∪ Parabola Studio

The parabola studio starts with vertex zero zero and focus zero point two five above it. The saved formula sets coefficient a to one divided by four times the vertical focal distance. That gives a equal to one and the curve y equal to x squared. At x one the height is one. At x two it is four. The plotted geometry samples this actual quadratic. The focus is a geometric construction point, not a point lying on the parabola. Move the focus vertically from point two five to point five while leaving the vertex at zero. The coefficient falls from one to one half. At x two the height falls from four to two. A more distant focus therefore widens the parabola. The two bars compare those heights using the saved inverse focal-distance relation. The source rejects focal distances with magnitude below point zero zero one instead of pretending division by zero is a valid quadratic. A vertex at one negative one and a focus at one negative point seven five still has focal distance point two five. Its equation becomes y equals x minus one squared minus one. Expanding gives a one, b negative two and c zero. The roots are zero and two, equally spaced around axis x one. The plotted curve uses the translated vertex form. Dragging changes the geometry; the source then derives the displayed coefficients and roots from it.