Graphing parametric equations

To graph parametric equations, you first use the parameter, $t$, to generate a set of $(x, y)$ coordinate pairs. Then, you plot these coordinate pairs on a standard Cartesian plane and connect them in order. This process reveals the path or relationship between the $x$ and $y$ variables, independent of time.

Example 1: For $x = 3t$ and $y=t+2$: if $t=1$, the point is $(3, 3)$; if $t=2$, the point is $(6, 4)$. Plot these points to see the line. Example 2: For $x = t^2$ and $y=t 2$: at $t=0$, you get $(0, 2)$; at $t=1$, you get $(1, 1)$; at $t=2$, you get $(4, 0)$. Connect them to form a curve. Example 3: Using a table for $x=120t, y=96t 16t^2$: calculate each $(x,y)$ for $t = 0, 1, 2...$ and then plot the points $(0,0), (120,80), (240,128)$ to see the parabola.

Time to play matchmaker! For each value of your parameter, $t$, calculate the corresponding $x$ and $y$. This gives you an $(x, y)$ point. Once you have a few of these points, plot them on a graph and connect the dots in order of increasing $t$. Voila! You have just revealed the secret path that $x$ and $y$ follow.