Back To Course

Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

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.

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.

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.

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.

Going the other way and converting from parametric form to rectangular form is more straightforward. What we need to do is first solve one equation for either *x* or *y* and then plug it into the other to find our rectangular equation.

*Convert x = t + 1 and y = 2t into a rectangular equation.*

Both of these equations are easy to solve for either *x* or *y*. So we will just pick one. Let's solve *x = t + 1 for t*. We get *t = x - 1*. Now, we plug this into *y = 2t*. We get *y = 2(x - 1)*. This becomes *y = 2x - 2*, a linear equation. We are done. Our rectangular equation is *y = 2x - 2*.

Converting from parametric form to rectangular form is more straightforward and requires straight substitution and solving.

Let's review what we've learned:

We learned that **rectangular equations** are the equations with several variables such as *x*, *y*, and *z* and **parametric equations** are our same rectangular equations rewritten with only one variable, the parameter. So, rectangular equations have more than one variable and parametric equations only have one. Also, while the rectangular equation only has one equation, a parametric equation has one definition equation for each variable.

To change from a rectangular equation into a parametric equation, we need to look at the equation and compare it to other equations we know that have the same shape. If it's a straight line, we can make a straightforward substitution such as *x = t* and then plug that into our equation to find out what *y* equals. For more complex rectangular equations, we may need to look at possible trig identities and other identities that we know.

To convert from a parametric equation into a rectangular equation, we solve one of the parametric equations for one of the rectangular variables, and plug it into the other equation to find a rectangular equation.

Following this lesson, you should have the ability to:

- Define rectangular equations and parametric equations
- Explain how to change from simple and complex rectangular equations into a parametric equation
- Convert a parametric equation into a rectangular equation

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
4 in chapter 24 of the course:

Back To Course

Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

- Graphing Functions in Polar Coordinates: Process & Examples 7:18
- Complex Numbers in Polar Form: Process & Examples 7:21
- Evaluating Parametric Equations: Process & Examples 3:43
- Converting Between Parametric & Rectangular Forms 5:33
- Parametric Equations in Applied Contexts 5:29
- Conic Sections in Polar & Parametric Forms 6:35
- Go to Polar Coordinates and Parameterizations

- Go to Continuity

- Go to Limits

- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- U.S. Politics & Civics Lesson Plans
- US History - Civil War: Lesson Plans & Resources
- iOS Data Analysis & Recovery
- Acquiring Data from iOS Devices
- Foundations of Digital Forensics
- Introduction to Mobile Forensics
- Examination of iOS Devices
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison
- TACHS Prep Product Comparison
- Top 50 Blended Learning High Schools
- EPPP Prep Product Comparison
- NMTA Prep Product Comparison
- Study.com NMTA Scholarship: Application Form & Information

- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Logical Thinking & Reasoning Questions: Lesson for Kids
- Human Geography Project Ideas
- Asian Heritage Month Activities
- Types of Visualization in Python
- Quiz & Worksheet - Frontalis Muscle
- Octopus Diet: Quiz & Worksheet for Kids
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Quiz & Worksheet - Dolphin Mating & Reproduction
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- 6th Grade Math Worksheets & Printables
- Classroom Management Strategies | Classroom Rules & Procedures

- Introduction to Statistics: Homework Help Resource
- Educational Psychology: Help and Review
- High School World History: Tutoring Solution
- Common Core ELA Grade 7 - Language: Standards
- Human Growth and Development: Tutoring Solution
- AP World History - The Enlightenment: Help and Review
- Reconstruction (1865-1877): Tutoring Solution
- Quiz & Worksheet - Characteristics of & Programs for At-Risk Students
- Quiz & Worksheet - Social Needs in Maslow's Hierarchy
- Quiz & Worksheet - Silent Mutation
- Quiz & Worksheet - Autonomy in Psychosocial Growth
- Quiz & Worksheet - Cell Membrane Analogies

- Non-Competitive Inhibition: Examples & Graph
- What is a Usability Study?
- 2nd Grade Florida Science Standards
- 4th Grade Colorado Science Standards
- ACT Accommodations for ELL Students
- How to Pass Multiple Choice Tests
- How to Pass the Real Estate Exam
- Fractions Lesson Plan
- Failed the USMLE Step 1: Next Steps
- Preschool Math Centers
- Nebraska State Math Standards
- Shays' Rebellion Lesson Plan

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject