Parabola Calculator
What is a Parabola?
A parabola is a U-shaped curve that is the set of all points equidistant from a single point (the Focus) and a line (the Directrix).
Every time you throw a ball, water shoots from a fountain, or you look at a satellite dish, you are looking at a parabola.
Standard Form vs. Vertex Form
Parabolas are typically written in one of two ways. Our calculator accepts the Standard Form.
1. Standard Form
This is useful for finding the Y-intercept ($c$).
2. Vertex Form
This is useful for graphing, because $(h, k)$ is the vertex (the peak or valley) of the curve.
The Anatomy of a Parabola
When you calculate a parabola, you are finding four key geometric features:
1. The Vertex (h, k)
This is the turning point. If $a > 0$, the parabola opens Up and the vertex is the minimum. If $a < 0$, it opens Down and the vertex is the maximum.
Formula for h: -b / 2a
2. The Focus
This is a specific point inside the curve. It has a magical property: Any ray entering the parabola parallel to the axis of symmetry will be reflected exactly into the Focus.
Satellite TV dishes are parabolas. The receiver is placed exactly at the Focus point. When radio waves hit the dish, they all bounce and converge onto that one point, amplifying the signal thousands of times.
3. The Directrix
A line outside the curve, perpendicular to the axis of symmetry. It acts as the mathematical anchor for the definition of the parabola.
4. Axis of Symmetry
The vertical line that splits the parabola into two perfect mirror images. It always passes through the Vertex and the Focus.
Calculating the "p" Value (Focal Length)
The distance between the Vertex and the Focus is called the Focal Length ($p$).
Formula: p = 1 / (4a)
- If $a$ is large (e.g., 5), the parabola is skinny, and the focus is close to the vertex.
- If $a$ is small (e.g., 0.1), the parabola is wide, and the focus is far away.