Parabola Equation Calculator: Vertex & Directrix

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.