Parabola Properties Calculator – Focus, Directrix & Vertex
A parabola is the set of points whose distance to a fixed point (the focus)
equals its distance to a fixed line (the directrix).
This calculator focuses on axis-aligned parabolas (no rotation / no \(xy\) term).
1) Standard focus–directrix forms
With vertex \(V=(h,k)\) and signed parameter \(p\):
-
Vertical axis:
\[
(x-h)^2 = 4p(y-k)
\]
Focus \(F=(h,k+p)\), directrix \(y=k-p\), axis of symmetry \(x=h\).
Opens upward if \(p>0\), downward if \(p<0\).
-
Horizontal axis:
\[
(y-k)^2 = 4p(x-h)
\]
Focus \(F=(h+p,k)\), directrix \(x=h-p\), axis of symmetry \(y=k\).
Opens right if \(p>0\), left if \(p<0\).
2) Vertex (function) forms
Dividing the standard forms by \(4p\) gives the vertex forms:
-
Vertical:
\[
y = \frac{1}{4p}(x-h)^2 + k
\]
This matches \(y=a(x-h)^2+k\) with \(a=\dfrac{1}{4p}\).
-
Horizontal:
\[
x = \frac{1}{4p}(y-k)^2 + h
\]
This matches \(x=a(y-k)^2+h\) with \(a=\dfrac{1}{4p}\).
3) From a quadratic polynomial to vertex, focus, directrix
For a vertical parabola:
\[
y=ax^2+bx+c,\quad a\ne 0.
\]
Completing the square gives:
\[
\begin{aligned}
y &= a\left(x^2+\frac{b}{a}x\right)+c \\
&= a\left[\left(x+\frac{b}{2a}\right)^2-\left(\frac{b}{2a}\right)^2\right]+c \\
&= a(x-h)^2 + k,
\end{aligned}
\]
where
\[
h=-\frac{b}{2a},\qquad
k=c-\frac{b^2}{4a}.
\]
Then \(a=\dfrac{1}{4p}\Rightarrow p=\dfrac{1}{4a}\), and the focus/directrix follow from Section 1.
The same steps apply to horizontal parabolas written as \(x=ay^2+by+c\) (swap x and y roles).
4) Latus rectum and focal length
The latus rectum is the chord through the focus perpendicular to the axis.
Its endpoints are:
-
Vertical: \((h\pm 2p,\ k+p)\)
-
Horizontal: \((h+p,\ k\pm 2p)\)
Its length is \(|4p|\). The focal length (vertex-to-focus distance) is \(|p|\).
Example
Given \(x=\dfrac{y^2}{4p}\) with \(p=3\), rewrite as:
\[
y^2 = 12x.
\]
This is a horizontal parabola with vertex \((0,0)\) and \(p=3\). Therefore:
\[
\begin{aligned}
F &= (3,0),\\
\text{Directrix} &: x=-3,\\
\text{Axis} &: y=0.
\end{aligned}
\]
