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Parametric equations with the same graph

Sal shows how different parametric equations can result in the same relationship between x and y (and therefore with the same graph). Created by Sal Khan.

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Video transcript

In the last video, we started with these parametric equations: x is equal to 3 cosine of t and y is equal to 2 sine of t. And doing a little bit of algebra, we were able to remove the parameter and turn it into an equation that we normally associate with an ellipse. We got x squared over 9 plus y squared over 4 is equal to 1. And we graphed it. And I'll do that. Actually I have it from the previous-- We graphed it and we got something that looked like that right there. And we said well why don't we always just represent it like this? Well this right here-- this equation just in terms of x and y --it just represents the shape. It doesn't represent the actual path that-- Let's say that t represents an object's point-- an object's position at some place and time. And we went over that. And as time progressed, we went around the ellipse, in that case. And as time goes forward, we kept going around this ellipse. But this begs the question: is this the only set of parametric equations that, when we do a little bit of algebra, gives this path? Can we think of another set that gives-- that would also, when you do the algebra, get here? And I think we can. So let's try a couple. I mean, we could do a whole bunch of them. Actually, maybe we'll do a whole bunch of them, just to see that there-- And what you'll see is that there's actually an infinite number of parametric equations that have this path. So one could be x is equal to 3 cosine of 2t. And the other one is y is equal 2 sine of 2t. Like we did in the last video, let's solve for cosine of 2t and solve for sine of 2t. So we just have to divide both sides of-- Well, both sides of this top equation by 3. You get x over 3 is equal to cosine of 2t. Divide both sides of this equation by 2. You get y over 2 is equal to sine of 2t. And we'll use the same trig identities we did before. When we did this problem, we said that sine squared of t plus cosine squared of t is equal to 1. Well if that's true, so is this true. Let me see. If that's true, then we can also say that sine squared of 2t plus cosine squared of 2t is equal to 1. In fact, sine squared of anything here plus cosine squared of anything, as long as it's the same anything as we put here-- I can't have a 2t here and a 3t here, but if this is a 2 and this is a 2 --then that's always going to be equal to 1. You can just fill in the black. Because this is always going to be the same angle. So if this is true, we can then just take this and substitute it in for cosine of 2t and take the y over 2 and substitute it in for sine of 2t. And we get the sine of 2t squared becomes y over 2 squared plus cosine of 2t is x over 3 squared. It equals 1. And that is exactly what we had to begin with. That is the same thing as x squared over 9-- I'm just switching the order here --plus y squared over 4 is equal to 1. So both of these parametric equations, both sets of parametric equations have the exact same path. So how are they different? Let's see. If we were to take-- So in the previous video, we took this yellow one. Actually I rewrote it here. And we did a little table and we plotted the points that it hits as you go around the circle. So let's do the same thing for our new set. So just since I lost it, let me write the two equations here. It was x is equal to 3 cosine of 2t and y is equal to 2 sine of 2t. Let me make a little table. There we go. And we're going to have a t. And we're going to have an x and a y. And let's pick the same values of t. 0 pi over 2 and pi. When t is 0, cosine of 2 times 0. That's cosine of 0, which is one, times 3, it's 3. When t is equal to pi over 2, 2 times pi over 2 is pi. Cosine of pi over 2 is minus 1. Minus 1 times 3 is minus 3. When t is equal to pi, cosine of 2 times pi is 2 pi. Cosine of 2 pi is the same thing as cosine of 0. So that's 1. 1 times 3 is 3. And then on the y side. When t is equal to 0, that's the same thing as sine of 2 times 0. Sine of 0 times 2 is 0. Pi over 2. Sine of 2 times pi over 2. That's the same thing as sine of 4. That's also 0. 0 times 2 is 0. And then pi. Sine of 2 pi. Well, that's also 0. So here we get a bunch of 0's. So what happened here? Let me do it in a different color. So in our first situation-- So they both-- We established that both this set of parametric equations and this set of parametric equations, they both have the same overall, I guess we could say, shape of their path. They're both this ellipse. But what we saw in the previous videos, when you plot these points at time t equals 0, we were right. This point right here is that right there. And then at t is equal to pi over 2, we got right there. And then at t is equal to pi, we got over here. And we did that to establish that the direction of motion. If these parametric equations really are describing some type of motion, it would be counterclockwise. And what happens in this scenario-- So when t is equal to 0, we're still at the point 3, 0. So they at least-- At t equals 0, you can kind of view them as starting at the same point, if we assume we're starting a t equals 0. You don't have to start at t equals 0. I did that a little bit in the last video, but you can start at t equals minus a google or t equals minus infinity. So you don't necessarily have a start point. But if we assume t equals 0 as a start point, you could say that they both started there. Then at-- let me pick a new color that I haven't used yet --then at t is equal to pi over 2-- Where are we? We're at minus 3, 0. We get all the way-- Oh. That's the same color I used before. Let me see this. Let me do this color. We're here. Now notice: in the first one, when we went from t equals 0 to t equals pi over 2, we went from here to there. We went kind of a quarter of the way around the ellipse. But now when we went from t equals zero to pi over 2, where did we go? we? Went halfway around the ellipse. We went all the way from there, all the way over there. And likewise, when we went from t equals pi over 2 to t equals pi with this set of parametric equations, we went another quarter of the ellipse. We went from there to there. But here, when we go from t equals pi ever 2 to t equals pi, we go all of this way. We go back to the beginning part of our ellipse. So what you see is that this set of parametric equations has the exact same shape of its path as this set of parametric equations. Except it's going around it at twice as fast of a rate. For every time when t increases by pi over 2 here, we go by-- we kind of go a quarter way around the ellipse. But when t increases by pi over 2 here, we go halfway around the ellipse. So the thing to realize-- and I know I've touched on this before --is that even though both of these sets of parametric equations, when you do the algebra, they can kind of be converted into this shape. You lose the information about where our particle is as it's rotating around the ellipse or how fast it's rotating around the ellipse. And that's why you need these parametric equations. We can even set up a parametric equation that goes in the other direction. Instead of having these-- and I encourage you to play with that --but if you instead of this, if you just put a minus sign right here. Cosine of minus t and 2 sine of minus t. Instead of going in that direction, it would go in this direction. It would go in a clockwise direction. So one thing that you've probably been thinking from the beginning is OK, I was able to go from my parametric equations to this equation of ellipse in terms of just x and y. Can you go back the other way? Could you go from this to this? And, I think you might realize now, that the answer is no. Because there's no way, just with the information that you're given here, to know that you should go to this parametric equation or this parametric equation or any of an infinite number of parametric equations. I mean anything of the form x is equal to 3 cosine of really anything times t and y is equal to 3 times cosine of-- As long as it's the same anything-- I drew the two squiggly marks the same. --as long as these two things are the same, then you will have-- they'll both converge to this shape. So if you have the shape, you don't know what parametric equation you can go back to. You could make up one, but you don't know which one it'll go back to. And just to kind of hit the point home of kind of making up a parametric equation. Sometimes that's asked of you. So let's just do a very simple example. Let's start with a normal function of x. So let's say that we have the equation y is equal to x squared plus x. I just made it up. And let's say that you were asked to turn this into a parametric equation. And this is often a very hard thing for people because there is not one right answer. You can turn into an arbitrary number of parametric equations. I just may do something really crazy and arbitrary. I could say x is equal-- When I have y defined explicitly in terms of x like this, I can make x into anything in terms of t. I could say it's cosine of t minus the natural log of t. I just made that up. That's just a random thing. But then if x is this, then y is going to be equal to-- I just substitute this back in --cosine of t minus ln of t squared plus cosine of t minus ln of t. And we're done. We just converted this into a parametric equation or a set of parametric equations. I could've also just written x is equal to t. And then what would y equal? y would be equal to t squared plus t. You might say well what's the difference between this parametric equation and this parametric equation? Well, they're both going to have the same shape of their paths. It's going to be something like a parabola. But the rate and the direction with which they progress on those paths will be very very different. It's actually a very interesting thing to think about. There are some paths that you could take where-- Let's say that the-- You could set up a parametric equation. Let's say that the shape is-- You know everything we've done so far, we've always been going in one direction, but you could have a-- There are scenarios-- And maybe if I have time, I'll make a video. And this isn't what-- Let's say the shape of the path is some type of, I dunno, let's say it's a circle of some kind. I'm just going to-- That's not this. I'm doing a completely different example right now. I'm just kind of-- Sometimes, with the ellipses we had paths that went counterclockwise and then paths that went clockwise. You can also have paths that kind of isolate in between, move around, move back and forth along the thing. So there's all sorts of parametric equations that you can define. And you can say if t equals from this to this, you'll use this set of parametric equations. If it's another set of t's, use another one. So there's all sorts of crazy things you can do to say what happens as you move along the path. So the difference-- Not that this is the path of that. This is actually more of a parabola. But the difference between this and this is how you move along the shape. Anyway, I hope you found that vaguely useful.