What is vertex form?
y = a(x−h)² + k with vertex at (h,k).
Parabola Calculator
Explore parabola properties from the focus–directrix definition or from vertex form.
Choose parabola input
Use focus & directrix y = d, or vertex-form coefficients a, h, k.
About this calculator
How to use Parabola Calculator, plus the formulas and logic behind the results.
Tool description
A parabola is the set of points equidistant from a focus and a directrix. Vertex form highlights the vertex (h,k) and opening scale a.
How to use
- Select vertex form or focus/directrix mode.
- Enter the required parameters.
- Select Calculate to view key properties.
- Note the focal length p = 1/(4a) in vertex mode.
Formulas & reference
Vertex form: y = a(x − h)² + kp = 1/(4a); focus (h, k+p); directrix y = k − pFocus/directrix: vertex midway; p = fy − vertex_y
Parabola basics
- a > 0 opens up; a < 0 opens down
- Axis of symmetry x = h
- Focal length |p| = |1/(4a)|
- Definition: PF = distance to directrix
Calculation logic
Vertex mode derives focus, directrix, and axis. Focus mode assumes a horizontal directrix y = d and a vertical axis.
Focus/directrix mode requires the focus not to lie on the directrix.
Parabola Calculator FAQ
Common questions about this calculator and how to use the results.
How do you find the focus?
From vertex form, focus is (h, k + 1/(4a)) for a vertical parabola.
What is the directrix?
The line y = k − 1/(4a) in the standard vertical case.
What if a = 0?
Then it is not a parabola; a must be nonzero.
Can the parabola open left/right?
This tool focuses on vertical (up/down) parabolas.
What is p?
The signed distance from vertex to focus.