Parabola Focus & Directrix Calculator – Plot & Properties
A parabola is the set of all points whose distance to a fixed point
(the focus) equals its distance to a fixed line (the directrix).
This tool treats axis-aligned parabolas (no rotation, so there is no \(xy\) term).
1) Standard focus–directrix forms
The most geometry-friendly (standard) forms are:
-
Vertical axis (opens up/down):
\[
(x-h)^2 = 4p(y-k)
\]
Vertex \(V=(h,k)\), focus \(F=(h,k+p)\), directrix \(y=k-p\), axis \(x=h\).
-
Horizontal axis (opens left/right):
\[
(y-k)^2 = 4p(x-h)
\]
Vertex \(V=(h,k)\), focus \(F=(h+p,k)\), directrix \(x=h-p\), axis \(y=k\).
The parameter \(p\) is a signed distance from the vertex to the focus:
• If \(p>0\), the parabola opens up (vertical) or right (horizontal).
• If \(p<0\), it opens down (vertical) or left (horizontal).
2) Vertex (function) forms and the link \(a=\dfrac{1}{4p}\)
Divide the standard forms by \(4p\). You get the vertex forms:
-
Vertical:
\[
y = a(x-h)^2 + k
\quad\text{with}\quad
a=\frac{1}{4p}
\]
-
Horizontal:
\[
x = a(y-k)^2 + h
\quad\text{with}\quad
a=\frac{1}{4p}
\]
So if you know \(a\), you immediately get:
\[
p=\frac{1}{4a}
\]
and then focus/directrix follow from the standard form rules above.
3) Polynomial form → vertex, focus, directrix (completing the square)
A vertical parabola can also be written:
\[
y=ax^2+bx+c,\quad a\ne 0
\]
Completing the square gives the vertex directly:
\[
h=-\frac{b}{2a},
\qquad
k=c-\frac{b^2}{4a}
\]
and the same parameter relation holds:
\[
p=\frac{1}{4a}
\]
Then:
\[
F=(h,k+p),\quad \text{directrix } y=k-p,\quad \text{axis } x=h.
\]
For a horizontal parabola:
\[
x=ay^2+by+c,\quad a\ne 0
\]
the vertex is
\[
k=-\frac{b}{2a},
\qquad
h=c-\frac{b^2}{4a}
\]
and again \(p=\dfrac{1}{4a}\). Then:
\[
F=(h+p,k),\quad \text{directrix } x=h-p,\quad \text{axis } y=k.
\]
4) Latus rectum (a useful extra property)
The latus rectum is the chord through the focus perpendicular to the axis.
In standard form, its endpoints are easy:
-
Vertical: endpoints \((h\pm 2p,\;k+p)\)
-
Horizontal: endpoints \((h+p,\;k\pm 2p)\)
The latus rectum length is:
\[
|4p|
\]
5) Worked example
Example input:
\[
y=\frac{x^2}{8}
\]
This matches \(y=a(x-h)^2+k\) with \(a=\frac18\), \(h=0\), \(k=0\). Then:
\[
p=\frac{1}{4a}=\frac{1}{4\cdot \frac18}=2
\]
So:
\[
V=(0,0),\quad F=(0,2),\quad \text{directrix } y=-2,\quad \text{axis } x=0.
\]
6) Common pitfalls
-
Sign of \(p\): If the parabola opens downward/left, then \(p<0\).
The directrix is always on the opposite side of the vertex from the focus.
-
Non-rotated assumption: This calculator is for axis-aligned parabolas.
If an \(xy\) term is present, the conic is rotated and requires a different method.
-
Degenerate case: If \(a=0\) (or \(p=0\)), the equation is not a parabola.
