The Equation of a Parabola

The Equation of a Parabola — Moosa Academy

A parabola is the set of all points that are equally far from a fixed point — the focus — and a fixed line — the directrix. That balance between a point and a line produces the single open curve behind projectile paths, satellite dishes and headlight reflectors.

Concept The defining property

For every point

P

on the curve, the distance to the focus equals the perpendicular distance to the directrix. The vertex sits exactly halfway between them.

The focus always lies inside the curve and the directrix outside it, on opposite sides of the vertex — a useful sanity check on any answer.

Concept The two standard forms

Vertical — opens up or down:

(x - h)^2 = 4p(y - k)

Horizontal — opens right or left:

(y - k)^2 = 4p(x - h)

In both, $(h,\ k)$ is the vertex and $|p|$ is the distance from the vertex to the focus — and equally, from the vertex to the directrix. Which variable is squared tells you the orientation: square the $x$ and it opens vertically; square the $y$ and it opens sideways.

Note Reading every feature off the equation

Feature	Vertical (x squared)	Horizontal (y squared)
Vertex	(h, k)	(h, k)
Opens	Up if p > 0, down if p < 0	Right if p > 0, left if p < 0
Focus	(h, k + p)	(h + p, k)
Directrix	y = k − p	x = h − p
Axis of symmetry	x = h	y = k

Notice the symmetry in the last two rows: the focus is $p$ units one way from the vertex, the directrix $p$ units the other way. Unlike the ellipse and hyperbola, there is no $a$ , $b$ or $c$ here, and the eccentricity is fixed at exactly 1.

Example A parabola at the origin

Analyse $x^2 = 8y$ .

The

x

is squared, so this opens vertically; the vertex is at

(0,\ 0)

Matching

4p = 8

gives

p = 2

. Since

p > 0

, it opens upward.

Focus:

(0,\ 0 + 2) = (0,\ 2)

Directrix:

y = 0 - 2

, that is

y = -2

⟹ vertex (0, 0), focus (0, 2), directrix y = −2, axis x = 0

Example A shifted parabola opening downward

Analyse $(x - 3)^2 = -12(y + 1)$ .

Vertex

(3,\ -1)

— the

(y + 1)

means

k = -1

4p = -12 \Rightarrow p = -3

. The negative sign means it opens downward.

Focus:

(3,\ -1 + (-3)) = (3,\ -4)

— below the vertex, as expected.

Directrix:

y = -1 - (-3) = 2

— above the vertex, on the opposite side.

⟹ vertex (3, −1), focus (3, −4), directrix y = 2, axis x = 3

The focus below and directrix above is exactly the arrangement you should expect for a downward-opening curve.

Example Completing the square

Convert $x^2 - 4x - 8y + 20 = 0$ to vertex form and find its focus.

Keep the

x

terms together and move everything else across:

x^2 - 4x = 8y - 20

Half of −4 is −2, and

(-2)^2 = 4

. Add 4 to both sides:

x^2 - 4x + 4 = 8y - 16

Factor the left and the right:

(x - 2)^2 = 8(y - 2)

Vertex

(2,\ 2)

;

4p = 8 \Rightarrow p = 2

, so it opens upward.

⟹ focus (2, 4), directrix y = 0

Check it on a point: at $x = 6$ the curve gives $y = 4$ . The distance from $(6,\ 4)$ to the focus $(2,\ 4)$ is 4, and its distance down to the directrix $y = 0$ is also 4 — the defining property holds.

Note The reflection property

Any ray arriving parallel to the axis reflects off a parabola straight through the focus — and the process runs equally well in reverse. This one fact explains nearly every practical use of the shape.

Satellite dishes and radio telescopes — incoming parallel signals concentrate on a receiver at the focus.

Headlights and torches — a bulb at the focus sends out a parallel, focused beam.

Projectile motion — a thrown object under gravity traces a parabolic arc.

Suspension bridges — a cable carrying an evenly distributed load hangs in a parabola.

Note Mistakes to avoid

Reading the coefficient as

p

when it is actually

4p

— always divide by 4 first.

Ignoring the sign of

p

, which is what decides the direction of opening.

Putting the focus and directrix on the same side of the vertex; they are always opposite.

Mixing up the vertex coordinates:

(y + 1)

means

k = -1

, not

+1

Confusing the two orientations — check which variable carries the square.

Summary

A parabola is the set of points equally distant from a focus and a directrix, with the vertex halfway between them.
(x − h)² = 4p(y − k) opens vertically; (y − k)² = 4p(x − h) opens horizontally — whichever variable is squared names the orientation.
The focus sits p units from the vertex and the directrix p units the other way; the sign of p gives the direction of opening.
From general form, complete the square on the squared variable to recover the vertex form.