>> As you might guess, in parametric equations time or T is not the only parameter that is available to us or possible in these equations. Here we have a pair of parametric equations in which an angle theta is the parameter. We're going to graph these in just a moment, but let's examine some aspects of the graph, what we might anticipate here. We're going to let theta take on values from zero to 2 pie. But because we're going through one complete cycle for theta, let's think about the values for X and Y that we can anticipate because we, a lot of times we want to set up our window before we actually look at a graph. And how would we set up our window for X minimums and maximums, Y minimums and maximums and so forth. Well we know that for cosign, the cosign of an angle those values will change from negative 1 to 1. So times negative 1 to 1 means that X will take on values from negative 3 to 3. In a similar way, the sign of angles changes from negative 1 to 1 as well. And so four times negative 1 to 1 means that Y will change from negative 4 to 4. So that allows us to set up our window. Now we're using radium mode here and, and we're going to look at the graph in just a second. You know if we want to though with this set of parametric equations, we could remove the parameter. We could go through a, a, the process of setting up an equation that is rectangular in form and look at that maybe at the same time. But maybe we could gain some insight about what we may expect from the graph of these parametric equations by looking at the rectangular equation. All right, let's, let's go through that process. We take our two equations here, our parametric equations and we solve them for something that they have in common. You know we can't very well solve for theta itself, but we can do this. We can take X and solve it for cosign theta. That's pretty easy, divide on both sides by 3 and we get this. For this other equation we can solve for sign theta, divide on both sides by 4 and we get sign theta is Y over 4. Now all we need is some relationship involving cosign theta and sign theta. Well, the Pythagorean identity gives us that very easily. So we take our Pythagorean identity and just pop the information in. Sign theta we know is Y over 4. So we have Y over 4 squared here. Cosign theta is X over 3 squared. Once we square, we see an equation that looks vaguely familiar. You see, this is the equation corresponding with an ellipse. Now we're a little more familiar I think from our experience with ellipses to see the, the X in the first numerator of our fractions, so I've rewritten here in a more familiar form. Well, from here we, we can make an analysis even before we look at the graph, we know that the larger of the two numbers is the 16 and so our major axis is vertical here. And let's see, the A value is 4, the B value is 3 from the 9 you see. So we have all of that information right away. By the way, notice that if the A value is 4, gee that means that our from the origin now and the, the origin is the center of our ellipse. We go a distance of 4 to our vertex points in both directions vertically. So we're going up 4 and down 4. That corresponds with those, the Y minimum and maximum that we talked about a few moments ago. And the, the B distance you see, the distance from the center of the ellipse to the ends of the minor axis, gee that's a horizontal idea here. And let's see, so we're talking about 3 to the left of the origin, 3 to the right of the origin, so our X values are going to go from negative 3 to 3. Gee, that's exactly what we found over here. And that's no idle coincidence that we can make an analysis like this and have a corresponding idea with the parametric equations as well as the rectangular form of the two equations together.