# Parabola: Formula, Focus, Directrix

How to solve parabola problems: definition, formula (equation, focus, directrix), proof, examples, and their solutions.

## Definition

A parabola is the set of the points

that are equidistant

from the focus and the directrix.

## Formula 1

Parabola: *y*^{2} = 4*px*

Focus: (*p*, 0)

Directrix: *x* = -*p*

## Proof: Formula 1

Set [distance between (*x*, *y*) and (*p*, 0)]

= [distance between (*x*, *y*) and *x* = -*p*].

Square both sides.

Left side

Square of a difference

Right side

Square of a sum

Cancel the gray terms on both sides.

Then *y*^{2} = 4*px*.

## Example 1

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

= 4⋅2⋅*x*.

So *p* = 2.

Then the focus is (2, 0).

And the directrix is *x* = -2.

See how the graph of *y*^{2} = 4⋅2⋅*x* looks like.

The graph is determined by

its focus (2, 0)

and its directrix *x* = -2.

## Example 2

Roughly draw the given conditions.

Focus: (3, 4)

Directrix: *x* = -1

The distance between

the focus (3, 4) and the vertex

is *p*.

And the distance between

the vertex and the directrix *x* = -1

is also *p*.

The focus, the vertex, and the diretrix

form a segment

whose length is 2*p*.

So 2*p* = 3 - (-1) = 4.*p* = 2

The focus is not (2, 0).

So the parabola is under a translation.

Translation of a point

So use *p* = 2

to find the changes of *x* and *y*:

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

So the parabola is under the translation

(*x*, *y*) → (*x* + 1, *y* + 4).

Write the equation of the parabola

using the above conditions.

(*y* - 4)^{2} = 4⋅2⋅(*x* - 1)

Translation of a function

See how (*y* - 4)^{2} = 8*x* = 4⋅2⋅(*x* - 1) looks like.

The graph is determined by

its focus (2 + 1, +4)

and its directrix *x* = -2 + 1.

## Formula 2

Parabola: *x*^{2} = 4*py*

Focus: (0, *p*)

Directrix: *y* = -*p*

## Proof: Formula 2

Set [distance between (*x*, *y*) and (0, *p*)]

= [distance between (*x*, *y*) and *y* = -*p*].

Square both sides.

Left side

Square of a difference

Right side

Square of a sum

Cancel the gray terms on both sides.

Then *x*^{2} = 4*py*.

## Example 3

*x*^{2} = *y*

= 4⋅(1/4)⋅*y*.

So *p* = 1/4.

Then the focus is (0, 1/4).

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

See how the graph of *x*^{2} = 4⋅(1/4)⋅*y* looks like.

The graph is determined by

its focus (0, 1/4)

and its directrix *y* = -1/4.

## Example 4

Switch both sides

to set the *x*^{2} term on the left side.

Move the constant term +9 to the right side.

And complete the square on the left side.

Complete the square

Factor the right side.

The parabola is (*x* - 1)^{2} = 4⋅1⋅(*y* - 2).

This is the image of *x*^{2} = 4⋅1⋅*y*

under the translation (*x*, *y*) → (*x* + 1, *y* + 2).

Translation of a function

So the focus is (+1, 1 + 2) = (1, 3).

And the directrix is *y* = -1 + 2, *y* = 1.

See how the graph of (*x* - 1)^{2} = 4⋅1⋅(*y* - 2) looks like.

The graph is determined by

its focus (+1, 1 + 2)

and its directrix *y* = -1 + 2.