Suppose we have two functions u = f(t) and v = g(t) defined on the same domain D. In this context we can call the variable t as the parameter and the equations u = f(t) v = g(t) the parametric equations.
Graph of Parametric Equations
In a given coordinate system the graph of the parametric equations u = f(t) v = g(t) on their common domain D is the set of all the points (f(t), g(t)) for t in D.
Such a graph is also called the parametric curve represented by the given parametric equations or by the function p(t) = (f(t), g(t)) for t in D (Note that the range of p(t) is composed of ordered pairs. The graph of the parametric equations is NOT the graph of the function p(t) which is a three dimensional curve).
To be more descriptive when graphing parametric equations in the xy Cartesian coordinate system, it is customary to use x and y instead of f and g for the names of the functions and plot all the points (x(t), y(t)) for t in D The parametric curve can easily be plotted by using parametric grapher which also animates the parametric graphing process.
For graphing in the polar coordinate system r and θ are used instead of f and g for the names of the functions: (r(t), θ(t)) for t in D You can use polar parametric grapher which is the only known parametric grapher to plot parametric curves in the polar coordinate system.