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 xt or a yt. 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 are doing this. Let's just jump right in. Alright. So 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. Alright. 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 xt and a yt. 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 xt. I'm going to get back to this in just a second, by the way. So I'm going to solve for xt. If t is equal to x, that means that xt is just t itself. Right? xt is just t. All I've done is swap the equation. So now that I've gotten one of my parametric equations, xt is t, how do I get the second one, the yt? All you're going to do is you're just going to substitute the xt expression into the original equation and then solve for your yt. So in other words, if I pop this into my original equation and every time I see x, I just replace it with a t, what I end up with for y is I just end up with 4(t+1). Alright. 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. Alright? So it turns out that setting t equal to x is pretty much almost always going to work in your problems. Alright? Now most problems, what they'll have you do is they'll have you sort of they'll prevent you from from choosing xt equals t because they want you to be a little bit more creative. Alright? 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. Alright? 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 4=y equals 4(t-1+1). The minus 1 plus 1 will cancel and leave me with just 4×t. 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 going to want to choose something really simple. Alright? 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 going to want to choose a t that avoids domain restrictions. So, for example, one of the best things to do here is either to choose x, a t equals x, or some kind of multiple or addition to x. You're really just going to want to pick variables that are odd powers of x because t can be any very t could be any number and then x could be any number There's no domain restrictions there What you want to avoid is you want to avoid even powers of x like x^2 or the square root of x. Because what's going to happen is when I solve this expression for x, I'm going to get the square root of t and then t can't be any negative numbers because then I'll get imaginaries. You usually gonna want to avoid even powers of x. Alright? So that's all really all there is to it. Let's go ahead and take a look at some examples because I want to walk through a step-by-step process of how to do this. Alright. So let's take a look. So without choosing xt to be t, we're going to have to find parametric equations for these rectangular equations. Alright? Let's take a look at the first one, example a. We've got that y is equal to 2x+5, and, actually, what's given to us already is what t equals. So they've already actually chosen t for us. So the first step in these problems is you're going to define t unless you're already given that what that t expression is going to be. So in this case, we don't have to worry about anything. So if t is equal to x+1, let's move on to the second step. What you're going to do here is you're going to solve that expression of t for xt. So if t is equal to x+1, that means that xt is just going to be I'm just going to have to subtract 1 from both sides. I'm just going to be t-1. So just imagine that you had this equation here, and you were just solving for x. You'd have to move this to the other side, and you just get xt. So that's what happens is you'll often see a lot of times these parentheses get added in once you get x as a function or an expression of t. Okay. The third step is we're now going to plug that xt into our original y of x equation to get y to get yt. So what happens here is if I plug this into my 2,x+5, every time I see x, I replace it with t-1. So this is going to be t-1 with a parenthesis, then +5. Alright? Now what happens here is I get y as a function of t. So this is just going to be 2t-2+5, and we can simplify this minus 2 plus 5 ends up being 2t+3. Alright? So that's my set of parametric equations. I ended up getting that xt is t-1 and yt is equal to 2t+3. And that's all there is to it. That's the 4th step. So let's go ahead and now take a look at this example b over here. I've got y equals, it's a little bit more complicated, x+2 squared minus 3. Let's go through the steps. We're going to have to define t unless it's given to us. Now there's some guidance here. You can always try t equals x. However, the problem told us we actually can't do that. We can't just choose xt to be t. But another perfectly valid option that you'll often see a lot is you can choose t to be whatever expression is inside of the parentheses. Not the not the power, just whatever is in the parentheses. So in this case, what I'm going to do here is I'm just going to set t to be equal to x+2. That's the thing that's inside the parentheses over here. Alright? It's another perfectly valid thing that you can do. Let's just see how it works. The second step is I'm going to solve this expression for xt. If I just flip this around, what I'm what I'm going to get here is that xt is equal to t-2. Alright? Now, Now let's move on to the 3rd step. I'm just going to plug this xt into my original y of x expression. So this ends up being y equals. And now what I have here is parenthesis squared, and the thing that goes in this parenthesis here is going to be this t-2. Right? Because every
Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
10. Parametric Equations
Writing Parametric Equations
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