Graph of Quadratic Functions (vertex/axis geometric construction)

This example demonstrates, through geometric construction, how a quadratic function’s graph (a parabola) is shaped by its parameters, emphasizing the vertex location (h, k), the axis of symmetry, and the relationships among characteristic points such as intersections and segment-defined lengths. Key concepts include quadratic functions and their graph properties, vertex form (“a h k form”), symmetry of parabolas, and the geometric interpretation of optimization ideas. Construction details: it uses GeoGebra commands Point, Line, Intersect, Midpoint, OrthogonalLine, Circle, and Segment to place and connect key points, compute intersection points, generate symmetry/auxiliary lines via midpoints and orthogonal lines, and employ circles and segments to encode parameter constraints. The setup also includes labeled elements (text objects) and interactive controls (buttons) to support dynamic exploration. Suitable for high-school to early-undergraduate learners studying function graphs, analytic geometry, and related visualization-based problem solving.