Example 1: Parametric curves provide us with a way to use our understanding of functions to describe graphs that fail the vertical line test. We've already done this once in this course! We parameterized the unit circle as follows:[br][br][math]x\left(t\right)=\cos t[/math][br][math]y\left(t\right)=\sin t\text{ }0\le t\le2\pi[/math][br][br]Here there is a sneaky route for eliminating the parameter in the set of parametric equations given above for the unit circle. Instead of solving for [math]t[/math] exploit the Pythagorean Identity for trig: [math]\sin^2\theta+\cos^2\theta=1[/math]

Example 2: Some invertible functions (i.e. functions that pass the horizontal line test) are really hard to invert using the algebraic strategy we developed earlier this semester. Parametric equations help solve this problem. Recall that inverting means switching the roles of [math]x[/math] and [math]y[/math]. [br]Consider the function [math]f\left(x\right)=x^3-2x^2-x+4[/math].[br]Parameterize the graph of this function by letting [math]x\left(t\right)=t[/math] (I call this the dumb parameterization).[br]Then switch the roles of [math]x[/math] and [math]y[/math] to achieve a parameterization of the graph of the inverse of [math]f\left(x\right)[/math].

In the GeoGebra applet above, we are using an animated blue dot to display orientation. The blue dot leaves behind a trace on the curve. Sometimes these traces are very close together and sometimes they are spaced apart. Why? Is there any way to predict by looking at the parametric equations which areas of a curve correspond to closely packed traces and which correspond to widely spaced traces?