3D Surface and Geometric Construction Visualization of Power Function x^y

Title (EN): 3D Surface and Geometric Construction Visualization of Power Function x^y Title (ZH): 幂函数 x^y 的三维曲面与几何构造可视化 Full English Description: This GeoGebra applet visualizes the three‑dimensional surface graph of the power function \(x^y\) together with a set of auxiliary geometric constructions. Using the commands Surface, Plane, and CurveCartesian, it builds the spatial surface that represents all points \((x,\,y,\,z)\) satisfying \(z = x^y\). The surface is complemented by Polygon and PolyLine objects that draw contour lines, grid curves, and reference planes, making the curvature and growth behavior easier to follow. Interactive parameters named showR and showTheta switch between Cartesian and polar/parametric views of the surface, allowing students to observe how the same function appears under different coordinate systems. The list objects listx and listy generate discrete sequences of base and exponent values; these sequences are used to plot sampled points and to animate the surface as the base or exponent varies, so learners can directly compare the effect of changing the input parameters. The applet combines three‑dimensional function plotting, solid‑geometry construction, and parametric representation in a single workspace. It is designed for high‑school advanced mathematics or first‑year calculus courses, where teachers can use it to demonstrate properties of power functions—such as continuity, asymptotic behavior, and monotonicity—for both positive and negative bases and exponents. Students can explore the graph by dragging sliders, toggling visibility of auxiliary curves, and watching the animated point‑cloud evolution, thereby developing an intuitive understanding of how the exponent influences the shape of the surface. The combination of static geometric constructions and dynamic parameter control makes this applet a versatile tool for both classroom demonstration and independent discovery learning in real analysis and multivariable calculus.