Sketching parametric curves by plotting points

Steps for sketching the parametric curve

To sketch a parametric curve, we’ll follow these steps:

1. Create a table where we find $x$- and $y$-values based on specific parameter values of $t$.

2. Eliminate the parameter to find a cartesian equation in terms of just $x$ and $y$.

3. Sketch the parametric curve.

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How to sketch a parametric curve by plotting individual points in the coordinate plane

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Plotting points to sketch the parametric curve over a particular interval

Example

Sketch the parametric curve.

$x=\sin{t}$

$y=\cos{t}$

$0\le{t}\le2\pi$

Let’s create a table of $x$- and $y$-values based on parameter values of $t$ inside the given interval. Since the interval is given as $0\le{t}\le2\pi$, we’ll choose well-known parameter values inside this interval so that they’re easy to plug into our equations for $x$ and $y$.

Now we’ll eliminate the parameter to find a cartesian equation that represents our parametric equation. If we remember that $\sin^2{t}+\cos^2{t}=1$, we can just square our parametric equations,

$x=\sin{t}$

$x^2=\sin^2{t}$

and

$y=\cos{t}$

$y^2=\cos^2{t}$

and then plug them into the identity.

$\sin^2{t}+\cos^2{t}=1$

$x^2+y^2=1$

Let’s create a table of x- and y-values based on parameter values of t inside the given interval.

Since we have a list of points to plot and we know our cartesian equation, we can sketch the parametric curve. When we plot the points following the direction of the parameter $t$, we’ll see that the parameter is moving counter-clockwise around the circle.

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