Exploration of Power Function Graphs, Tangents, and Equilateral Triangles

This applet demonstrates the graphs of power functions and their connections to geometry and calculus. Users can dynamically observe the behavior of the function f(x) and explore tangent lines at specific points on the curve. The construction incorporates equilateral triangles and a sequence of discrete points, integrating the function graph, slope calculations, and geometric properties into a unified visual model. By interacting with buttons and parameter controls, users can vary the exponent of the power function and watch in real time how the graph shape changes, how tangent lines evolve, and how the equilateral triangle configuration responds. This interactive approach vividly illustrates the relationships among sequences, calculus concepts such as derivatives and rates of change, and coordinate geometry. The applet serves as an effective teaching aid for helping students intuitively grasp the geometric meaning of derivatives and understand how function graphs are constructed and analyzed. It bridges algebraic manipulation, numerical exploration, and geometric visualization, making abstract calculus concepts more concrete and accessible for learners studying power functions and their properties.