Dynamic Construction and Reflection Properties of Parabolas

This GeoGebra example dynamically demonstrates the definition of a parabola, the relationship between its focus and directrix, and how points on the curve are constructed using arithmetic mean and reflection principles. Core mathematical concepts include the standard equation of a parabola, the focus-directrix definition, the reflective property (rays parallel to the axis reflect through the focus), and parametric curves. Construction highlights: the Parabola command generates the curve, while If, Segment, and slider-like parameters (hBound, kBound, vertex, p, option) control the vertex and opening direction. The focus and directrix are defined via objects focus and direct, and a button (button1) toggles display modes. Points A, B, and i illustrate dynamic loci. Suitable for high school analytic geometry or introductory calculus, this interactive tool helps students visualize the geometric essence of conic sections and their parametric representation.