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17Calculus - Parametric Equations

Single Variable Calculus
Multi-Variable Calculus

A parametric equation is a way to describe a curve by specifying the \(x\) and \(y\) values separately. These equations are linked together using a parameter, which is just another variable. It is simple but deceptively powerful.

Unfortunately, most calculus textbooks just touch on parametrics and leave a lot to be picked up along the way. However, we think that a thorough coverage of parametrics will help you understand polar, cylindrical and spherical coordinates, as well as other calculus concepts in the future. Therefore, this site contains the most complete discussion of parametrics that you will find anywhere. And we continue to add new material and practice problems.


Topics You Need To Understand For This Page


Related Topics and Links

Pauls Online Notes - Parametric Equations and Curves

Wikipedia - Parametric Equations

If you want a full-length lecture on this topic, we recommend this video.

Prof Leonard - Introduction to Parametric Equations [1hr-38min-25secs]

video by Prof Leonard

For starters, we will use the parameter \(t\). What we do is specify two equations where \(x=f(t)\) and \(y=g(t)\). Choosing a \(t\)-value gives us the point \((x,y)\) in the xy-plane for that specific \(t\). This point can also be written \((f(t), g(t))\). You may also see this written as \((x(t),y(t))\).
Here is an example.

\( x = \cos(t) \)

\( y = \sin(t) \)

Notice that, although we have two separate equations, the \(x\) and \(y\) are linked by the value of \(t\). In this example, we take a value for the parameter \(t\) and plug that value into both equations, to get a corresponding point \((x,y)\). When \(t=0\), we get \(x=\cos(0)=1\) and \(y=\sin(0)=0\). Writing this as an ordered pair, we have \((x,y)=(1,0)\). So, the point \((1,0)\) corresponds to the value \(t=0\).
Before we go on, let's watch a good introductory video.

Michel vanBiezen - What is a Parametric Equation?

video by Michel vanBiezen

If we want to describe a curve in three dimensions, we just add a third equation \(z=h(t)\) to get the point \((x,y,z) = (f(t),g(t),h(t)) = (x(t),y(t),z(t))\).

Advantages of Using Parametric Equations

One of the advantages of using parametric equations is that we can describe many more graphs than we could when we had only \(x\) and \(y\). Also, we are not going to require the graphs to be functions now, i.e. they do not have to pass the vertical line test.

Additional Information We Get From Parametric Equations

Another piece of information we get from parametric equations is the direction of the curve. This information is automatically included in the equations.

The use of the variable \(t\) as a parameter is not random. Often, we assign a meaning to the parameter and sometimes that meaning is time. When you graph a set of parametric equations, the graph is swept out in a certain direction. This is an inherent feature of parametric equations. We will often start at \( t=0 \) and increase \(t\), giving the idea that time is passing. By adjusting the parametric equations, we can reverse the direction that the graph is swept. You can get more information on the next page, where we discuss parametric curves in more detail.

Something that we need to watch for with parametric equations is, for closed curves, how many times a curve is swept out based on the range of \(t\). This is important since there will be times that we need to restrict \(t\) so that a curve is swept out only once. Three practice problems below discuss this idea in their solutions, 4426, 4427 and 4428, which can be found on the parametric graphing page.

What We Can Do With Parametric Equations

There are lots of things you can do with parametric equations, most of which we cover on other pages. However, here are a few practice problems to get you thinking more deeply about parametric equations and how they work.


Unless otherwise instructed, solve these problems giving your answers in exact form.

Find all points of intersection of the parametric curves \(C_1: x=t+1; y=t^2\) and \(C_2: x=3t+1; y=t^2+1\).

Problem Statement

Find all points of intersection of the parametric curves \(C_1: x=t+1; y=t^2\) and \(C_2: x=3t+1; y=t^2+1\).


PatrickJMT - 53 video solution

video by PatrickJMT

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Getting Started

Advantages of Using Parametric Equations

Additional Information We Get From Parametric Equations

What We Can Do With Parametric Equations

Practice Search

Practice Instructions

Unless otherwise instructed, solve these problems giving your answers in exact form.

Do NOT follow this link or you will be banned from the site!

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