How to Use the Parabola Calculator
This calculator takes the coefficients a, b, c of the quadratic equation y=ax²+bx+c and instantly returns the three values that define the parabola. The vertex is the turning point of the parabola, found with x=-b/(2a), y=c-b²/(4a), and it marks the curve's maximum or minimum value.
The focus and directrix are the geometric definition of a parabola itself — every point on the curve is equidistant from the focus and the directrix. This property is why parabolas are widely used in satellite dishes, flashlight reflectors, and solar collectors: they focus signals or heat onto a single point, or spread light from a single point into parallel rays.
This calculator computes the focus as (vertex x, vertex y + 1/4a) and the directrix as y = vertex y - 1/4a. If a is 0, the quadratic term disappears and the curve becomes a straight line rather than a parabola, so the calculator immediately shows a guidance message instead of computing a result.
Frequently Asked Questions
Every point on a parabola is equidistant from the focus and the directrix. This property is why parabolas are used in satellite dishes, car headlights, and solar collectors — anything that needs to focus signals or light onto a single point.
When a is 0, the quadratic term disappears and the equation becomes the straight line y=bx+c, which is no longer a parabola. The vertex, focus, and directrix formulas all involve dividing by a, so they're undefined when a=0.