Parabola Equation Calculator

Find the standard form equation of a parabola when you know its vertex and directrix. Visualize the parabola and easily copy the equation.

Vertex X (h):

Vertex Y (k):

Directrix Equation (y=c or x=c):

Parabola Equation:

Parabola Visualization

Understanding Parabola Equation from Vertex and Directrix

A parabola is a U-shaped curve that can be defined geometrically. One definition involves a point called the vertex and a line called the directrix. The vertex is the point where the parabola makes its sharpest turn, and the directrix is a line that is a certain distance away from the vertex.

For a parabola opening upwards or downwards, the equation is in the form $$ (x-h)^2 = 4p(y-k) $$ , where $$ (h, k) $$ is the vertex and $$ p $$ is the distance from the vertex to the focus (and also from the vertex to the directrix). If $$ p > 0 $$ , the parabola opens upwards; if $$ p < 0 $$ , it opens downwards. The directrix is a horizontal line given by $$ y = k - p $$ .

Similarly, for a parabola opening rightwards or leftwards, the equation is $$ (y-k)^2 = 4p(x-h) $$ . If $$ p > 0 $$ , it opens rightwards; if $$ p < 0 $$ , it opens leftwards. The directrix is a vertical line given by $$ x = h - p $$ .

This calculator helps you find the standard equation of a parabola given the vertex $$ (h, k) $$ and the directrix equation (either $$ y = c $$ or $$ x = c $$ ). Use it to quickly determine the equation and visualize the parabola.