Geometric Construction of the Tangent Function

This example demonstrates the key shape features of the tangent function by using geometric relationships and a ratio-driven construction. With a suitable parameterization, it turns a right-triangle (or right-triangle-like) configuration into a mapping from geometric lengths to function values, producing the characteristic tan(x) curve behavior, including its repeating pattern and vertical asymptote structure. The core mathematical ideas are trigonometric ratio interpretation (tan as opposite/adjacent), function graphing from geometric constraints, and the way geometry encodes functional dependence. Construction highlights from the provided GeoGebra summary: objects d, a, c, b are used to define the essential points/segments that set up the ratio reference; T1, T3, T4 perform the required geometric transformations/placements to obtain the correct tangent-related configuration; and f is used to generate the final function and its graph. Suitable for high-school trigonometry and function visualization lessons, as well as for learners who want a strong geometric intuition behind tangent graphs.