Parabola Input Panel
Choose one input form, set the graph window if needed, and calculate all major parabola properties.
Formula Used
Standard form
y = ax² + bx + c
Vertex coordinates
h = -b / (2a) and k = f(h)
Vertex form
y = a(x - h)² + k
Discriminant and roots
D = b² - 4ac, then x = (-b ± √D) / (2a)
Focus and directrix
For y = a(x - h)² + k, p = 1 / (4a), focus is (h, k + p), directrix is y = k - p.
Latus rectum length
|4p| = |1/a|
How to Use This Calculator
- Pick the input form that matches your known parabola data.
- Enter the values for coefficients, vertex terms, or roots.
- Set custom graph limits only when you want a fixed view.
- Choose the number of plotted points for curve smoothness.
- Press Calculate Parabola to display results under the header.
- Review equations, focus, directrix, axis, roots, and sample points.
- Use the CSV or PDF buttons to export the calculation report.
Example Data Table
Example based on y = x² - 4x + 3.
| x | y | Notes |
|---|---|---|
| 0 | 3 | Y-intercept |
| 1 | 0 | First x-intercept |
| 2 | -1 | Vertex x-value |
| 3 | 0 | Second x-intercept |
| 4 | 3 | Symmetric with x = 2 |
FAQs
1. What does the parabola calculator solve?
It calculates the vertex, focus, directrix, roots, axis of symmetry, latus rectum length, y-intercept, range, and graph from common vertical parabola forms.
2. Which forms can I enter?
You can enter standard form, vertex form, or intercept form. The calculator converts your selection into equivalent forms automatically.
3. Why must a not be zero?
When a equals zero, the x² term disappears. The equation becomes linear, so the curve is no longer a parabola.
4. What happens when the discriminant is negative?
The parabola has no real x-intercepts. The calculator shows complex roots and still reports the vertex, focus, graph, and other properties.
5. How is the focus found?
First convert to vertex form. Then compute p = 1/(4a). The focus lies p units above or below the vertex, depending on opening direction.
6. What does the graph window control?
It sets the horizontal x-range used for plotting and sample points. Leave it blank to let the calculator choose a reasonable view automatically.
7. Can I export the results?
Yes. CSV export saves summary values and generated points. PDF export creates a neat report with the key parabola properties and points table.
8. Does this calculator cover sideways parabolas?
No. This version is designed for vertical parabolas written with y as a function of x. Sideways forms need a different equation setup.