# Parabola: Equation

How to use a parabola equation to find the focus and the directrix (and vice versa): definition, formula, 6 examples, and their solutions.

## Definition

A parabola is the set of the points

that are equidistant

from the focus (blue point)

and the directrix (dashed line).

## Formulay^{2} = 4px

The black curve is the graph of a parabola

y^{2} = 4px.

For y^{2} = 4px,

the focus is (p, 0)

and the directrix is x = -p.

## ExampleFocus

Change the parabola to y^{2} = 4px form.

y^{2} = 4⋅2⋅x

y^{2} = 4⋅2⋅x

Then the focus is

(2, 0).

So

(2, 0)

is the answer.

## ExampleDirectrix

You just found that

the given equation y^{2} = 8x is

y^{2} = 4⋅2⋅x.

y^{2} = 4⋅2⋅x

Then the directrix is

x = -2.

So

x = -2

is the answer.

## ExampleParabola Equation

The focus is (3, 0).

And the directrix is x = -3.

Then the equation of the parabola is

y^{2} = 4⋅3⋅x.

4⋅3⋅x = 12x

So

y^{2} = 12x

is the answer.

## Formulax^{2} = 4py

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

the focus is (0, p)

and the directrix is y = -p.

## ExampleFocus

Change the parabola to x^{2} = 4py form.

Switch both sides.

y = 4⋅[1/4]⋅y

x^{2} = 4⋅[1/4]⋅y

Then the focus is

(0, 1/4).

So

(0, 1/4)

is the answer.

## ExampleDirectrix

You just found that

the given equation y = x^{2} is

x^{2} = 4⋅[1/4]⋅y.

x^{2} = 4⋅[1/4]⋅y

Then the directrix is

y = -1/4.

So

y = -1/4

is the answer.

## ExampleParabola Equation

The focus is (0, 2).

And the directrix is y = -2.

Then the equation of the parabola is

x^{2} = 4⋅2⋅y.

4⋅2⋅y = 8y

So

x^{2} = 8y

is the answer.