Graphs of Parametric Equations

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  • 0:02 Parametric Equations
  • 1:08 Graphing Parametric Equations
  • 2:03 Eliminating the Parameter
  • 3:09 The Graph
  • 4:29 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can go about graphing parametric equations. Learn the simplest way to graph parametric equations by eliminating the parameter.

Parametric Equations

Picture a two-headed snake making its way through a garden and you can see how even though the snake has two heads, it's still one snake. It still glides effortlessly through the garden with precision. Parametric equations are like that two-headed snake.

Parametric equations define an equation in two or more variables with just one variable called the parameter. Most times, you will see parametric equations defining equations with two variables, usually x and y. If our x and y equation such as x^2 + y^2 = 1 is like our snake, then our two heads are the parametric equations of x = sin (t) and y = cos (t) for the same equation.

Yes, the parametric equations x = sin (t) and y = cos (t) describe the same equation as x^2 + y^2 = 1. So, even though we have two equations, they are still working together in unison just like the two-headed snake gliding through the garden.

Graphing Parametric Equations

Just like with other equations in math, we likewise want to be able to graph parametric equations. We will use the same Cartesian coordinates that we are used to. However, the steps to graph parametric equations is just slightly more involved.

The best way to graph parametric equations is to find a way to eliminate the parameter and get the equations back to x and y form. Once we have our single x and y equation, then we can go ahead and graph our equation like we usually do, using all the graphing skills that we have learned.

If we can't eliminate the parameter, then our other option is to calculate some different points and then graph those points on our graph. This is similar to graphing our usual equations by finding a series of points and then plotting those points to see what kind of shape our graph takes. For parametric equations though, we plug in different values of t to find our x and y values.

Eliminating the Parameter

Let's take a look at how we can eliminate the parameter. Let's look at the parametric equations of x = t + 1 and y = 3t. Looking at this pair of equations, we can go ahead and eliminate the parameter by solving x = t + 1 for t and then substituting that into y = 3t. Solving x = t + 1 for t, we get t = x - 1. Substituting this into y = 3t, we get y = 3(x - 1) = 3x - 3. So our x and y equation is y = 3x - 3.

While we didn't need to here, we sometimes need to use other math identities such as trig identities to help us eliminate the parameter. For example, comparing x^2 + y^2 = 1 to the trig identity sin^2 (t) + cos^2 (t) = 1, we see that the parametric equations are x = sin (t) and y = cos (t).

The Graph

Looking at this equation (y = 3x - 3), we see that it is an equation written in slope-intercept form. It has a y-intercept of -3 and a slope of 3. We can easily graph this using the graphing skills we already know. We get a straight line graph:

Parametric Graph
parametric graph

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