Intermediate Alg Section 9.4 Ellipse And Hyperbola

Hello and welcome to Bay College's Intermediate Algebra online lectures. Today we're going to look at section 9.4 which deals with the ellipse and the hyperbola. The first thing we're gonna look at is the ellipse, now what I have here an ellipse is similar to a circle, except it's not perfectly round, it is stretched either in the x or in the y or compressed in the y or compressed in the x. it's not nice and round, ok, it's a circle that's been stretched out. One thing about that, is this is the general equation of an ellipse if it's centered at the origin, there's no h and k values, h and k in this case, would be 0, just like in a circle. But we notice that we have this a squared in the denominator for our x term and our b squared in the denominator of our y term. Essentially what this means is x has a different coefficient then y. That's what causes our stretch or compression in y or x. So, this is standard form of a circle. Now what we have to do is, we have to define a few things about an ellipse. Any point on an ellipse, is the sum of the distances from these points here, to any point on this circle. It's the sum of the distances. That's why we have this addition in here. Now interesting enough is these have a special word, they're called the focus. This is a focus and this is a focus. Together multiple, we call them foci. So there are two foci in this ellipse, and the sum of their differences, or the sum of their distances, is always a constant no matter where we are on this ellipse. So this is our equation. So one thing we could do is if this is on a graph, and for this particular parabola, or excuse me, ellipse. We're saying that it's centered at the origin. h and k are 0. So what we have to do, is we can use something to find the x intercepts and y intercepts to have some points of reference from this center here, to any other value on this ellipse. Essentially to find the x intercepts this is our tool right here. The a value in the denominator. We have to think, what is a? Well a is plus or minus the square root of whatever number is down here, because it's squared we'd want to undo it by taking its square root. So we have a negative a and a positive a. It's x intercepts are going to be some distance to the right that's our positive a. Some distance to the left, our negative a. So the y value for any x intercept is zero so we can find the x intercepts, this point here would be a, o. This point here would be negative a 0. The reason why it's plus or minus, well if I square negative a I get a squared. If I square positive a, I get a squared. So we have to remember that plus or minus value. If we want to find the y intercepts this value here, and this value here, well we can do that simply by understanding what an intercept is. Well the y intercept is when x is 0. We are on the y axis. Well that's going to give us plus or minus b. Because negative b squared is b, or excuse me, negative b quantity squared is b squared. Positive b squared is b squared. Kind of a redundant statement. So this point here is my 0 positive b and this value down here is my 0 negative b. So now we are going to tie all this together, and actually graph, or sketch the graph. If we want to move the camera over here, we'll see we'll find our a and b values. Well if we look at this, I say wait a second, I don't see a b value. Well we have to think about it in terms of a fraction. Alright, if it's just a value, we can always right over one to have some value there. The first thing we want to do is determine what are these a and b values? What value squared would give me 9? Well I know it's a plus or minus 3. What value squared would give me 1. Well plus or minus 1, 1 times 1, right, is going to give me 1. Now notice in order to graph an ellipse, it has to be sent equal to 1. That wasn't the case in a circle. So here we want this to be 1. If it's not 1, we'll see in the next example, we have to do something extra to make it 1. So now with these values, essentially, I'm gonna plot four points. I'm gonna plot my x intercepts for this ellipse which are going to be plus and minus 3 when y is 0. So I have plus 3 and minus 3. In the x direction. And then I'm going to plot my y intercepts plus 1, minus 1, and notice this one is centered at the origin 'cause there is no h value. There is no k value. So now I can go ahead and graph my ellipse. Pretty simple. If it's in standard form, we can go ahead and we can just find these a's and b's and use those to go from the center a values to the right, a values to the left, from the center of my ellipse, b value up, b value down. Alright let's look at the next example here. If we look at this one we notice hey, this is not set equal to 1. It's not in standard form, so our first step is to say let's set this equal to 1. that's easy enough to do by dividing all the terms by this constant out here and then we'll do a little bit of simplifying. So I'm gonna write right here. This here doesn't simplify, it's x squared over 16, but 4 over 16 reduces to a 1/4 So I have y squared over 4 equals 16 divided by 16 is 1. Now it's set equal to 1. Again, I assess this and say, ok I'm not adding or subtracting anything to my x's or y's so it centered at the origin. Let's find that a value. a is plus or minus 4. The square root of this is plus or minus 4. My b value, if this is b squared, then b must be plus or minus 2. And now I can go ahead and graph these centered from the origin, I'm gonna go 4 to the right. I'm gonna go 4 to the left. I'm gonna go up 2 with my b value and down 2, and now I can graph my ellipse. Pretty simple. Alright let's look at one more example here. we have this ellipse here we notice it's addition. We have x minus 3 quantity squared over 9. y plus 3 quantity squared over 16 equals 1. Well it's set equal to 1, but this is different than the other ones. There is an h value in here. something we're adding or subtracting from x before we square it and there's a k value here something we're adding or subtracting from y before we square it. So let's do a couple of things; First thing I want to do is assess, this is essentially our ellipse. Right? Some x value squared over a squared y value squared over b squared equal to 1. So I want to determine what is that h k value? This is the center of my ellipse. So I look at this and say it's always the opposite of what's in there, so i'm gonna write a positive 3. Here, a negative 3. So this is not centered at the origin. It's elsewhere. So I found my h, k value. So I'm going to go over to the graph here, and I'm going to plot the center. So I'm gonna go 3 to the right for x and down 3 in my y direction. This is the center of my ellipse. A very important concept. Now looking at this equation, I can determine what a is, and what b is. Well since a is 9, or a squared is 9. I know that a is plus or minus 3. Those two values will give me 9 when I square it. my b value here is 16. So I know, plus or minus 4 squared will give me 16. Now, how this is different from the examples we looked at that were center at the origin, now we're not finding x intercepts or y intercepts anymore, we're finding some point on our graph relative to the center. So what I'm going to do is from the center, I'm going to go a units left and right. Positive 3 to the right. So I'm gonna go 1, 2, 3. Negative 3 units to the left 1, 2, 3 and there we are. So I have those two values. Now for my b I have to go up 4 and down 4 from the center of my reference point. So I'm going to go up 1, 2, 3, 4. And I'm going to go down, 1, 2, 3, 4. And now I can draw my ellipse. And you can kinda see this is similar to a circle, but it's stretched a little bit in y. And if we go back to this. Well why is it stretched more in the y? Our b value's larger than our a value. So it's pulling it further in the y. So we have our ellipse. The next thing we're going to look at is a hyperbola. Now if we look at the equation of a hyperbola we see it's very similar to that of an ellipse. There is only one significant difference. And this is one way we can differentiate between what is an ellipse and what is a hyperbola. The sign in between the terms is subtraction. So we're not adding, we're finding the difference between their foci, instead of their sum between their foci. So if we look at this, we still see our a squared and our b squared's, but the difference is subtraction. Well we can have two versions of this because it does matter, we don't want to make a sign error. What if we had y squared over b squared minus the x squared term over the a squared? So we have subtraction there. This tells us something significant about the hyperbola. It tells us the direction in which this hyperbola opens. If we look at this value, if the x squared term is first, it means it opens in the x direction. So we can see here, it opens to the left in x and it opens to the right in x. If we have the y squared term first our positive value. That tells me it opens in the y direction. It opens either up or it opens down. So when you first see this ellipse, recognize that subtraction, and you say, okay, now I'm dealing with an ellipse. What direction does it open? Which of these values is the positive value? Which one has the positive coefficient? In this case, it's the x. So it opens in the x directions. This one is the y. It opens in the y directions. Now we still have to find these values a and b, in order to graph this, but we have to use them a little bit differently. The points we're gonna graph are going to be plus or minus a, and plus or minus b. This actually gives us four points. We have positive a positive b. We have negative a positive b. We have positive a negative b, and we have negative a negative b. Four points that we can graph. And I'm just going to look at this one right here. This is the four points we can graph. And in order to graph a hyperbola, we have to put these four points on our graph. And we have to do a little bit with them. So this is my value a, b, this is my value negative a, b, this is my value negative a, negative b, and this is my value negative a positive b. So what we do once we plot these four points from our a's and b's that we can determine, we can draw a little rectangle. That gives us some idea, some reference point, for the center of our hyperbola. In these cases, and in the hawks learning system, you'll see that the center is going to be 0, 0 for your hyperbolas. It gets a little more complicated when it's not, because you have to move these points all over, so for simplicity sake, your h, k will always be 0 for this course. Now, once we draw that rectangle, we can draw diagonals through the corners of our rectangle, through our points. What this tells us is the end behavior of the graph. As x gets really big as we go to the right or left our graph is gonna approach these lines. It's gonna get closer and closer as x goes to infinity. And since we know that this example, it opens left or right, in the x directions. I can start right at the edge of my box in between the two points, and I can draw my parabola type shape towards these lines. And these lines are defined as asymptotes. It tells us the end behavior of the graph. What does it look like as it goes further and further? Well it starts to look like this line. So as we get further away, same thing with this, the only difference is it opens up or down. So we plot our a and b points, we draw a rectangle so that we can connect the corners with a diagonal dashed line our asymptotes. And then we could say, okay, well this one opens up and down so I start at the center of this little box here, and I go up towards that asymptote, up towards this asymptote. Start right here at the center and move left and right towards our asymptotes. So let's look at an example. Here we have x squared over 9 minus y squared equals 1. I see that subtraction, I know immediately we're dealing with a hyperbola, and because the x squared term is the positive value its first here, it opens left and right in the x direction. So I'm going to determine, what my a and b values are. Well a is the value squared that gives me 9. That's plus or minus 3. We look at this, and I say, wait a second, there's no denominator. Well we could think of it is being over 1, and it's set equal to 1, that's good news, so I say ok, plus or minus 1 will give me this value if I square it. Alright, so now we're ready to start drawing our rectangle. So I'm going to go plus 3 and up 1. That is the value 3, 1 It's in the first quadrant Both values are positive. And then I can go negative 3 positive 1. And I can go negative 3 negative 1. And I can go positive 3 negative 1. So now I have these four points that I can make a little dotted rectangle. There's my rectangle. Now I have to draw my asymptotes through the diagonal. So I'm just going to go right through here. And I apologize for that not being perfect, but it is what it is. And then I can draw my diagonal this way, and I know that this parabola opens left or right that x term is first so I'm gonna start right here in the middle of the box, the box. And approach those asymptotes, and make this line nice and dark, so we can distinguish it from all the other stuff I had to put on the graph, in order to graph it. And that is my hyperbola. hyper bola. It's more than one parabola, right? One going this way, and one going that way. And if we think about ellipses, they are similar to circles. They are similar to parabolas, when we look at hyperbolas, and the reason why this is, is because, all of these things: parabolas circles and hyperbolas They're essentially conic sections. They're how we would slice a series of cones. And if we think about what an ellipse is, well if we have an x squared term, usually that's a..... parabola. And if we have a y squared term, well that's just another parabola. What would happen if we added those two together? Well if I have a parabola and a parabola, I add them together, I end up with an ellipse. What would happen if I subtracted two parabolas? Well if I subtract them, they're gonna go in opposite directions, which is the shape of my hyperbola. So you can think of it that way as well because we are dealing with these squared terms. Alright, let's look at this example here. This is a hyperbola. I know this because I see the subtraction of these squared terms. But it's not in standard form. We have to get it set equal to 1. So in order to do that I'm gonna use this black marker here. In order to get this set equal to 1, I need to divide everything by 36. This needs to be 1. So I'm going to divide by 36 to all the terms. And I'm going to do a little bit of simplifying. So here I get y squared over 9 minus x squared over 36 equals 1. 36 divided by 36. And now I can say, okay well it's centered at the origin, as they will always be in, for this class. Let's determine what our a and b is. Now this is where we have to be very careful, because this value is not the a squared value. a squared is our x value. So my a is going to be plus or minus 6. The square root of 36. My b value is plus or minus 3. Square root of 9. Now to graph this. And I should have put my little graph on here before I started. I'm gonna plot the four points that I can make with the combination of these. I'm gonna go positive 6 to the right and 3 up 3 down. I'm gonna go negative 6 to the left 3 up and 3 down. Now I'm gonna draw my rectangle. Here's my rectangle. And my diagonals. And I know this is a little sloppy but, it is what it is. Now what direction does it open? Let's go back and assess. The y squared value is the positive one. It is first here. So we know it opens in the y. Up and down. So I'm gonna start right at the center of my box in between these two points and I'm gonna approach these asymptotes. Very stretched out here. Notice that's a very large coefficient so it's stretching it out in the x directions. So it's not getting very big in the y right away. And that is our hyperbola. Alright so let's do a little assessment of what we've covered so far in sections 9.1 9.2, 9.3 and 9.4. What we're gonna look at here is our equations. What have we seen so far? Well when it comes to parabolas, we can have two types of parabolas. We can have a parabola as a quadratic. This is a parabola that either opens up or down depending on what a is. We could write this in another form, if we complete the square we're going to get that value right there. Well there's actually another parabola that we discovered in 9.2 Which was what if the y value is the squared value? Ok, so this would be a parabola that opens either to the left or to the right, because this is a parabola on it's side. It's the y value that's being squared and we could do the same thing to that except it would be x and y. They're inverses. Alright, if we look at circles, well we know that there are two versions of the circle. We've seen that we can have x minus h quantity squared plus y minus k quantity squared equals r squared. This is our circle in standard form. It can also be in general form, where we'd have to complete the square for the x terms, complete the square for the y terms, and write it back into this form for graphing purposes. So we have our standard form, and our general form. Which is just expanded, we have to unexpand it to get it back to that. We've seen with the ellipse x squared over a squared plus y squared over b squared. Actually what we should do, is realize that not all of our ellipse are going to be centered at the origin, but we do want them set equal to 1. This is our standard form for an ellipse and if it's not equal to 1, we divide everything by this constant, to get it into this form, reduce any fractions. The hyperbola is essentially the same as an ellipse, the only difference is it's subtraction. We're subtracting these two squared values. This one would open up in the x direction, either left or right. And this one here, oops make sure I get my right values in there. This one here would either open up or down. So we can see that our two versions of hyperbolas in standard form, but notice they are always set equal to 1. If they're not equal to 1, make sure you divide through by that constant to put it into standard form. Now one thing about this class is, you're going to be asked on tests and on the final exam to be able to identify the shape of an equation, by its form here. So if you see something in any of these forms, you gotta say, hey I recognize that as a parabola. I recognize either of these as a circle. I recognize this as an ellipse, or I recognize one of these as a hyperbola. One opens in the x directions, one opens in the y direction. So this has been section 9.4 ellipses and hyperbolas. Thank you for watching.