Parabola — Equation, Focus, Directrix & Graph

A parabola is the set of all points that are the same distance from a fixed point, called the focus, and a fixed line, called the directrix. That one rule explains the parabola equation, where the graph opens, and how to find the focus and directrix from the equation.

A parabola is often drawn as a U-shape, but that picture is only part of the idea. The more useful fact is this: every point on the curve satisfies the same distance condition.

Key Parts of a Parabola

The vertex is the turning point of the parabola. It lies halfway between the focus and the directrix along the axis of symmetry.

The axis of symmetry is the line that cuts the parabola into two mirror halves. If the parabola opens up or down, the axis is vertical. If it opens left or right, the axis is horizontal.

The parabola always opens toward the focus and away from the directrix.

Parabola Equation in Standard Form

If the vertex is at the origin, there are two standard forms.

For a vertical parabola,

$$x^2 = 4py$$

The focus is $(0, p)$ and the directrix is

$$y = -p$$

If $p > 0$, the parabola opens upward. If $p < 0$, it opens downward.

For a horizontal parabola,

$$y^2 = 4px$$

The focus is $(p, 0)$ and the directrix is

$$x = -p$$

If $p > 0$, the parabola opens right. If $p < 0$, it opens left.

The important detail is that the coefficient is $4p$, not $p$.

Shifted Parabola Equations

If the vertex is at $(h, k)$, the forms become

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

and

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

For

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

the parabola has vertex $(h, k)$, focus $(h, k + p)$, and directrix

$$y = k - p$$

For

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

the parabola has vertex $(h, k)$, focus $(h + p, k)$, and directrix

$$x = h - p$$

These formulas assume the equation is already written in one of these standard forms.

Worked Example: Find the Vertex, Focus, and Directrix

Consider

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

Match it to

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

So

$$h = 2, \quad k = -1, \quad 4p = 12$$

which gives

$$p = 3$$

Now the main features are easy to read:

• Vertex: $(2, -1)$
• Axis of symmetry: $x = 2$
• Opening: upward, because $p > 0$
• Focus: $(2, -1 + 3) = (2, 2)$
• Directrix: $y = -1 - 3 = -4$

So the graph is a vertical parabola with vertex at $(2, -1)$, opening upward toward the focus $(2, 2)$.

How to Graph a Parabola Quickly

Start by finding the vertex. Then look at which variable is squared.

If the squared part is $(x - h)^2$, the parabola is vertical. If the squared part is $(y - k)^2$, the parabola is horizontal.

Next, find $p$ from the factor $4p$. This tells you both the opening direction and how far the focus and directrix are from the vertex.

Plot the vertex and focus first, then draw the directrix. Once those three features are in place, the curve is much easier to sketch correctly.

Common Mistakes With Parabolas

Confusing $4p$ with $p$

In

$$(x - h)^2 = 12(y - k)$$

you should read $4p = 12$, so $p = 3$. A lot of errors come from treating $12$ as $p$ directly.

Mixing up the two standard forms

If $x$ is the squared variable, the parabola is vertical. If $y$ is the squared variable, the parabola is horizontal. Swapping those gives the wrong focus and directrix.

Missing the sign

If $p$ is negative, the parabola opens down or left, not up or right. The sign controls direction.

Assuming every parabola has vertex at $(0, 0)$

That is only true for the simplest form. Shifted equations move the vertex away from the origin.

When a Parabola Is Used

Parabolas show up in coordinate geometry, quadratic graphs, and conic sections. They also appear in motion models, such as projectile motion, but only in the idealized case of constant gravity and negligible air resistance.

They matter in applications because a parabola has a reflection property: rays parallel to its axis reflect through the focus in the ideal geometric model. That is why parabolic shapes appear in some dishes, reflectors, and mirrors.

A Simple Way to Remember It

If you forget the formulas, remember the geometry first: a parabola is the set of points equally far from the focus and the directrix. The vertex sits in the middle, and the curve opens toward the focus.

From there, the equations are easier to rebuild instead of memorizing them blindly.

Try a Similar Problem

Try your own version with

$$(y + 3)^2 = -8(x - 1)$$

Find the vertex, focus, directrix, and opening direction before you sketch the graph. Then check whether your focus lies on the side where the parabola opens.