Writing Parametric Equations - Online Tutor, Practice Problems & Exam Prep

To convert rectangular equations into parametric equations, start by defining a parameter \( t \) based on either \( x \) or \( y \). For example, if \( t = x + 1 \), solve for \( x(t) \) and substitute into the original equation to find \( y(t) \). Use the Pythagorean identity for equations like \( x^2 + y^2 = 1 \) to derive \( x(t) = 2 \cos(t) \) and \( y(t) = \sin(t) \). Always ensure your choice of \( t \) avoids domain restrictions, particularly with even powers of \( x \).

Parameterizing Equations

So in recent videos, we've learned how to eliminate the t parameter from parametric equations to get back to equations just involving x and y. Well, some questions will actually have you do the opposite. They will give you a rectangular equation, and they'll ask you to find parametric equations for it. In other words, they'll ask you to write an x of t or a y of t. This is also sometimes called parameterizing an equation. It's really just the opposite of eliminating the parameter. So, I'm going to walk you through a step-by-step process of how to do this because there are a couple of things you want to keep in mind as you're doing this. Let's just jump right in.

Remember how when we eliminated the parameter, the basic idea was that we would solve one of these equations for t, then plug it into the other one and get rid of the t variable. Now we're doing the opposite. I'm going to go from an equation that involves just y and x, and I'm going to end up with an x of t and a y of t. So the first thing that you have to do here is you're going to have to choose an expression for t. You're going to have to pick something, an expression for t, either involving x or y. So what you're going to do here is if this is your equation, the first thing I can do for choosing t is I could just choose t to be x itself. It sounds kind of silly because I'm just choosing one variable to be another, but what happens? Well, what I'm going to do here is once I've chosen my t, is I'm going to solve for x of t. If t is equal to x, that means that x of t is just t itself. Right? x of t is just t. All I've done is swap the equation. So now that I've gotten one of my parametric equations, x of t is t, how do I get the second one, the y of t? All you're going to do is you're just going to substitute the x of t expression into the original equation and then solve for your y of t. So in other words, if I pop this into my original equation and every time I see x, I just replace it with t, what I end up with for y is I just end up with y equals 4(t+1). So all I've done here in this equation is I ended up with the same exact y equation. I've just traded the x variable for a t. It seems kind of silly because I just pick one variable to be another. But, in fact, if you actually were to plot out a bunch of t values, you would get the same exact x and y pairs as this equation. They actually just describe the same exact line.

So it turns out that setting t equal to x is pretty much almost always going to work in your problems. Now most problems, what they'll have you do is they'll have you sort of they'll prevent you from choosing x of t equals t because they want you to be a little bit more creative. Now this only worked because I chose t to be x, but I also could have chosen t to be something else. So let's come up with a different set of parametric equations for this. So again, I'm going to have to choose t, and it's going to usually involve some expression of x. So if I look through this equation over here, one of the things I can do is I could set t to be everything that's inside of this parenthesis over here. So I can set t not to just be x, but x + 1. Again, I'm just going to solve my x expression for t. So all I'm going to do is just subtract t-1. And then if I pop this into the original expression, what I'll end up with is that y = 4(t-1+1). The -1 +1 will cancel and leave me with just 4t. So, again, this is another perfectly valid set of parametric equations. If you were to plot out a bunch of values, you'll get the same x y pairs as this equation and also your first set as well. So it just depends on what you chose t to be. There's an infinite number of possibilities, but, usually, you're gonna want to choose something really simple.

That's really all there is to it. There's one thing that I haven't mentioned yet for choosing t, which is that you're gonna want to choose a t that avoids domain restrictions. So, for example, one

Parameterizing Equations Example 1

Welcome back, everyone. So in this example, we're going to take a look at this equation, y=2x3, and we're going to write 2 sets of parametric equations that describe this rectangular equation. So in other words, they want us to parameterize this y=2x3 equation. So let's take a look at the steps of how to do this. The first thing you're going to do in these problems is you're going to have to define t unless it's given to you already. In this case, it's not. So in this case, what we're going to have to do is choose a t to parameterize our equation, and there's always some guidance to this. In this case, you can always just try t=x or t=x equals whatever the thing in the parentheses is if you're given some kind of a parentheses in the equation. And in this problem, we're not told that we can't use x=t or t=x. So option 1 is that we just choose x=t or t=x. Right? So if we let t=x, then the second step is we're going to have to solve for x(t). So in other words, we're going to have to get x in terms of t. And if t=x, so you just flip the equation around. In other words, x(t) is just equal to t itself. That's the first equation, the first parameterized equation. Now to get the y equation, we're just going to have to plug that x(t) into the original equation for x. In other words, the y=2x3. And let's go ahead and do that. So if x(t) is equal to t itself, that means that y is equal to 2. And instead of now plugging in x cubes, we're really just going to replace this with t3 because that's what x is equal to. Right? So in other words, this really just becomes 2t3. Alright? So this is your x(t) equation, and this is your y(t) equation. So again, it's always kind of silly how these problems work out because if they're not if you're not told that you can't use this option or this definition of t, then this is always a viable set or a solution to your parametric equations. You can always just basically replace x with t, and then your y equation just becomes the original equation, but just you've replaced x with t. Alright? So that's always one of your options. So that's one of your sets of parametric equations. Or what we can also do is we're going to have to pick another sort of definition for t and sort of parameterize the equation a different way. Remember, there's always a ton of different ways that you can do this. There's no really correct answer, but there are some easier ones, for defining t. Alright? So let's go ahead and take a look at the second one here. You're going to define t unless it's given to us. In this case, we can't use t=x, obviously, because we just use that in the first one. So let's just try to pick a different definition for t. What we can do here is whenever you're just exhausted with you know, t=x or whatever the parenthesis is, you can start to just try to set t in terms of powers of x if you have t if you have powers of x. And so, again, so what you can do here is you can set t=x3. You always want to avoid even powe

Equations of Circles & Ellipses

So in the last couple of videos, we saw how to take parametric equations involving trig functions and, using this Pythagorean identity, rewrite them to have only just x's and y's. Well, just as we've done with previous videos, lots of problems will ask you to do the opposite. They'll give you equations involving x's and y's and ask you to write parametric equations for them. What I'm going to show you is a step-by-step process of how to do this, and it's really just the opposite of what we did with eliminating the parameter. We're still just going to use this Pythagorean identity to write two parametric equations for this equation involving x's and y's. Let's go ahead and get started here. We're just gonna jump right into this example. Alright? So whenever you have an equation that contains something like x² + y², remember that's an equation of a circle or an ellipse. Right? Obviously, there are other numbers involved here, but the basic idea here is that you have x² + y² = 1. So what you're gonna do in these problems is, for example, if you have something like (x/2)² + y², we have obviously some other numbers, but we have x² + y² in there. So we can parameterize this. The key idea here is we're gonna rewrite this equation to be in the form we have some kind of function of x² + some kind of function of y² = 1 on the right side. So if you take a look at this equation here, we've got something of x² + something of y² = 1. And if you sort of just sort of match these things, you'll end up seeing that this actually is a function of x that's squared + some kind of function of y that's squared, and that equals 1. So this is in this form so we can parameterize the equations using this step. Alright?

These steps. And so here's how this was gonna work. Alright? How does this how how do we use this to get back to parameterized equations involving just t? Well, the key idea is we're still just gonna use this Pythagorean identity. So we have cosine of something² + sine² something = 1. And in this case, that something is just the parameter t. Alright. So what you're gonna do here is notice how these equations actually look kind of similar. We have something that's squared + something that's squared = 1. So the key idea here is that you're just going to set that equation involving x to be cosine². You're gonna set that cos that f that f of x function to equal cosine², and you're gonna set that function involving y to be sine². And then what you're gonna do here is you're just gonna go ahead and try to isolate x and y. Okay? So here's how this works. I'm gonna take my expression involving x, which is gonna be (x/2)², and I'm gonna set that equal to cosine² of t. Now, what I have to do is I just have to get x by itself. Well, if I take the square root of both sides, I just get that x/2 is equal to cosine of t, and if I get x by itself just by moving the 2 over to the other side, I end up with 2 cosine of t. Notice how I have just x by itself equal to some kind of expression involving t. That's why you'll see this written as x of t. We're gonna do the exact same thing with the y equation. So, in this case, what happens is I got y² is equal to my sine² t. So this is gonna be sine² t, and I'm just gonna go ahead and isolate y and get it by itself. Take the square root of both sides. You'll end up with that y is equal to sine of t, and you don't actually have to do any other steps here, so you could just repeat because y of t is equal to sine of t. Notice how we ended up with two parameterized equations, x of t and y of t, that both involve the parameter. This actually makes sense that we kinda ended up back at the two equations that we started out with over here on the left side, because all we did was we just did the reverse process of eliminating the parameter.

Remember, if you're ever unsure of how you've parameterized your equations, you can always just eliminate the parameter to get back to the original equation, and you should get back to where you started from. Alright? Now problems would always be this straightforward, so I wanna go ahead and walk you through a step-by-step process of how to do that using our next example. Alright? Let's take a look. So we're gonna write parametric equations for this example over here. We've got 9 x² + y² = 9. So let's take a look here. This is an equation just like the previous one that involves x² + y². There are other numbers, but the key thing is that there's something like x² + y² in there. Alright? So let's stick to the first step here. If the equation isn't already written in this form, you're gonna go ahead and write this as f of x² + g of y² = 1. So you want something of x² + something of y² = 1. And in this case, what happens is we actually have an equals 9 on the right side. So we're gonna have to get that to be equal to 1. We've seen how to do this before in other videos, but, basically, you're just gonna divide everything on the left and right sides by 9. And what you'll end up with is you'll have the nines canceling on the left. You'll just end up with x² + y²/9 = 1. So notice how here we have something like something of x² + something of y² = 1. And, of course, there are other numbers involved here. Let me go ahead and rewrite this. Alright? So now we can move on to our next step. What we're gonna do here is we're gonna set the f of x, that equation involving x or the expression involving x, to be equal to cosine² t and you're going to solve. Alright? The key idea here is we're basically just going to take this x² and we're just going to set this equal to cosine² t. So how do I get x by itself? Well, in this case, it's pretty straightforward. All I have to do is just undo the squares by taking the square roots, and I just get that x is equal to cosine of t, and I'm done there. So in other words, x is equal to cosine of t. That's it. And that's your first parametric equation. Now, what you're going to do is you're going to do the same thing for the y equation. So what I'm going to do here is step 2 or that was step 2. We're going to do step 3 which is that now we're just going to set the g of y expression. So, in other words, y²/9 equal to the sine². And now, we're just gonna go ahead and try to solve for y. So, in this case, we're gonna have to do is we're gonna move the 9 over to the other side. So this becomes y² = 9 times the sine² of t. And then we're gonna have to take the square root of both sides. So if I take the square root of both sides over here by doing this, then the square root of 9 just becomes 3, and the square root of sine² just becomes 3 sine t. And there you go. There you go. So these are your two parametric equations that describe this ellipse, over here that we have. Alright? So that's it for this one, folks. Let me know if you have any questions, and I will see you in the next video. Let's get some practice.

Write parametric equations for the rectangular equation below.

$x^2+y^2=25$

• A. $x=25\sin t$; $y=25\cos t$
• B. $x=25\cos t$; $y=25\sin t$
• C. $x=5\sin t$; $y=5\cos t$
• D. $x=5\cos t$; $y=5\sin t$