Parabola: Equation

A parabola is the set of the points
that are equidistant
from the focus (blue point)
and the directrix (dashed line).

The black curve is the graph of a parabola
y² = 4px.

For y² = 4px,
the focus is (p, 0)
and the directrix is x = -p.

Change the parabola to y² = 4px form.

y² = 4⋅2⋅x

y² = 4⋅2⋅x

Then the focus is
(2, 0).

So (2, 0) is the answer.

You just found that
the given equation y² = 8x is
y² = 4⋅2⋅x.

y² = 4⋅2⋅x

Then the directrix is
x = -2.

So x = -2 is the answer.

The focus is (3, 0).
And the directrix is x = -3.

Then the equation of the parabola is
y² = 4⋅3⋅x.

4⋅3⋅x = 12x

So y² = 12x is the answer.

For the parabola x² = 4py,
the focus is (0, p)
and the directrix is y = -p.

Change the parabola to x² = 4py form.

Switch both sides.

y = 4⋅[1/4]⋅y

x² = 4⋅[1/4]⋅y

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

So (0, 1/4) is the answer.

You just found that
the given equation y = x² is
x² = 4⋅[1/4]⋅y.

x² = 4⋅[1/4]⋅y

Then the directrix is
y = -1/4.

So y = -1/4 is the answer.

The focus is (0, 2).
And the directrix is y = -2.

Then the equation of the parabola is
x² = 4⋅2⋅y.

4⋅2⋅y = 8y

So x² = 8y is the answer.