Calculator Input
Example Data Table
| Case | Equation | Focus | Directrix | Axis | Opening |
|---|---|---|---|---|---|
| Vertical vertex form | (x - 2)^2 = 8(y - 1) | (2, 3) | y = -1 | x = 2 | Upward |
| Horizontal vertex form | (y + 1)^2 = -12(x - 4) | (1, -1) | x = 7 | y = -1 | Leftward |
| Quadratic y-form | y = 0.5x^2 - 2x + 3 | (2, 1.5) | y = 0.5 | x = 2 | Upward |
| Quadratic x-form | x = -0.25y^2 + y + 5 | (5, 2) | x = 7 | y = 2 | Leftward |
Formula Used
Vertical axis parabola
(x - h)^2 = 4p(y - k)
Focus: (h, k + p)
Directrix: y = k - p
Axis of symmetry: x = h
Horizontal axis parabola
(y - k)^2 = 4p(x - h)
Focus: (h + p, k)
Directrix: x = h - p
Axis of symmetry: y = k
From quadratic form y = ax^2 + bx + c: vertex x-coordinate is h = -b / (2a), vertex y-coordinate is k = ah^2 + bh + c, and p = 1 / (4a).
From quadratic form x = ay^2 + by + c: vertex y-coordinate is k = -b / (2a), vertex x-coordinate is h = ak^2 + bk + c, and p = 1 / (4a).
Latus rectum length: |4p|. The sign of p determines opening direction.
How to Use This Calculator
- Select the equation style that matches your given parabola data.
- Enter the known coefficients or vertex parameters.
- Set decimal precision and add any preferred unit label.
- Press Calculate Focus to generate focus, vertex, directrix, axis, and latus rectum.
- Review the graph, compare with the example table, and download CSV or PDF if needed.
FAQs
1. What does the focus of a parabola represent?
The focus is a fixed point used in the parabola definition. Every point on the curve is equally distant from the focus and the directrix.
2. What is the role of p in the standard equation?
The value p is the signed distance from the vertex to the focus. It also determines the directrix position and the direction in which the parabola opens.
3. Why can p be negative?
A negative p means the parabola opens downward in vertical form or leftward in horizontal form. The sign carries direction, while the magnitude gives focal length.
4. Can this calculator work from expanded quadratic equations?
Yes. Use either y = ax^2 + bx + c or x = ay^2 + by + c. The calculator converts the equation into vertex-based geometry automatically.
5. What is the directrix?
The directrix is a fixed line paired with the focus. A parabola consists of all points whose distances to that line and to the focus are equal.
6. What is the latus rectum?
The latus rectum is the focal chord perpendicular to the axis of symmetry and passing through the focus. Its length equals |4p|.
7. Why is a equal to zero invalid in quadratic form?
If a becomes zero, the equation is no longer quadratic. Without the squared term, the graph is a line, not a parabola.
8. Does the graph help verify the answer?
Yes. The graph shows the parabola, focus, vertex, and directrix together. It helps confirm the opening direction, symmetry axis, and geometric placement visually.