# How do you graph parametric equations?

Feb 6, 2015

When graphing, we are comparing two variables (usually $x$ and $y$). Parametric equations connect these two variables through a third variable, the parameter (often designated $p , t$ or $\theta$). To graph parametric equations, we need to combine them in such a way as to eliminate the parameter and produce a single Cartesian equation. Here's a classic example:

$x = \cos t \to \left(1\right)$
$y = \sin t \to \left(2\right)$

Square both equations:
${x}^{2} = {\cos}^{2} t \to \left(1 a\right)$
${y}^{2} = {\sin}^{2} t \to \left(2 a\right)$

Now add equations $\left(1 a\right)$ and $\left(2 a\right)$:
${x}^{2} + {y}^{2} = {\cos}^{2} t + {\sin}^{2} t$
${x}^{2} + {y}^{2} = 1 \to \left(3\right)$

Now we've eliminated the parameter, $t$. So, equation $\left(3\right)$ is the Cartesian equation that corresponds to the original parametric equations, $\left(1\right)$ and $\left(2\right)$. Incidentally, this shows that using the basic trigonometric functions as the basis for our parametric equations produces the unit circle.