Discovering Function Equations: From Graphs to Algebraic Forms

① This GeoGebra example demonstrates how to derive function equations from geometric figures such as parallelograms, rhombi, polygons, sine curves, and tangent lines, revealing the correspondence between visual shapes and algebraic expressions. ② Core mathematical concepts include function definitions, solving equations, sine function graphs, tangent slopes, coordinates of polygon vertices, and using turning points to identify extrema. ③ Construction highlights: Based on the summary, the TurningPoint command is used to mark critical turning points, with point objects (A, P, J, etc.) and function objects (f, g) created; text objects (text1–text4) display equations or explanations. Objects include k, a, h, k_1, c, b, d, p, C, j, J, P, and F, forming an interactive environment for exploring graphs and equations. ④ Suitable for upper secondary to pre-university students, this example supports teaching on functions and coordinate geometry, and serves as a resource for inquiry-based lessons where learners connect graphical features with algebraic representations.