Parabola Calculator

About Parabola Calculator

Parabola vertex form: y = a(x-h)² + k. Focus is at (h, k+1/(4a)). All points equidistant from focus and directrix.

How to use this calculator

1. Identify the parabola form in the calculator input.
2. Enter the vertex coordinates and the value of p, or the equivalent standard form information.
3. Click calculate to get the vertex, focus, and directrix instantly.
4. Use the displayed formula and example to check your setup if needed.

The formula explained

• \(x\) = horizontal coordinate
• \(y\) = vertical coordinate
• \(h\) = vertex x-coordinate
• \(k\) = vertex y-coordinate
• \(p\) = distance from the vertex to the focus and to the directrix
• \(\text{vertex}\) = turning point of the parabola
• \(\text{focus}\) = fixed point inside the parabola
• \(\text{directrix}\) = fixed line used to define the parabola

Step by step method

1. Start by writing the parabola in standard form so you know whether it opens up, down, left, or right.
2. Read off the vertex from the form as (h, k).
3. Use p to move from the vertex to the focus in the opening direction.
4. Use p again in the opposite direction to write the directrix.

Worked example

Suppose a parabola has equation \((x-2)^2 = 12(y+1)\).

1. Compare \((x-2)^2 = 12(y+1)\) with \((x-h)^2 = 4p(y-k)\). This gives \(h=2\), \(k=-1\), and \(4p=12\), so \(p=3\).
2. The vertex is \((2,-1)\).
3. Because the parabola opens upward, the focus is \((2,-1+3) = (2,2)\).
4. The directrix is \(y=-1-3=-4\).

Tips and common mistakes

• Check whether the squared term is x or y, because that tells you whether the parabola opens vertically or horizontally.
• Remember that p is signed by direction, but the calculator usually reports the geometric result directly.
• Do not mix up the focus and the directrix. The focus is a point, while the directrix is a line.
• If your equation is not already in standard form, rewrite it first before using the calculator.

Frequently asked questions