Parabolas: Focus, Directrix & Equations

Q: Why does the equation sometimes look like $y = \dfrac{x^2}{4p}$ instead?

That is the same equation solved for $y$. Dividing $x^2 = 4p\,y$ by $4p$ gives $y = \dfrac{x^2}{4p}$. With $p = 2$, for instance, $x^2 = 8y$ and $y = \dfrac{x^2}{8}$ describe the identical curve.

In Algebra 1 a parabola was whatever $y = x^2$ happened to draw. Algebra 2 gives it a definition: a parabola is the set of all points that are the same distance from a fixed point, called the focus, as they are from a fixed line, called the directrix. That definition is not decoration. It is the reason a satellite dish is parabolic, because every signal coming straight down bounces to the focus.

Once you accept the definition, writing the equation is short. The vertex sits halfway between the focus and the directrix, and the distance from the vertex to either one is called $p$ . Find $p$ , note which way the curve opens, and drop the numbers into $(x - h)^2 = 4p(y - k)$ or $(y - k)^2 = 4p(x - h)$ .

Focus, directrix, and the number p

Pick any point on a parabola. Measure straight to the focus, then measure straight down to the directrix. Those two distances are equal, and that is true for every point on the curve. The vertex is the one point where this is easiest to see: it lands exactly halfway between the focus and the directrix.

That half-gap is $p$ . If the focus is $(0, 3)$ and the directrix is $y = -3$ , the vertex is $(0, 0)$ and $p = 3$ , because the vertex is $3$ units from the focus and $3$ units from the directrix. Every equation in this lesson runs on $p$ , so find it first and write it down before touching the formula.

The two orientations

A parabola with a horizontal directrix opens up or down, and its equation is $(x - h)^2 = 4p(y - k)$ , where $(h, k)$ is the vertex. A parabola with a vertical directrix opens left or right, and its equation is $(y - k)^2 = 4p(x - h)$ . The squared variable tells you the orientation: square the $x$ and the curve opens vertically, square the $y$ and it opens horizontally.

A parabola always opens toward its focus and away from its directrix. That fixes the sign of $p$ : positive $p$ opens up or right, negative $p$ opens down or left. The axis of symmetry is the line through the vertex and the focus, so it is vertical for the first form and horizontal for the second.

Reading a parabola backwards

Problems also run the other way: you are handed an equation and asked for the focus and the directrix. Match the equation to a form, read $4p$ off the right side, and divide by $4$ . For $x^2 = 8y$ the vertex is the origin, $4p = 8$ , so $p = 2$ , the focus is $(0, 2)$ and the directrix is $y = -2$ , the same distance on the other side.

You will also see the up or down form solved for $y$ . Dividing $(x - h)^2 = 4p(y - k)$ by $4p$ gives $y - k = \dfrac{(x - h)^2}{4p}$ , so a parabola with its vertex at the origin is $y = \dfrac{x^2}{4p}$ . That is the same equation wearing a different coat, and it is the shape most answer choices are written in.

Worked examples

Example 1: vertex and focus given

A parabola has vertex $(0, 0)$ and focus $(0, 3)$ . Write its equation.

The focus is above the vertex, so it opens up

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

Distance from the vertex to the focus

p = 3

Substitute

h = 0

k = 0

4p = 12

x^2 = 12y

Answer: $x^2 = 12y$

Example 2: vertex and directrix given

A parabola has vertex $(2, -1)$ and directrix $x = -1$ . Write its equation.

The directrix is vertical, so the curve opens left or right

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

Distance from the vertex to the directrix

p = 2 - (-1) = 3

The directrix is on the left, so it opens right and

p

is positive

4p = 12

Substitute

h = 2

k = -1

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

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

Example 3: equation to focus and directrix

For $x^2 = 8y$ , find the focus and the directrix.

Match the up or down form

(x - 0)^2 = 4p(y - 0)

Read

4p

and divide by

4

4p = 8 \Rightarrow p = 2

The focus is

p

above the vertex

(0, 2)

The directrix is

p

below the vertex

y = -2

Answer: Focus $(0, 2)$ , directrix $y = -2$ .

Example 4: focus and directrix, vertex not given

A parabola has focus $(1, 4)$ and directrix $y = -2$ . Write its equation.

The vertex is halfway between them

y = \dfrac{4 + (-2)}{2} = 1

So the vertex is

(1, 1)

Distance from the vertex to the focus

p = 4 - 1 = 3

The focus is above, so it opens up

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

Answer: $(x - 1)^2 = 12(y - 1)$

Try one yourself

Common questions

How do I know which way the parabola opens?

It opens toward the focus and away from the directrix, always. If the focus is above the vertex it opens up; if the directrix is a vertical line to the left of the vertex, the focus is on the right and the curve opens right.

What exactly is p?

$p$ is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. It shows up in the equation as $4p$ , so after you find $p$ you still have to multiply by $4$ before substituting.

Why does the equation sometimes look like

y = \dfrac{x^2}{4p}

instead?

That is the same equation solved for $y$ . Dividing $x^2 = 4p\,y$ by $4p$ gives $y = \dfrac{x^2}{4p}$ . With $p = 2$ , for instance, $x^2 = 8y$ and $y = \dfrac{x^2}{8}$ describe the identical curve.

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