Horizontal and Vertical Transformations: Exploring af(b(x-c))+d

① This interactive GeoGebra example dynamically demonstrates how parameters a, b, c, and d in the general function y = af(b(x-c))+d affect the graph through horizontal and vertical transformations, including stretching, compressing, shifting, and reflection. ② Core mathematical concepts include affine transformations of function graphs, amplitude, period, phase shift, vertical translation, and mirror effects caused by sign changes. ③ Construction highlights: using GeoGebra commands If and Text to create dynamic labels, defining sliders for parameters a, b, c, d, constructing function objects g and f, and using text objects like gtext and ftext to display the transformation formula in real time, with vscale and hscale controlling axis scaling for interactive exploration. ④ Suitable for high school and pre-university students learning function transformations, as well as teachers for classroom demonstrations, helping learners understand how parameters alter graph shape and position and fostering visual-mathematical thinking.