# 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

### Definition

A parabola is the set of the points

that are equidistant

from the focus (blue point)

and the directrix (dashed line).

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

### Formula

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.

## Example 1: Focus

### Example

### Solution

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.

## Example 2: Directrix

### Example

### Solution

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.

## Example 3: Equation

### Example

### Solution

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.

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

### Formula

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

the focus is (0, p)

and the directrix is y = -p.

## Example 4: Focus

### Example

### Solution

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.

## Example 5: Directrix

### Example

### Solution

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.

## Example 6: Equation

### Example

### Solution

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.