Square Root Function Example 2: Geometric Construction and Optimization

①This example demonstrates the square root function combined with geometric constructions, using right triangles, line segments, and special points to illustrate the relationship between the function graph and geometric objects, while addressing optimization problems. ②Core mathematical concepts include the square root function, properties of right triangles, segment lengths, special points (such as midpoints and feet of perpendiculars), continuity, and optimization. ③Construction details: Based on the summary, objects g, h, k, h_1, k_1, f, f_1, text1, text2, and parameters a, b, c, d are used. Likely, functions f and f_1 are defined, line segments and right triangles are drawn, and text labels annotate key points or values, possibly with sliders or parameters to explore extrema. ④This is suitable for high school or pre-calculus students learning function graphs, geometry-algebra connections, and optimization, as well as for teachers demonstrating dynamic geometry and function relationships in class.