Stretching and Shifting a Circle into an Ellipse

①This example demonstrates how a standard circle can be transformed into an ellipse through horizontal/vertical stretching (scaling) and translation, visually revealing the geometric relationship between the two conic sections. ②Core mathematical concepts include parametric equations of circles and ellipses, coordinate transformations (translation and scaling), general quadratic curve forms, and the roles of parameters a, b, h, k in controlling shape and position. ③Construction: Using GeoGebra's CurveCartesian command, parameters a and b control the semi-major and semi-minor axes, h and k shift the center, and c may adjust the stretch ratio or angle; sliders allow real-time exploration of how the circle morphs into an ellipse and moves. ④Suitable for high school analytic geometry or introductory college mathematics, this interactive tool helps students understand parametric representation of conics and coordinate transformations, and serves as a dynamic teaching aid.