## 17Calculus - Parametric Equations

Parametrics is a way to plot a graph by specifying the x, y (and z for three dimensions) values separately. We do so using a parameter, another variable used to link equations together. It is simple but deceptively powerful.
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. 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

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 may not pass the vertical line test.

Another piece of information we get from parametric equation is direction. 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 the 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.

Graphing Parametric Equations

As you learned when graphing functions, you need some experience with the equations and what the graphs look like in order to be able to know what a graph looks like just from the equation. As you are learning, I suggest using winplot to graph parametric equations. It is easy to use, has a very short learning curve and, best of all, it's free. You can find out more information on the Tools page.
Here is a video showing how to plot parametric equations.

### MIP4U - Introduction to Parametric Equations

video by MIP4U

Eliminating The Parameter

Sometimes we are given the set of parametric equations and we are asked to write the equation without the parameter by eliminating the parameter. This not always possible but with some equations there are ways to do it.

The easiest technique to try is to solve one of the equations for the parameter and then substitute the result in the other equation. Here is an example.
$$x = 2t$$ and $$y = t^2$$
Solve the first equation for t giving $$t=x/2$$ and substitute into the second equation.
$$y = (x/2)^2 = x^2/4$$

A second technique involves the use of trig identities. For example, given the parametric equations $$x=\cos(t)$$ and $$y=\sin(t)$$, we know that $$\cos^2(x) + \sin^2(x) = 1$$. So we can write $$x^2 + y^2 = 1$$ which eliminates the parameter.

A third way is by inspection. Sometimes it is obvious what a substitution might be. For example, if we have $$x=e^t$$ and $$y=e^{3t}+1$$ then we can rewrite $$y=(e^t)^3+1$$. Then we can replace $$e^t$$ with $$x$$ in the last equation (because our first parametric equation was $$x=e^t$$) to get $$y=x^3+1$$. Of course, the first technique would have worked too by solving $$x=e^t$$ for $$t$$ and then substituting and simplifying but by standing back and looking at the equations more carefully, the solution was much easier.
Okay, time for some practice problems.

Practice

Basic

Eliminate the parameter $$t$$ from $$x=3t$$ and $$y=t^2$$.

Problem Statement

Eliminate the parameter $$t$$ from $$x=3t$$ and $$y=t^2$$.

$$y = x^2/9$$

Problem Statement

Eliminate the parameter $$t$$ from $$x=3t$$ and $$y=t^2$$.

Solution

Solve the first equation for t giving $$t=x/3$$ and substitute into the second equation.
$$y = (x/3)^2 = x^2/9$$

$$y = x^2/9$$

Log in to rate this practice problem and to see it's current rating.

Eliminate the parameter $$t$$ from $$x=3\cos(t)$$ and $$y=2\sin(t)$$.

Problem Statement

Eliminate the parameter $$t$$ from $$x=3\cos(t)$$ and $$y=2\sin(t)$$.

$$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{4} = 1 }$$

Problem Statement

Eliminate the parameter $$t$$ from $$x=3\cos(t)$$ and $$y=2\sin(t)$$.

Solution

We will use the trig identity $$\cos^2(t) + \sin^2(t) = 1$$.

$$x = 3\cos(t) \to x/3 = \cos(t)$$

$$y = 2\sin(t) \to y/2 = \sin(t)$$

$$(x/3)^2 + (y/2)^2 = 1$$

$$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{4} = 1 }$$

Log in to rate this practice problem and to see it's current rating.

Eliminate the parameter t from $$x=5\cos(t)$$ and $$y=3\sin(t)$$.

Problem Statement

Eliminate the parameter t from $$x=5\cos(t)$$ and $$y=3\sin(t)$$.

$$9x^2 + 25y^2 = 225$$

Problem Statement

Eliminate the parameter t from $$x=5\cos(t)$$ and $$y=3\sin(t)$$.

Solution

In the video, she shows a really difficult way to solve this problem. Her solution is correct but it is overly tedious and unnecessarily complicated. Here is an easier and more common solution.
In this problem, you are given that $$x=5\cos(t)$$ and $$y=3\sin(t)$$. We solve each of these for the sine and cosine terms since we use $$\cos^2(t)+\sin^2(t)=1$$. So we have $$x/5=\cos(t) \to (x/5)^2=\cos^2(t)$$ and $$y/3=\sin(t) \to (y/3)^2=\sin^2(t)$$.
$$\begin{array}{rcl} 1 & = & \cos^2(t)+\sin^2(t) \\ 1 & = & (x/5)^2+(y/3)^2 \\ 1 & = & x^2/25+y^2/9 \\ 225 & = & 9x^2 + 25y^2 \end{array}$$

### 719 video

video by Krista King Math

$$9x^2 + 25y^2 = 225$$

Log in to rate this practice problem and to see it's current rating.

Eliminate the parameter from the parametric equations $$x=t-1$$, $$y=4t+1$$.

Problem Statement

Eliminate the parameter from the parametric equations $$x=t-1$$, $$y=4t+1$$.

Solution

### 48 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Eliminate the parameter from the parametric equations $$x=\sqrt{t+1}$$, $$y=3t+2$$.

Problem Statement

Eliminate the parameter from the parametric equations $$x=\sqrt{t+1}$$, $$y=3t+2$$.

Solution

### 49 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Graph the parametric equations $$x=1+\sqrt{t}$$, $$y=t^2-4t$$ on $$0\leq t\leq4$$.

Problem Statement

Graph the parametric equations $$x=1+\sqrt{t}$$, $$y=t^2-4t$$ on $$0\leq t\leq4$$.

Solution

### 51 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Graph the parametric equations $$x=\sqrt{t}$$, $$y=1-t$$ and eliminate the parameter.

Problem Statement

Graph the parametric equations $$x=\sqrt{t}$$, $$y=1-t$$ and eliminate the parameter.

Solution

### 52 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Eliminate the parameter t from $$x=\ln t$$ and $$y=\sqrt{t}$$.

Problem Statement

Eliminate the parameter t from $$x=\ln t$$ and $$y=\sqrt{t}$$.

Solution

### 2466 video

video by Michel vanBiezen

Log in to rate this practice problem and to see it's current rating.

Intermediate

Eliminate the parameter $$t$$ from $$x=e^t-1$$ and $$y=e^{2t}$$.

Problem Statement

Eliminate the parameter $$t$$ from $$x=e^t-1$$ and $$y=e^{2t}$$.

$$y=x^2+2x+1$$

Problem Statement

Eliminate the parameter $$t$$ from $$x=e^t-1$$ and $$y=e^{2t}$$.

Solution

### 2464 video

video by Michel vanBiezen

$$y=x^2+2x+1$$

Log in to rate this practice problem and to see it's current rating.

For the parametric equations $$x=1+\sin(t)$$, $$y=-2+\cos(t)$$, find the point that corresponds to $$t=\pi/2$$, graph the equations and eliminate the parameter.

Problem Statement

For the parametric equations $$x=1+\sin(t)$$, $$y=-2+\cos(t)$$, find the point that corresponds to $$t=\pi/2$$, graph the equations and eliminate the parameter.

Solution

### 47 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Eliminate the parameter from the parametric equations $$x=4e^{t/4}$$, $$y=3e^t$$.

Problem Statement

Eliminate the parameter from the parametric equations $$x=4e^{t/4}$$, $$y=3e^t$$.

Solution

### 50 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

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 $$C_2: x=3t+1; y=t^2+1$$.

Solution

### 53 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

You CAN Ace Calculus

Wikipedia - Parametric Equations

### Calculus Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on books

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.
 Getting Started Graphing Parametric Equations Eliminating The Parameter Practice

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.