Parabola Equation Calculator

Find the equation of a parabola when you know its vertex and focus. Visualize the parabola and get the equation in standard form.

Vertex X (h):

Vertex Y (k):

Focus X:

Focus Y:

Parabola Equation:

Visualization will appear here after calculation.

Understanding Parabola Equations

A parabola is a U-shaped curve that is symmetric about a vertical or horizontal axis. It is defined as the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus).

The standard form of a parabola equation depends on its orientation:

Vertical Parabola: \( (x-h)^2 = 4p(y-k) \), opens upwards if \( p > 0 \) or downwards if \( p < 0 \). Vertex is \( (h, k) \), and focus is \( (h, k+p) \).
Horizontal Parabola: \( (y-k)^2 = 4p(x-h) \), opens right if \( p > 0 \) or left if \( p < 0 \). Vertex is \( (h, k) \), and focus is \( (h+p, k) \).

In this calculator, you input the vertex \( (h, k) \) and the focus. The tool then determines if the parabola is vertical or horizontal and calculates the standard equation. The visualization helps to understand the orientation and shape of the parabola based on your inputs.

Use this tool to quickly find the equation of a parabola for your math problems, homework, or educational purposes.