 Welcome back. In the previous video, we saw that we can parameterize a parabola using, you know, using a parameter t, right? x equals yada-yada, y equals yada-yada. We can do this with these parametric equations. Can we do it for the other conic sections, like circles, hyperbolas, ellipses, and such? And the answer is yes, we can do it. So consider the, consider our first parameterization right here, where we're gonna take x equals 3 cosine of t and we're gonna take y equals 3 sine of t, and so what, so this is a parameterization. What type of curve does this form in the plane, and can we remove the parameter? Well, as x and y are defined using trigonometric functions, it turns out we can utilize trigonometric identities to remove and eliminate the parameter here. In particular, we're going to be using the fact that cosine, cosine squared of t plus sine squared of t is equal to 1. So using that observation, notice the following. If I take x squared plus y squared, this will equal 3 cosine squared of t plus 3 sine of t squared. Notice we can distribute here. We're gonna get 9 cosine squared t plus 9 sine squared t. Factoring out the 9, we get 9 times cosine squared t plus sine squared t, and there you are. Cosine squared plus sine squared is equal to 1, this is equal to 9. And therefore, if we take this parameterization, x equals 3 cosine of t, y equals 3 sine t, if we remove the parameter, we're gonna get back the usual equation of a circle, x squared plus y squared equals 9. So this is a circle centered at the origin who was a radius of 3. Notice the 3, of course, was the coefficient in front of the cosine and sine. We'll come back to that in just a second. Now, one does have to also be careful here that as we parameterized the circle here, we end up with a counter-clockwise orientation. Now, I know clocks might seem like this weird thing. Like when you're looking down at your Apple watch right now, are you seeing a watch? I mean, you've seen a watch, but are you seeing a clock? Just so you remember back in the day of the dinosaurs, right? People told time by this thing called a clock, right? And as the hands spin around the clock in this direction, this is what we call clockwise. And therefore, if the hands were to spin backwards, that would be called counterclockwise. The way that this circle is oriented using cosine and sine right here is that we're going to start when t equals 0. We're going to start here at the point 3 comma 0. And as t gets larger, cosine will get smaller, sine will get bigger until it reaches t equals pi halves. And then as you go from pi halves to pi, sine will get smaller, cosine will get bigger until we reach here at this point negative 3 comma 0, which will correspond to pi, t equals pi. Then as t increases from pi to 3 pi halves, you'll see that sine continues to get smaller, cosine got smaller. And then as t will range from 3 pi halves to 2 pi, let's see, cosine will get bigger and sine will get bigger. And so you get this counterclockwise rotation, the way we usually orient angles in trigonometry. It's for this reason that we orient things counterclockwise. It's not just because mathematicians want to mess up our game of uno and go the wrong way. Well, I take that back. We do want to do that. But generalizing what we saw in this example right here, I want to mention that if you take the orientation x equals r cosine t and y equals r sine t, if you remove the parameter, this will become x squared plus y squared equals r squared. This gives us an orientation, a parameterization of the circle, and that orientation will be counterclockwise. Now, if you just want to be a little bit dicey, if you want to swap the roles of cosine and sine, if you make x equal to r sine of t and y equal to r cosine of t, that'll give you the same circle, the exact same circle here, but it'll give you a clockwise orientation in case you're curious with that. All right, another thing we could do is take in this parameterization, we're going to take x equal to 3 cosine of t plus 1 and take y to equal 3 sine of t minus 2. How does that affect the parametric curve? Well, kind of like we did before, if we saw for cosine, right, subtract 1 for x, so x minus 1 is equal to 3 cosine of t. Divide both sides by 3, you're going to get a cosine of t is equal to x minus 1 over 3, right? Do a similar thing for y, y plus 2 is equal to 3 sine of t, and then divide by 3, you're going to get that sine of t is equal to y plus 2 over 3, like so, and so if we play around with our identity from before, 1 equals cosine squared t plus sine squared t. This would equal x minus 1 over 3 squared plus y plus 2 over 3 squared, which sets equal to 1, which of course you can square at the top and bottom here. I'm going to leave the top factored, so we get x minus 1 squared over 9 plus y plus 2 over 3, over, excuse me, go back there, y plus 2 squared over 9, and so if you times both sides of the equation by 9, so we times it by 9 over here to clear the denominators times y 9 over here, you're going to end up with this equation right here, x minus 1 squared plus y plus 2 squared equals 9. This of course is going to be a circle whose center is 1 comma negative 2 and whose radius is still 3, and so what we saw going on here is if you add or subtract from these functions, it has the effect of moving the picture around, so some general principles we can see from this is the following, right? If you take the parameterization x equals r cosine of t plus r, or y equals r sine of t plus k, this will produce for you a circle whose center will be h and k, so we shifted the graph h to the right k up, of course if those are negative values, it'll shift the other way around, the radius is still going to be r and it's still oriented counterclockwise. Of course if you want to clockwise rotation swap the sine and cosine, that's a possibility here, but this plus k and plus h correspond to horizontal and vertical shifts, and I want to mention that this is actually completely general, right? So given the parametric curves x equals f of t and y equals g of t, like so, then x equals f of t plus h comma y equals g of t plus k, this right here, this will correspond, this graph right here will correspond to a horizontal shift by h, so adding h to the x equation causes it to have a horizontal shift, and adding k to the y equation causes a vertical shift, a vertical shift by k, and that's really kind of a cool thing right there that to, because when we learn about shifting functions of the form y plus f of x, it feels like like the stuff that happens to x works backwards than what happens to y, but in the parametric setting the two things actually work together in harmony, plus h to x moves h in the positive, or moves x in the positive direction, adding k to y moves y in the positive y direction, a k vertical right here, and I actually want to show you that when we have worked with the type of functions we've worked in the past, that is y equals f of x, this really is a parametric function because you can always parameterize in the following sort of cheap shot way, x equals f of t, and here f means the exact same thing as it means over here, and then y just equals t, this is a parameterization of this curve, so again parametric curves generalize the notion of function we've seen before, now in this setting if you want to do a horizontal shift of some kind, all one has to do is just add these things together, you know we can add, let's say we want to add a k right here, we can also add to h, notice what happens though it's like we did above like here, in order to move the parameter we had to kind of get rid of everything attached to the cosine of t, we kind of had to free up the parameter over here, and when we do that you end up getting things like the following, this one down here becomes y minus k is equal to t, and this one over here would look like x minus h is equal to f of t, like so, and so what I mean by this is if we, when you start plugging these things in here it looks like, because you know when you first learn about these graph transformations like if I want to shift things up I just have to add k over here, but really you're not, and then you're like if I want to shift things to the right I'm supposed to, I'm supposed to replace x with x minus h, why do you do a minus h instead of a plus k right here, because the issue is if you solve for the function you really should be putting k on the other side, y minus k is equal to f of x, like something like that, and so in this situation, because in the usual function situation we want y equals whatever, we always push everything to the right hand side. In the case of a parametric equation we don't care about that, we have x equals over here, we have y equals over here, and we get a nice symmetry going on when it comes to shifting. Let's look at another example and actually kind of get an idea about stretching and scaling of some kind. If we take the parametric equations x equals 3 cosine of t and y equals 2 sine of t, oh you see what we did there, it was crazy, we switched it up, and we put a 2 instead of a 3. Well what does that do for us in this discussion here? Well like we kind of did in the in the previous example solve for cosine, you're going to get x over 3 equals cosine of t, and if you do that for our sine you're going to y over 2 equals sine of t, and now utilizing the Pythagorean relationship 1 is going to equal cosine squared t plus sine squared t, and making the substitutions that come from this parameterization we get x over 3 squared plus y over 2 squared, that is to say we're going to end up with this equation right here, x squared over 9 plus y squared over 4 equals 1, and this gives us an ellipse, an ellipse in this case it looks like a circle but someone sat on it that they shouldn't have, an ellipse, an ellipse is generally given by the following form x squared over a squared plus y squared over b squared equals 1, where this is an ellipse centered at the origin zero zero, and the distance from the center of the ellipse to this horizontal point right here would be a distance of a, that's this number underneath the x squared, and then the distance upward from the center to this point right here would be b, and so I often refer to this as the horizontal and vertical radii of an ellipse, a circle is in fact just an ellipse which has a uniform radius, the horizontal and vertical radius are the same, but for for a general ellipse these numbers could be different right here, this still is going to be a counterclockwise orientation, and generally speaking what we get here from this parameterization a cosine of t for x, b sine of t for y, this will give us the ellipse x squared over a squared plus y squared over b squared equals 1, so the coefficients right here determine the horizontal and vertical radii of this ellipse, we can also throw in some vertical shifts going on here plus h and plus k, and this will actually shift the ellipse as well so we can shift the center to be h and k if we're so inclined, now this idea of throwing in different coefficients a and b actually comes to the idea of vertical and horizontal shifting, so if we have a function given x equals f of t and y equals g of t, then if we replace the graph with x equals a f of t and y equals b g of t, what this effect to the graph is, what the effect of the graph is going to be is that the a right here is going to do a horizontal or horizontal stretch, we're going to stretch the graph horizontally by a factor of a and this coefficient b right here, this will have the effect of a vertical stretch, a vertical stretch by a factor of b, okay, and so that's exactly what an ellipse is, an ellipse is just a circle which has been distorted, you stretch the horizontal one way or you stretch the vertical in another way and so we can do these type of these, we can do these stretches and curves stretching and scaling and shifting of parametric functions in a very simple way and so by transforming a circle we can get all these different types of ellipses. Let's also take a look at our last conic section a hyperbola, a hyperbola turns out we can find out by just using other trigonometric functions if we take x equals to secant of t and y equals four tangent of t, well the Pythagorean relationship we want to use here of course is that secant squared of t is equal to tangent squared of t plus one and so if we solve for secant and tangent we can make the substitution in here, secant is going to become x over two squared, this is equal to tangent which tangent is going to be y over four squared plus one, if we move the y to the other side we end up with x squared over four minus y squared over 16 is equal to one and this gives us the equation of a hyperbola which the hyperbola if you take the center this one centered at the origin you're going to get this nice little box here where you go over here this is a distance of a if you go the distance of b right here you get this nice little box that kind of helps you determine how big your hyperbola is going to be there we go so this is a this is a 2a by 2b box right here the edges of the box will touch the vertices of the hyperbola right here so this one has the point two comma a and this one has the point negative two comma said sorry two comma zero and negative two comma zero um uh hyperbola also has asymptotes that is it people when you see a hyperbola it looks like it's two parabolas concave and away from each other um that's not exactly true because as you get far away from the center of the hyperbola this thing will asymptotically approach these diagonal lines you see right here these are oblique asymptotes for this hyperbola one interesting thing about the hyperbola is that if you look at the diagonals of this box I drew this a by b box the hyperbola the asymptotes of the hyperbola are the diagonals of this box and so that's how this box kind of represents the the dimensions of this hyperbola going to erase some of this stuff right here and so this gives us this would give us an example of a horizontal hyperbola and so in some greater generality if we take if we take the orientation x equals a secant of t y equals b secant of t you're going to get something like this x squared over a squared minus y squared over b squared equals one uh this is going to give you a horizontal hyperbola whose horizontal radius will be a whose vertical radius will be b um it is centered at the origin and it'll have the asymptotes y equals plus or minus b over a x we can incorporate some shifts into this which of course i see another typo yikes how embarrassing x equals a secant t of h and y equals b tangent of t plus k this will give you another horizontal hyperbola right here x minus h squared over a squared minus y minus k squared over b squared equals one uh this gives you the asymptotes of the hyperbola y minus k equals plus or minus b over a times x minus h this will be centered at the origin right um if you want to have a vertical hyperbola what you do is you're just going to swap tangent and secant swap those things around uh so you're going to get x equals a tangent of t and y equals b secant of t and so what that does to get this vertical hyperbola i'm gonna erase this right here the vertical hyperbola is going to be this region right here so it's going to go through the vertex four like so and it'll go through negative four right here the asymptotes will stay the exact same uh but you get this blue one right here this is our vertical hyperbola that is it concaves up and down as opposed to the horizontal one which concaves left and right it'll have the formula y squared over b squared minus x squared over a squared equals one and you can also shift it h and k as well so you can actually describe all of the conic sections using uh these trigonometric parameterizations like the circle the ellipse uh the hyperbola you can do it for parabolas as well although you don't need you don't need trigonometry to get a parabola here um if this is of use tangent secant to get a hyperbola right if you want to mess up the orientation because this one's flowing this way if you want to mess up the orientation switch to like secant and cotangent those will also give you hyperbolas but uh it'll give you a different orientation if you want to and so as one last quick example here this one's not a conic section but i did want to mention since we did all of these uh all of these trigonometric parameterizations you know the those parameterizations like the x equals cosine the y equals sine we were doing like a cosine a and b sine that kind of changed that changed the radii of the ellipse or the hyperbola and you know for like in a trigonometry class you would talk about those as amplitude changes right and the plus k and the plus h that we saw uh those would be like a shift a vertical shift up and down on the graph uh again something also we deal with in when we when we graph a sinusoidal waves one thing we haven't really talked about yet is if you have a trigonometric parameterization what if you were to change the period uh what happens to a period change right here and so you get something that starts off like a circle right it's going to start off at the point uh one comma zero but what happens is a sine uh is changing faster now than cosine you don't quite you don't have the usual circular arc like this instead you know x is changing at the same pace but y changes at the at a faster pace and so you end up getting this like bow type type region and so as you start getting exotic uh with your parameterizations doing some period changes and things you can get some really interesting looking graphs which again these are functions of time or in this case you can think of t as your angle with respect to the x axis this is a function of t even if it's not a function of the usual sense and I should mention that most graphing calculators and computer graphing programs can be used to graph curves defined by parametric equations typically this involves switching a setting uh somewhere in your in your mode or setting menu from the cartesian or rectangular mode to the parametric mode feel free to use a graphing utility when you're working on problems with parametric functions it can be very difficult to graph these things if we're not used to them uh so if you have a graphing character you can use those for these if you don't my recommendation would go to a to a free website like desmos.com it's a free online graphing calculator and you can graph these things very nicely just in the box where you're supposed to type something you're just going to type in for your x coordinate the function you'd so you type in cosine of t comma and then you type in sine of two t like so you just have to type that in there and then once you hit enter it'll then create a domain for you t by default it'll go from zero to one and you can change these numbers if you want so you could change this from like zero to two pi or something like that because this graph this graph will be two pi periodic because of the cosine and the two the sine of two t there and separates the end of our lecture 29 about parametric functions we're going to talk some more about parametric functions of course in upcoming videos so as to take do stay tuned for those and I will see us next time bye everyone