Converting Between Parametric & Rectangular Forms

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  • 0:01 Rectangular Equations
  • 0:39 Parametric Equations
  • 1:26 From Rectangular to Parametric
  • 3:22 From Parametric to Rectangular
  • 4:20 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.

There are all kinds of ways of writing equations. Learn about parametric equations in this video lesson. Learn how you can convert between parametric equations and their equivalent rectangular equations.

Rectangular Equations

We are all familiar with rectangular equations. These are the equations with several variables such x, y, and z. We are used to these equations because we come across these all the time in math classes. Examples of rectangular equations include equations for lines such as y = 4x - 3 and equations for circles such as x^2 + y^2 = 1. Notice how all these equations have more than one variable.

Parametric Equations

In addition to our rectangular equations, we also have what are called parametric equations. These are our same rectangular equations rewritten with only one variable, the parameter. Yes, we define each of our variables with only one variable, with one parameter, we call it. Even though we can use any letter, we will stick to the letter t for our parameter in this lesson. And we will stick to the letters x and y for our rectangular equations.

The difference between parametric equations and rectangular equations is that where a rectangular equation only requires one equation, a parametric equation is made up of one definition equation for each variable. For example, x = t + 1 and y = 2t is an example of one parametric equation for a rectangular equation with only x and y variables. Yes, we need one equation to define each variable in terms of our one parameter.

From Rectangular to Parametric

How do we come up with our parametric equation from a rectangular equation? Let's take a look. Are you ready?

Convert y = 4x to parametric form.

We have been given a problem to convert the rectangular equation y = 4x to parametric form. This is a simple linear equation, so we can make the easy parametric substitution of x = t. Then to find out what y is in terms of t, we plug in t for x. We get y = 4t.

So our parametric equation is x = t and y = 4t. Since our rectangular equation only has the variables x and y, our parametric equation will have only two equations, one for x and one for y.

Sometimes, our problem is a little bit more complicated.

Convert x^2 + y^2 = 1 to parametric form.

To convert this rectangular equation to parametric form, we make use of our knowledge of trigonometry and its identities. Looking at this equation, we see that it looks a lot like the Pythagorean identity in trigonometry, sin^2 (t) + cos^2 (t) = 1. Seeing how they compare, we can make the parametric substitution of x = sin (t) and y = cos (t). So our parametric equation is x = sin (t) and y = sin (t).

While simple rectangular equations can have a simple parametric substitution, some more complicated rectangular equations will make use of our extended math knowledge. The key is in knowing the shape of the graph and the other equations that also have the same shape graph. For example, both x^2 + y^2 = 1 and sin^2 (t) + cos^2 (t) = 1 graph into a circle.

If you haven't already noticed, with this process, it is definitely possible to get different parametric equations that will produce the same rectangular equation.

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