How this calculator works
An axis-aligned parabola is the set of points equally distant from a focus and a directrix. Completing the square converts a quadratic equation to vertex form. With independent coordinate u and dependent coordinate v, v-kv = a(u-hu)^2 and p=1/(4a). Positive p opens toward increasing v; negative p opens the other way. Three-point mode fits a quadratic only when all independent coordinates differ and the resulting quadratic coefficient is nonzero.
Formula
(u-hu)^2 = 4p(v-kv); focus=(hu,kv+p); directrix v=kv-p; latus rectum length=4|p|. For v=a u^2+b u+c, hu=-b/(2a) and kv=c-b^2/(4a).
Worked example
For y = x^2, the vertex is (0, 0), p = 1/4, the focus is (0, 0.25), the directrix is y = -0.25 and the latus rectum length is 1.
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Frequently asked questions
Can it handle a rotated parabola?
No. Choose a vertical or horizontal axis. A general rotated conic with an xy term needs a different transformation.
Why must the focus align with the vertex?
For the chosen orientation, the axis through the vertex is vertical or horizontal. An off-axis focus would define a rotated parabola.
Is a negative coefficient valid?
Yes. It reverses the opening direction while keeping the same geometric definitions.