Welcome to our lecture on Parametric Equations. The first thing I want to talk about is a very common shape called the circle. Let's talk about the circle for a minute. It's a beautiful object. It's got lots of symmetry, however, up to this point, we've thrown it away. What do you mean? If I take the circle and if I graph it on the xy-plane, there's something that you've noticed or maybe you haven't noticed before that, we never talk about it, why? Because it completely fails the vertical line test. This thing fails the vertical line test. Therefore, what does that mean? It's not the graph of a function. You cannot be written in the form of y equals f of x. Everything we've done so far has been about functions. I want to take the derivative of functions. I want to take the integrals of functions. Unfortunately, we have to take the circle and throw it out. You can imagine that any study or discipline that takes the circle and completely ignores it is not a very complete theory. Now our job in this last section here is to bring it back. We want to describe things that fail the vertical line test that are not graphs. In particular, they'll be called curves. We'll use the more umbrella term called curves. What I want to do is now describe x and y not of each other with y as a function of x, but instead as a function of a new variable called the parameter. We're going to write x as a function of t, usually time, and y is a new function of time as well. They can be the same. They can be different. It does not matter. But in this case, when you do this, they can be called parametric equation. We're introducing t, which is the parameter. Everything here is a dummy variable, so everything can be changed to be whatever you want, but t is pretty common as we move things along in time. The set of points that we're going to obtain, this is called the graph of the parametric equation and the curve itself is the plane curve or parameterized curve described by this. Let's look at an example starting with the circle again. For example, if I describe x as cosine of t and y as sine of t, each one as a function of t, I can start making a table. Let's start making a t-values. Whatever t is completely determines the values for x and y. For example, when t is 0, then x is cosine of 0, which of course is 1, and y is sine of 0, which of course is 0. You can go ahead and make your table. How about when t is like Pi over 4? Well, when x then is cosine Pi over 4, that's root 2 over 2, and y is root 2 over 2. From this you're getting these x and y values that you can then plot. It's just start looking like trig. So you can see where I'm going hopefully with this. When t is Pi over 2, x of cosine of Pi over 2 is just 0 and y is 1. I will do one more because I'm sure you've seen when this is going here. X is Pi, cosine of Pi, which is negative 1, and sine of Pi which is 0. It goes on and on. So you can restrict the values of t. You can not, doesn't matter. For this example here, let's let t go from 0-2Pi. You can certainly fill in more values of the table if you want. But here's the point. Imagine like a little bug traversing here. It starts at t equals 0. When time begins, when we begin our measurements, it starts at cosine of 0, which we said was 1 and sine of 0 here. The coordinate in the xy-plane is the point 1, 0. When time is Pi over 4, we move over the square 2 root 2. This corresponds to the time equals Pi over 4. When time is Pi over 2, we move to the point 0, 1. We're starting to trace out this arc of a circle. Hopefully, you realize by now that when x is cosine of t and y sine of t, this is the good old unit circle from trig. My little bug is moving. If I let it go all the way to 0 to Pi, I get to the other side over here. So t is Pi. Then of course it completes the circle if we let it go all the way to. This is the idea. If something is moving, you see an objects in motion, you let time go by and you've introduced a new parameter, a new variable, that in addition to the xy-coordinate, also gives you time value associated. There's another variable that dictates the rest. Given a fixed time, then you're at a certain location. Let's do one more example. For example here lets define the parametric equations will have x of t defined by 1. We'll let y of t be defined by t squared minus 2t. This is just an example where maybe you don't know where this is going off the bat. We're going to let t be any real number. Now people say to me like, "Can time be negative?" Sure, yesterday. It's okay if t is negative, you can restrict the domain. You cannot just pay attention to what the question is asking. We don't know really what this is. This is going to handle on it before we jump over to technology for a minute, let's look at some values of t. Then let's look at our values of x and we can look at our values of y. Now it's just a little bit of plug and chug to find some values. Find the parametric equations when t is minus 2. Check my math here, but x is negative 1, y is 8. When t is negative 1, then x is 0, y is 3. We could do this for awhile. When you're 0, you get 1, 0. When t is 1, plug that in you get 2 minus 1. Keep going. When it's 2, then it's 3, 0. I'm doing a lot here so that I can graph this. Where I pick a value of t and then I get an x and y coordinate. Each one of these two rows, the x row and the y row correspond to points in the x, y plane that I can then go ahead and graph. If I start to graph these, what do we get? If I graph negative one comma eight, let's draw the cycle. It's way up there and it's a picture not to scale, but negative one comma eight, it's way up there. Then at zero, I want to graph at three. Maybe we'll put a zero comma three, and then at, when t is zero, I'm at the 0.1 comma zero. That'll go through that point there and we can keep working this out. You can start to see that this does actually trace a shape that I might know, although maybe it wasn't obvious at first. This is why graphing this thing, being able to manipulate these things is important and this will turn out to just be the parabola. Sure, you could describe this as a graph. I could write this as y equals f of x and use my skills to get the equation on the graph, but realize you can do both. Parametric equations is the umbrella term for a broader class of graphs. Parametric equations are the ones that may or may not fail the vertical line test. But graphs themselves, graphs of functions, these are the ones that pass the vertical line test. Every graph is a parametric equation, but not vice versa. Every graph of a function is a parametric equation but not vice versa, so this is a strict inclusion here. It turns out that this will be a nice parabola. Now the question is, let's say it's not something so nice, how do I do this? Things to note, you should know how to graph more complicated ones using your calculator. If you're not sure how to do that, I'll show you a quick search on the Internet can do that or just ask me. The other one of course is that there are online websites, particularly one that's pretty good is Desmos. There's some links included that you can go and they have graphing calculators. A lot of times when the shape is more complicated, especially if you're modeling something in nature, it is not as easy to parabola or a circle. A lot of things move in orbits and go back on themselves, so you want to get out the machinery when you need to. Whenever you talk about parametric equations, the classic example for great historical significance is the cycloid. Now this is a link from GeoGebra that I wanted to walk you guys through. But here's a wheel, and wheels are old. They've been invented, been around since at least 1986. If I animate this thing, watch what's going to happen as I move the slider and this website again, it's posted in your course. If you move the parameter, we're going to move the wheel, and if I chase the point on the wheel where it's dragged out, then you can see I form the shape and this shape is not a parabola. It's a little different and it keeps going obviously and would repeat itself. But this shape was studied and caused some confusion and lots of architecture confirms this, this is a cycloid. It is the shape traced out by one rotation of the wheel. Let me do it one more time for you. Heck, we can even animate this and see it in motion. To describe this thing in Cartesian coordinates is very difficult and that was the historical difficulty of it. If we describe it though however, in parametric equations, it has a nice form you can see them here, x as a function of time is negative one sine t plus t, and then you can play around with different values of t and grow the wheel as you move as well. In parametric equations, this is the right way. This is the healthy way to think about this thing. If you do it in rectangular coordinates, well then you're going to get the historical fights with people over how to describe this thing and what it actually is. This is a great little interactive to see one of these things in motion. Please work on this, play around with this and have fun with the buttons on this as you go through with the cycloid. Graphing parametric equations is not as intuitive as graphing single equations in x and y. Looking at equations it's not easy just to look at it and know what you're looking at. For example, in the last example did you know you were looking at a parabola? Probably not. What you can do is you could try to draw the curve by plotting several points. You could plug in values of the parameter to both equations and generate some points. But a quicker way to do these is to either do one of two things, throw algebra at this problem and convert the equation. That's called eliminating the perimeter. We want to eliminate the parameter. For example, if I gave you a parametric equations, x equal to t cubed and y of t is equal to t squared, what we could do? Step 1, one way to graph this just to throw it in a calculator. Use a calculator to graph this thing and that's fine. Another way to do this is to eliminate the parameter. Here's what we're looking for. We're looking for a relationship between the x function and the y function, and sometimes you can solve for that explicitly. In this case, for example, we're going to write x, we're not going to write x with t just to simplify notation. But we write x of t is equal to t cubed and therefore, take the cube [inaudible] size. The cube root of t or t to the one-third, however you want to write it, is equal to x to the one-third, of course, we get t is equal to x. We solve for t. Then take your variable and plug it into the equation for y. If y is equal to t squared and t is equal to x to the one-third, or I substitute and I get that x is equal to two-thirds. That's my equation. You've solved our y in terms of x. Notice we have eliminated the parameter. There is no t involved anymore and this is a more familiar form that we're used to, perhaps an x equals t cubed and y equals t squared. Let's do another one. Now let's look at the equation, x equals e to the t and let's say that y is 3e to the 2t. Once again, I'd like to write this if possible, in the form of eliminating the parameter of the form y equals f of x. Let's look at this again, let's try to solve for one of the terms of the other. This is not always possible, but if they're asking you to do it and then it is, so this is when a parametric equation is a graph of a function. Let's take natural log of both sides. If we do that, you get t is the natural log of x. We plug that in to our equation for y. In that case, we get y equals 3e_2 times the natural log of x. Now this is important, I've seen people make this algebra mistake. They see e_lnx, but they try to cancel. You can't do that because that two is in the way. Friendly reminder, you can use law of logarithms to make it x squared, to move the two up as a power on the x. Now e_ln are next to each other, they're inverse functions, so they cancel and you get 3x. This is where this parabola here is not necessarily obvious that it is a parabola in the current parametric form. But at the end of the day, you're staring at the graph of y equals 3x squared. Now you can start talking about things like its concavity and its intercepts and more familiar terms in this. Let's do another one where we're going to eliminate the parameter. How about we start with x equals eight cosine of t, and we'll let y equal four sine of t. Now let's let t run from zero to two pi. In this form, I'm not quite sure what this is however, and I don't want to solve for t in terms on the ones. This is another trick that you can do to get a relationship between x and y. I know this relationship between sine and cosine. That's our good old Pythagorean identity. Sine squared plus cosine squared of t is one. I know this relationship, so let's use that relationship. In particular, cosine t through a little rearranging is just x over eight, so let's write this as x over eight squared. Then sine of t is y over four. If I solve for sine, I get y over four, and I can plug that into my known Pythagorean trig identity. This is all equal to one. Now I have a relationship between x and y. I can bring the squares in and of course you get y squared over 16 plus x squared over 64, and that's one. Now in this case here I can't solve for y. If you try to solve for y, you going to get to take a square root and that gives you plus or minus. What this is though, maybe you recognize this from a conic. This is the equation of an ellipse, this is equation of an ellipse centered at the origin, it's a little fatter than it is taller, so it goes something like this. That's coming from the fact that the four and x are your major and minor axis, but this is equation of an ellipse. If you notice it's not a graph of a function, it fails. This fails the vertical line test, which is why you can't write this as y equals, but now you can see something like this in bold term. In parametric equations, this is what an ellipse looks like compared to what's x and y coordinate equation, x squared. Let's do one more and let's use some trig functions just because I know that those tend to give people a hard time. Let's do one where x is sine of t, and we'll let y equal cosecant of t and we'll let t go from zero to t, actually just strict inequality to pi over two. Now we have a trig function and solving for one variable and substituting is going to get messy, so we're looking for best way to do it when you have algebraic relations. But for trig relations, you're trying to find some identity that you can plug x and plug y into. If you think about your traditional trig identities, Pythagorean identities, things where square, sine squared plus cosine squared of their versions, you actually don't have one between sine and cosecant. If you're scouring the page therefore, relation between sine and cosecant, they are hard to find. However, now you take a step back a little bit, but cosecant, wait a minute, this is just one over sine of t, so this is just another way to do it. There's not one way, but you have to be a little clever when you look at these and practice them. Now it's weighted y equals one over sine. That's perfect, because from here I can just write it as one equals one over x. Now you have the one over x. Now be careful though. Y equals one over x, the entire graph, the graph of this function it looks something like this, it has asymptotes at the origins, it's not defined at zero. But you got to remember what this is, I don't want the whole graph, I've restricted my domain of the parametric function, t is only going from zero to pi over two, so If I think of just what x is doing, the sine starts at the origin, pi over two when it goes to one so normally would go down and go through a period up and down forever. I just want the point of where x goes from zero to pi over two. In particular, just where x goes from 0-1. This is the graph of x of t, is sine of t. Instead, they actually like you draw this curve, you don't draw the entire function, you only draw the part where the parameters are. When t is zero, that's why it's strict inequality, you want the part of the curve that comes down and only goes where x equals one, so you're looking at the points, where t goes between zero and pi over two. You can also analyze the y coordinate to see this as well, but you want where x goes from 0-1. That's basically the first part of the curve. As it comes down from the x-axis, it goes up in a part. Just the part of the curve is what this parameter is chasing. If they didn't get restrictions on the parameter they gave larger ones, we would obviously adjust the graph that we draw back but the parameter, the values of the parameter matter when you're trying to draw the graph or parts of it. All right, good job on this tricky little way to think of things. It's a nice way to do things that opens the door to many, many great examples and has a good historical significance. Make sure you have calculators ready, you know how to graph them, that'll help them working through some more complicated problems. All right, great job, and I'll see you next time.
