Last updated: 2026-07-02

The focus-directrix definition of a parabola

A parabola is the set of all points that are the same distance from a fixed point — the focus — as they are from a fixed line — the directrix. That single balancing condition is what gives the parabola its shape. The turning point of the curve, the vertex, sits exactly halfway between the focus and the directrix.

For a parabola that opens vertically (upward or downward) with vertex $(h, k)$, the standard form is:

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

The number $p$ is the focal length — the signed distance from the vertex to the focus. From it, every feature of the parabola follows:

$\text{focus} = (h,\ k + p) \qquad \text{directrix: } y = k - p \qquad \text{axis: } x = h$

This calculator takes the vertex plus either $p$ or the leading coefficient $a$, and returns the focus, directrix, axis of symmetry, focal length, and latus rectum, with the working shown step by step.

Focal length, focus, and directrix

Because the vertex is midway between the focus and the directrix, the focus and directrix are each a distance $|p|$ from the vertex — on opposite sides. Moving $p$ units along the axis $x = h$ from the vertex lands on the focus; moving $p$ units the other way lands on the directrix line.

$\text{focus} = (h,\ k + p) \qquad \text{directrix: } y = k - p$

The sign of $p$ sets the direction the parabola opens. When $p > 0$ the curve opens upward and the focus sits above the vertex; when $p < 0$ it opens downward and the focus sits below.

The latus rectum

The latus rectum is the chord that passes through the focus perpendicular to the axis of symmetry — the "width" of the parabola measured at the focus. Its length is:

$L = |4p|$

The endpoints of the latus rectum are $(h - 2p,\ k + p)$ and $(h + 2p,\ k + p)$, which is a quick way to plot two extra points at the height of the focus.

The coefficient form

Many textbooks write a vertical parabola as $y = a(x - h)^2 + k$. This is the same curve — just solved for $y$. Comparing it with the standard form shows how the leading coefficient $a$ and the focal length $p$ relate:

$a = \frac{1}{4p} \qquad \Longleftrightarrow \qquad p = \frac{1}{4a}$

A large $|a|$ gives a narrow parabola with a short focal length; a small $|a|$ gives a wide parabola with the focus far from the vertex. Since $a$ and $p$ always share the same sign, either one tells you which way the parabola opens.

Worked example

Problem: Find the focus, directrix, axis of symmetry, and latus rectum of $(x - 1)^2 = 8(y + 2)$.

Step 1 — read the vertex. Matching against $(x - h)^2 = 4p(y - k)$ gives $h = 1$ and $k = -2$, so the vertex is $(1, -2)$.

Step 2 — find the focal length. The coefficient on the right is $4p = 8$, so:

$p = \frac{8}{4} = 2$

Step 3 — find the focus. Move $p = 2$ up the axis from the vertex:

$\text{focus} = (h,\ k + p) = (1,\ -2 + 2) = (1,\ 0)$

Step 4 — find the directrix. Move $p = 2$ the other way:

$y = k - p = -2 - 2 = -4$

Step 5 — axis and latus rectum.

$\text{axis: } x = h = 1 \qquad L = |4p| = |4 \cdot 2| = 8$

So the parabola opens upward with focus $(1, 0)$, directrix $y = -4$, axis $x = 1$, and a latus rectum of length $8$. In coefficient form the same curve is $y = \tfrac{1}{8}(x - 1)^2 - 2$, since $a = \tfrac{1}{4p} = \tfrac{1}{8}$.

Reading the graph

The plot frames the vertex $V$, the focus $F$, and the directrix line $d$ together so you can see the defining balance: pick any point on the curve, and its distance down to the directrix equals its distance across to the focus. The vertex always lands midway between $F$ and $d$, and the whole figure is mirror-symmetric about the axis $x = h$.

Notes and limitations

This calculator handles vertical parabolas of the form $(x - h)^2 = 4p(y - k)$, which open upward or downward. Horizontal parabolas $(y - k)^2 = 4p(x - h)$, which open left or right, swap the roles of $x$ and $y$: their axis is horizontal, the focus is at $(h + p, k)$, and the directrix is the vertical line $x = h - p$. The focal length $p$ must be non-zero — at $p = 0$ the curve degenerates and there is no parabola.