Rectangle Under a Parabola: Dynamic Area and Optimization

This GeoGebra example demonstrates a rectangle inscribed under a parabola, allowing users to drag a point and observe how the rectangle's area changes. The core mathematical concepts involve quadratic functions, area formulas, optimization, and the interplay between algebraic functions and geometric shapes. The construction uses GeoGebra commands such as Point, OrthogonalLine, Intersect, and Polygon: first define a parabola f, place a point A on it, draw a perpendicular line to the x-axis to get point B, construct symmetric points C and D, and finally form the rectangle poly1 using the Polygon command. By dragging point A or using a slider, learners can visually explore how the area varies and determine the maximum area using properties of quadratic functions. This resource is suitable for upper middle school to high school students, helping them understand quadratic function extrema, function-geometry integration, and serving as an intuitive introduction to optimization ideas in calculus.