Calculated Results
The computed output appears here above the form after submission.
Standard Form
Explicit Form
Source Form
Plotly Graph
Sample Points Used for Plotting
| # | x | y |
|---|
Calculator Inputs
Choose an input mode, enter known values, then compute the directrix and related properties.
Example Data Table
These sample cases show how different input modes lead to the directrix, focus, and axis outputs.
| Mode | Input summary | Directrix | Focus | Axis |
|---|---|---|---|---|
| Vertex and p | Vertical, h = 2, k = 1, p = 3 | y = -2 | (2, 4) | x = 2 |
| Vertex and focus | Vertex (0, 0), Focus (0, -2) | y = 2 | (0, -2) | x = 0 |
| Vertex and directrix | Vertex (4, -1), Directrix x = 1 | x = 1 | (7, -1) | y = -1 |
| Expanded equation | x² - 4x - 8y + 12 = 0 | y = 1.5 | (2, 0.5) | x = 2 |
Formula Used
1) Vertical parabola
(x - h)² = 4p(y - k)
Focus: (h, k + p)
Directrix: y = k - p
Axis: x = h
2) Horizontal parabola
(y - k)² = 4p(x - h)
Focus: (h + p, k)
Directrix: x = h - p
Axis: y = k
3) From vertex and focus
If the focus is (h, yf), then p = yf - k and y = k - p.
If the focus is (xf, k), then p = xf - h and x = h - p.
4) From vertex and directrix
If the directrix is y = d, then p = k - d.
If the directrix is x = d, then p = h - d.
5) From expanded form
For A x² + B x + C y + D = 0, rewrite as y = a x² + b x + c, complete the square, then use p = 1 / (4a).
For A y² + B y + C x + D = 0, rewrite as x = a y² + b y + c, complete the square, then use p = 1 / (4a).
6) Extra measures
Focal length from vertex to focus: |p|
Latus rectum length: |4p|
Eccentricity of any parabola: 1
How to Use This Calculator
- Choose the input mode that matches the data you already know.
- Enter vertex values, focus values, directrix values, or equation coefficients.
- Set the number of plot points and graph spread for a denser or wider view.
- Click Calculate Directrix to show results above the form.
- Review the directrix, focus, axis, standard form, and explicit form.
- Inspect the sample plotting points and the generated graph.
- Use the CSV button to export numeric output.
- Use the PDF button to capture the displayed result block.
FAQs
What does the directrix represent in a parabola?
The directrix is a fixed line used with the focus to define the parabola. Every point on the curve is equally distant from the focus and the directrix.
How is p related to the directrix?
The parameter p is the signed distance from the vertex to the focus. The directrix lies the same distance from the vertex on the opposite side.
Can this calculator handle upward, downward, left, and right openings?
Yes. Positive p opens upward or rightward, depending on orientation. Negative p opens downward or leftward, and the graph updates automatically.
What equation forms are supported here?
You can calculate from vertex and p, vertex and focus, vertex and directrix, or an expanded non-rotated quadratic form for vertical or horizontal parabolas.
Why must the focus align with the axis?
For a standard non-rotated parabola, the focus must sit directly above, below, left, or right of the vertex. Otherwise the curve would not match these forms.
What happens when p equals zero?
A parabola cannot exist when p equals zero because the focus and vertex collapse together. The calculator blocks that case as invalid.
Does the calculator return other properties besides the directrix?
Yes. It also returns the focus, vertex, axis, opening direction, latus rectum length, focal length, standard form, explicit form, and graph.
Can I export the computed results?
Yes. Use the CSV button for tabular output and the PDF button for a shareable report image of the result section.