Variations of a Function: Interactive Tangent and Normal Constructions

① This GeoGebra example demonstrates how tangent lines, normal lines, and associated geometric objects vary along the graph of a function, providing a dynamic visualization of local function behavior. ② Core mathematical concepts include derivatives (slope of tangent), tangent function, normal lines, vectors, area, volume, and geometric transformations of function graphs, integrating calculus, vector geometry, and geometric construction. ③ Construction highlights: based on the summary, the example uses GeoGebra commands such as Tangent, OrthogonalLine, Circle, Intersect, Segment, Polygon, Vector, Textfield, and Ray, with numerous objects (t, a, b, c, f, T1, T3, A, T4, etc.) forming an interactive network of geometric relations. ④ Suitable for high school to early college students to explore the geometric meaning of derivatives and rates of change, and for teachers to use as a classroom demonstration or for student-led inquiry.