 Welcome back to our lecture series, Math 1220, Calculus II, for students at Southern Utah University. As usual, I am your professor today, Dr. Andrew Missildine. In this lecture, we're going to start Chapter 10 in James Stewart's Calculus textbook and talk a little bit about parametric equations, or that is functions, curves that we can define from using these parametric equations. Now, to begin our discussion here, we have to kind of first talk about what does one mean actually by a function? We've seen many examples of functions where, you know, we take our x and our y-axis and we draw a picture. We're like, oh, whoops, that's a line. That's a function because it passes the vertical line test. A parabola as a function passes the vertical line test. A circle is not a function because it doesn't pass the vertical line test. And things like that. The problem with that discussion is that's an argument that shows that y is a function of x or y is not a function of x. There's other options, of course. I mean, if we have a concave right parabola, like so, although this graph does fail the vertical line test, it does pass the horizontal line test. And so while y is not a function of x, we can express x as a function of y. That's perfectly acceptable. And so, again, our perspective of function has to kind of evolve as we go forward here. So imagine we have a little ant that walks through a pool of red ink. And then as the ant walks around the floor, it has a, you know, it leaves behind a little red trail that represents the path of our bug. And, you know, one could ask, is this path a function? Maybe our little ant leaves this mess on our homework here. Is this path a function? Well, you know, the floor represents a flat plane. And so we could actually think of, you know, the path of this ant, as you can see drawn on the screen here, as a curve in the xy plane. And so is this path a function, right? Well, it certainly doesn't pass the vertical line test. It doesn't pass the horizontal line test. And I'm not sure you can find any diagonal line that would intersect this graph in only one location. So with the traditional sense of these line tests, it's not going to be a function. But the graph is a function. If we think of it as a function, a function of time, a function of time, let's say t here. Because at any, at any moment in the journey, we could determine where our ant was located. So let's say when t equals five seconds or whatever, the ant was here. And then later on, when t equals say like 10 seconds, the ant was here. And then at the beginning of the journey, when t equals zero seconds, the ant was right here. And so we can say that the ant's current location is a function of time, and hence the path transverse by the ant is a function of time. But when we start talking about that this point in space here, it has two coordinates, x and y. So we could describe x as a function of time. So we might say something like x equals f of t. And we might also describe the y coordinate as a function of time. We'll say g of t right here. And so this right here is what we refer to as the parametric equations for x and y. And we refer to this letter t, this variable t as its parameter. We often use the letter t because we like in the example of this ant here, we can think of this parameter as a variable of time. And so the location of the ant, we can think of as a parametric function, a function for which the x and y corner are determined by some third independent parameter, which in this case we call t. And so the location of any point in space we can describe as f of t. And g of t. Like so. And so this gives us a parametric function and then the graph of a parametric function we refer to as a parametric curve. Now a parametric curve has a lot more liberty than your traditional graphs that we've seen in like a calculus or an algebra class or trigonometry class. Because this parametric curve does allow the graph to violate the vertical line test and it stools a function. It can violate the horizontal line test. And so we get a lot more liberty in this type of situation. Now I want to kind of mention that this idea of parametric curves and parametric functions really is a precursor to multivariable type calculus questions. Because what we're doing right now is we're just considering a function, some function f here, which would take in one real variable and output to real variables. And so this really is a precursor to higher multivariate settings here. But we'll keep it pretty basic for right now. So let's consider an example of a parametric function and as it's very likely we haven't seen something like this before. Let's pretend like we have no idea what this would look like. Let's sketch and identify the curved function. There's a little type of there. Identify the curve defined by the parametric function. x will satisfy the equation x equals t squared minus 2t and y will satisfy the relationship y equals t plus one. Now we can graph this thing like we do any other graph, right? I mean, when we first learned how to graph functions, it really just comes down to plain connect the dots. We pick certain x values and find their corresponding y coordinates and if we get enough dots, we can connect the dots. Just like we learned in kindergarten, which apparently is college level mathematics here. So let's start with some easy things. If you take your parameter t to be zero, we'll plug it into this equation above, you'll get x equals zero squared minus two times zero, which will of course be zero. Which you can see right here. And if we do that in the other one, y equals zero plus one, you're going to get one, which gives us this coordinate right here. So the way we want to interpret this line of our table is that when t equals zero, x will equal zero and y will equal zero. And so we get this point right here. x equals zero means we're on the y-axis and y equals zero means we're one above the x-axis. Well, what if we try t equals one? Well, take this right here. You plug one into the x equation, plug one into the y equation. If you plug one into the x equation, you'll get one minus two, which is negative one. If you plug one into the y equation, you'll get one plus one, which is two. And that then produces this point right here on the, in the plane, x is negative one, y is two. Try this again when t equals two. Well, plug t equals two into the first equation, you'll get four minus four, which is zero. And then if you plug t, oops, into this one right here, if you plug t equals two into that equation, you get y equals two plus one, which is three, right? And in which case we get this point right here. Isn't that an adorable point? Zero comma three. We can keep on doing this right. Plug in t equals three into the first equation. For x, you'll get nine minus six, which is three. Plug it into the y equation, you'll get three plus one, which is four. And in which case we get this point right here. Three comma four. Try it for x equals four, or t equals four, excuse me. Plug it in here. You'll get 16 minus four. Is that right? No, 16 minus eight, which is eight. And plug in for y, you're going to get four plus one. I think we can all handle that last one there. In which case we then get the point. There you are. Eight comma five. We don't have to be so optimistic right now. We can be negative for a little bit. Let's take tdb negative one, you plug it into here. Well, that works out just fine. You're going to get one plus two, which is three for x-coordinate. And plug it in here, negative one plus one, we still can handle adding one to a number. We're going to get zero, which produces this number right here. This point I should say, three and zero. And then just as one last example, take t equals the negative two, plug it into x. You're going to get four plus four, which is eight. And plug it in here, you're going to get negative one. Thus producing this point right here. So we've now collected a good number of points. If we try to play connect the dots going like this, this is extreme connect the dots because we have negative dots as well. We get something like that. And you know, if we kind of blur vision a little bit, it's like, wow, it looks like a parabola. Now, admittedly, when I blur my vision, I see two parabolas. And so that's sort of a problem. Maybe I should go talk to my optometrist about. But if we connect these dots together in a more smooth fashion and go like a smooth criminal here. We actually do get some that kind of looks like a parabola. Now, if we were to complete this graph with more refined points as we have a lot more points, or if you just use a computer to assist us, we'll get a graph that looks something like the following. We actually do get a parabolic shape like this. And in fact, this right here, this yellow parabola is the graph of the function x equals y squared minus four y plus three. Now, how does one accomplish that? Well, notice before we had y equals t plus one, right? If we were to solve for t, this tells us that t equals y minus one. We also know that x equals t squared minus two t. If we make this substitution in here, we end up with y minus one squared minus two times y minus one. Simplifying here, right? We're going to foil out the y minus one squared that gives y squared minus two y plus one. Distribute the negative two here. We get negative two y plus two in combined like terms. Our favorite part, y squared minus four y plus three. And there you go. That's where we get this thing right here. We've found a parameterization of this curve right here. And so I want to make a comment about that. x equals t squared minus two t. That's what it was. And then y equals t plus one. So when we go from the equation to this, this is what we refer to as a parameterization. Oh, no, I'm not going to fit it parameterization. A parameterization of the equation of this locus right here. And that can be a very useful thing. Sometimes it's helpful to parameterize a curve, especially if the curve isn't sort of a traditional y equals f of x type of situation. And going the other way around, if you have a parameterization, you go back to the original equation for which the parameter t is now absent. We often refer to this as eliminating the parameter. Can we talk about a parametric curve without actually the parameter whatsoever? And so it can be useful and difficult to be able to go back and forth between the parameterization and the removed parameter, like in this situation.