General Parabola with Vertex at Origin

This example demonstrates the general equation and geometric construction of a parabola whose vertex is located at the origin. The core mathematical concept is the definition of a parabola as the locus of points equidistant from a fixed point (focus) and a fixed line (directrix), leading to the standard forms y²=4px or x²=4py. The construction uses GeoGebra objects such as Focus, Directrix, and Vertex to define the parabola, with parameters d, c, and e controlling the focus coordinates, directrix equation, and overall shape. By adjusting these sliders, users can dynamically observe how the parabola's opening direction, width, and position change. This resource is suitable for high school analytic geometry lessons or introductory college mathematics, helping students visualize the relationship between the parabola's definition and its equation parameters, and explore conversions between general and standard forms.