Simple sketch to support students in developing an understanding of parameterized curves. You can move the parameter t maually or press play, with the trace on or off.
Change the functions for the x- and y-coordinates and observe the effects.

The classic parametric equations to start with is x(t)=cos(t) and y(t)=sin(t). Try changing the phase, period or amplitude of those functions and observing the effects. Or have x(t) and y(t) both be polynomials in t. How could you predict the results? Another classic task is to parameterize a curve. Take the simple y=x2. Can you parameterize it so that the point accelerates going towards the origin and decelerates going away? So that the point comes from the right instead of from the left?