Calculator Inputs
Choose one solving mode, enter the values, and submit to compute the equation, graph, focus, directrix, and other properties.
Example Data Table
| Case | Input | Standard Form | Focus |
|---|---|---|---|
| Vertex and p | Vertex (1, -2), p = 3, vertical | (x - 1)² = 12(y + 2) | (1, 1) |
| Vertex and point | Vertex (0, 0), point (4, 2), vertical | x² = 8y | (0, 2) |
| Focus and directrix | Focus (3, 1), directrix x = -1 | (y - 1)² = 8(x - 1) | (3, 1) |
| General coefficients | y = 0.5x² - 2x + 1 | (x - 2)² = 2(y + 1) | (2, -0.5) |
Formula Used
Standard Forms
Vertical axis: (x - h)² = 4p(y - k)
Horizontal axis: (y - k)² = 4p(x - h)
From Focus and Directrix
If directrix is y = d, then h = fx, k = (fy + d) / 2, p = (fy - d) / 2.
If directrix is x = d, then k = fy, h = (fx + d) / 2, p = (fx - d) / 2.
From Vertex and Point
Vertical axis: p = (x₁ - h)² / [4(y₁ - k)]
Horizontal axis: p = (y₁ - k)² / [4(x₁ - h)]
From General Coefficients
For y = ax² + bx + c: h = -b / (2a), k = c - b² / (4a), p = 1 / (4a).
For x = ay² + by + c: k = -b / (2a), h = c - b² / (4a), p = 1 / (4a).
How to Use This Calculator
- Select the solving mode that matches the information you already know.
- Enter the values carefully. Negative values are allowed when the parabola opens left or downward.
- Click Solve Parabola to compute standard form, explicit form, general form, focus, directrix, axis, and latus rectum.
- Review the Plotly graph to confirm the curve orientation and position.
- Use the CSV button for spreadsheet export or the PDF button for a clean report.
FAQs
1. What does p mean in a parabola equation?
p is the signed distance from the vertex to the focus. It also determines the directrix position and how wide or narrow the parabola appears.
2. Which mode should I choose first?
Choose the mode that matches your known data. Use vertex and p for direct construction, vertex and point for fitting, focus and directrix for geometry, or general coefficients for quadratic equations.
3. Can this calculator solve both vertical and horizontal parabolas?
Yes. The tool handles vertical and horizontal axes. It automatically adjusts the standard form, explicit form, focus, directrix, domain, and graph orientation.
4. Why is my point rejected in vertex and point mode?
The point cannot create a zero denominator. For vertical parabolas, its y value cannot equal the vertex y. For horizontal parabolas, its x value cannot equal the vertex x.
5. What does the general coefficients mode solve?
It solves equations written as y = ax² + bx + c or x = ay² + by + c. From those coefficients, the calculator finds vertex form and other parabola properties.
6. How is the directrix shown in the graph?
The directrix is plotted as a dashed line. Its position is calculated from the solved vertex and p value, so it always matches the final parabola model.
7. Why does the curve sometimes look wider or narrower?
The shape depends on the magnitude of p. Larger absolute p values produce wider parabolas, while smaller absolute p values produce narrower curves.
8. Can I export the solved result?
Yes. The calculator includes CSV export for tables and PDF export for the solved result panel, including the graph and property summary.