I want to have a conversation about transforming parameterized curves, but before I do I need you to familiarize yourself with the transformations of graphs you ideally learned in your previous classes. Experiment with the applet below and then note the effects of the four constants [math]A,B,C,[/math] and [math]D[/math].

In the GeoGebra applet below I wanted the sliders for the four constants to have the [i]same effect[/i] on the image curve of the path [math]\vec{c}\left(t\right)[/math] as they did on the graph of the function [math]f\left(x\right)[/math] above. Note how I had to write the component functions for [math]\vec{c}\left(t\right)[/math].

So the take-away should be this:[br][list][*]When transforming a graph the transformations of the [math]x[/math] variable were indirect and the transformations of the [math]y[/math] variable were direct. [/*][*]When transforming the image curve of a path the transformations of both variables are direct.[/*][/list]