + 1, *y* + 4).

Translation of a point

*p* = 2

The directrix is [*x* = ...].

The parabola is under the translation

(*x*, *y*) → (*x* + 1, *y* + 4).

So the equation of the parabola is

(*y* - 4)^{2} = 4⋅2⋅(*x* - 1).

Translation of a function

4⋅2 = 8

So (*y* - 4)^{2} = 8(*x* - 1).

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

Its focus is (2 + 1, 0 + 4),

which is (3, 4).

And its directrix is *x* = -2 + 1,

which is *x* = -1.

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

For a parabola

whose focus is (0, *p*)

and whose directrix is *y* = -*p*,

the equation of the parabola is*x*^{2} = 4*py*.

## Example 3: Focus (0, 2), Directrix *y* = -2, Parabola?

The focus is (0, 2).

And the directrix is *y* = -2.

So *p* = 2.

*p* = 2

And the directrix is [*y* = ...].

Then the equation of the parabola is*x*^{2} = 4⋅2⋅*y*.

4⋅2 = 8

So *y*^{2} = 8*x*.

This is the graph of *x*^{2} = 8*y*.

The points on the parabola are equidistant

from the focus (0, 2) and the directrix *y* = -2.