Under and to the right of the rectangular graph is a faint extension of the curve. You can grasp it to move the rectangular curve up/down/left/right to show changes in vertical and phase shifts affect the polar graph. The equation also changes as you move the graph. With vertical shifts, students begin to see the source of inner loops and the transition of limacons from circles, through inner loops, to cardioids and dimples. It also shows where the "extra" petals go in roses.
My thanks to several contributors to the NCTM Teaching Mathematics journals over the years for ideas, insights, and Geogebra examples that led me to this combined script.
