Parabola: Formula

Parabola: Formula

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

Formula: y2 = 4px

For a parabola whose focus is (p, 0) and whose directrix is x = -p, the equation of the parabola is y^2 = 4px.

For a parabola
whose focus is (p, 0)
and whose directrix is x = -p,

the equation of the parabola is
y2 = 4px.

Parabola - Focus

Parabola - Directrix

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

Write an equation of the parabola whose focus and directrix are below. Focus: (3, 0). Directrix: x = -3

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
y2 = 4⋅3⋅x.

4⋅3 = 12

So y2 = 12x.

This is the graph of y2 = 12x.

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?

Write an equation of the parabola whose focus and directrix are below. Focus: (3, 4). Directrix: x = -1

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, 2p
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: x2 = 4py

For a parabola whose focus is (0, p) and whose directrix is y = -p, the equation of the parabola is x^2 = 4py.

For a parabola
whose focus is (0, p)
and whose directrix is y = -p,

the equation of the parabola is
x2 = 4py.

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

Write an equation of the parabola whose focus and directrix are below. Focus: (0, 2). Directrix: y = -2

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
x2 = 4⋅2⋅y.

4⋅2 = 8

So y2 = 8x.

This is the graph of x2 = 8y.

The points on the parabola are equidistant
from the focus (0, 2) and the directrix y = -2.