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About this Lesson
 Type: Video Tutorial
 Length: 13:34
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 146 MB
 Posted: 06/26/2009
This lesson is part of the following series:
College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Algebra Prerequisites (60 lessons, $69.30)
College Algebra: Polynomial & Rational Functions (23 lessons, $35.64)
College Algebra: Graphing Rational Functions (4 lessons, $5.94)
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
 Thinkwell
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11/13/2008
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Let's put together these notions of finding the asymptotes to actually start graphing some of these rational functions. So, if someone gives you a rational function and wants you to graph, the first thing you should do is probably just find the asymptotes, because that will cut the world up into little pieces. Then you can just figure out, by maybe plotting maybe one or two points, how the curve is going to fit in between those pieces. So, for example, let's suppose we have a function f(x) and its equation is 3 divided by (2x  4). I want to graph that.
The first thing I'm going to do is to find the asymptotes. Now, there are two types of asymptotes. There are just vertical asymptotes and horizontal asymptotes. What I tend to do first, is find the vertical asymptotes first. So, let's just do that, just for fun. Vertical asymptotesnow suppose you don't even remember what they are. Let's just think about it. A vertical asymptote, well I hope you know what vertical means. It means like this. So, what would it mean to have a vertical asymptote? It would mean somehow the curve would come up to it, but would not want to touch it. That would be an x value. Vertical means an x value that the function can't touch. Well, that would mean we'd look where the denominator is equal to 0 and that would be a candidate.
Here, there's no factoring. Here there's no factoring I can do. Everything is just in lowest terms. So, where does the bottom equal 0? Well, the bottom equals 0 when 2x  4 = 0. Which means that x would equal 2. So, we have a vertical asymptote at x = 2. Now, I want to point out a really great mistake. It doesn't quite make my top ten list, but it's a really great mistake. That is, you get confused with vertical asymptotes being x equal something. Sometimes people write vertical asymptotes means y equals something, because vertical, right. It's like the yaxis. Well, that's a great mistake, but you've got to think through what's going on. A vertical line is an x equal something, because everywhere on that line x is the same value. So, vertical asymptotes are always in the form x equals a number. A horizontal line is always at the same height and that means that y equals a number.
Now, how do I find the horizontal asymptote? For the horizontal asymptote I do the race thing. Right? I want to see what's happening as I go up to the horizon. So, I let x get really, really big. It's the horse race. What happens? Well, this bottom is going to infinity really, really fast. The top is not moving at all. So, this whole thing is squashing down to 0. The bottom is getting really big. The whole thing goes down to 0. So, we have a horizontal asymptote, since horizontal means y. So, add y = 0. Which is a fancy way of saying the xaxis. Now, those two pieces of information actually will allow us to put together a reasonably accurate curve.
Let's see how this is going to play out. You might want to actually mark in your asymptotes. So, there's a vertical asymptote at x = 2. That means if I go over to 2, I have a vertical line at two units over. You could even right in here, if you really want to be fancy, and say, x = 2. That's a vertical asymptote. There's a horizontal asymptote at y = 0. Which is just the xaxis. I don't know how you'd label that, but you draw like this. This is y = 0 and that's going to be the horizontal asymptote. What that means is, our function is going to somehow live in some of these regions.
Since it's a function it can't live both here and here, because then it would fail the vertical line test. So, it's either going to be here and here, here and here, here and here, here and here. Now, how am I going to find out where it lives? Just plot a couple of points and see. For example, suppose we plot in 0. That's an easy point to put in. If I let x be 0, let's see what's y is, or f(0) is. So, f(0), if I put in 0, it's just . So, I see that at 0 I'm at . I have to go down to right around here. So, there's the point. Now, for me that is enough, believe or not, to tell me what's going. If there's a point there, what that tells me is that probably my curve is going to be somewhere in here. I want to hug up against here and hug up against there. So, really, it must look like this.
You see how one point just gave me that? Once I knew which of the regions I was in, I was able to sort of put the rest of the graph in, hugging up against the asymptotes I found. Now, what about here? You see, I just can't say, "I live here." I may live here. So, I've got to take another point. Let's just try 3. So, let's look at f(3). If I plug in 3 here, I see 3 divided by, and this would be 6  4, which is 2. So, I see ³/[2]. I see that x = 3, I go up ³/[2], which is 1. That's enough to tell me what happens. This is not going to zoom down. It's going to have to hug up to here and I also have this asymptote here. So, I must be hugging just like that. That's a pretty neat sketch, a pretty accurate sketch of this rational function.
Let's try another one. Let's say we have f(x) = 5 + 3x and I divide the whole thing by x² + 4x 5. Okay, now what do I do? Well, first of all, I will find the vertical asymptotes. The order doesn't make a difference here, by the way. So, this is just how I do things. The vertical asymptotes, I have to factor the denominator. So, let's factor that denominator. Let's hope it can be factored. X, x, opposite signs, 5 and 1 looks very well. Where does the bottom equal 0? At x = 1 and at x = 5. Now I've got to make sure that none of those points make the top 0. If I plug in a 1 there, I see a 5 + 3. That's not 0. If I plug in a 5, I see 5  15. Not 0. So, there are going to be two vertical asymptotes. You can have more than one vertical asymptote. So, you have vertical asymptote at x equal, remember vertical asymptotes are always x's, x = 1 and at x = 5. Cool. This is going to be a real interesting one.
What about horizontal? This is going to be a y equals something. What do you do? Now you have the race. You let the x's go off into infinity and race. What happens? The highest power here is just the first, x^1. The highest power here is an x². Who wins out the race? Well, the x² does. It goes to infinity at a faster rate. So, that pulls this whole thing down, because this is going to infinity really, really quickly. Which means this whole thing is getting smaller, and smaller, and smaller. Since this denominator has a higher power than the numerator, the horizontal asymptote is y = 0. The bottom wins. The bottom wins, horizontal asymptote is at y = 0.
Well, that gives us a sense of what's going on. Let's now graph it and see if we can put in, fill in the pictures here. So, let's see, we have a vertical asymptote at x = 1. I'll mark those in. So, x = 1, that's one unit this way. Then I've got one at 5. Also, y = 0, which is a fancy way of saying the xaxis is going to be one. So, now we actually see I have six regions. A region over here, a region over here, a region over here, region over here, region over here, region over here and I've got to figure out how the thing is going to play out. So, again, I'll just pick points and see what happens.
Let's take a look, for example, at 0. So, what's f(0)? If I plug in 0 for x, I see 5 on top and 5 on the bottom. That's just 1. So, at 0, I'm at 1. What about at 1? Where is 1? So, let's look at f(1). Here on the top, I see 5  3, that's just a 2. At the bottom I see what? 1² is 1 and then I have a 1  4, which would then be a3 and 5 is 8. Here I see a ^1/[4]. Here at 1, I'm at ^1/[4]. Now, let's see, what's going to happen here? How am I going to inch up? It's plain that I'm going to come down this way, but then what do I o on this side? Well, one thing I can do, is try to plug in a number that's actually near 5. Like I could plug in for example, 4, if you want to see what's going on. Let's just try that.
4, if I put in a 4 here, this becomes 12. 5 + (12) is about 7. What do I have on the bottom? 4² is 16, but then I have a 16. So, that's 0. So, I see a 5. Look, that's a + ^7/[5]. So, there I see + ^7/[5]. So, at 4 did I do that correctly? Let's just see. That looks good. This is going to be a 16  16. Then I have a 5. That looks good. Looks good to me. So, I see 1 ^1/[5], basely. It looks like this. Okay? All right, so, what I'm seeing now is that this thing seems to have an interesting shape. There's some stuff on top and there's some stuff on the bottom. Let's see what happens now over here.
Let's take a point that's bigger than 1. Let's pick a point here, let's say f(2). What's that? I put a 2 in here, I get 6 and 5 is a 11. On the bottom, what do I have? I have a 4 and then an 8. 4 and 8 is 12. I subtract off 5 and I get 7. So, here I get ^11/[7], which is 1 something. This is the only region. So, I can sort of sketch this in here. It must look like that. Quite nice. What happens in this part of the world here? Let's put in 6. The numbers are getting pretty big. You might need a calculator. I might need a calculator, you probably don't.
Let's see, 6, that's a 18, 5  18 is going to be 13 Then what do we have here? Here we see a 36. Then I subtract off 24. So, I have 36  24. What in the world is that? That's going to be like 12. Then I subtract another 5. So, that's like another 7. So, I see ^13/[7]. So, at 6 I am at basically 2. That's enough for me to tell me that what must happens is I inch up like this. This looks like that. So, the only mystery thing here is what's happening in this middle. So, is it going to go like this, or how? Well, one way of thinking about this, by the way, is just looking at the factorization and asking exactly how that bottom looks. We factored that already. It's going to be (x + 5)(x  1).
Now, you just think about it. As I get closer and closer to 5, but always a little bit less than 5, what happens? On the top, I'm certainly negative, because I'm putting in values that are negative here, but pretty close to 5. This is actually getting pretty close to 10ish. What's happening here? If I put in something that's just negative, but just a little bit larger than 5, this is positive. When I put this in, this is negative. So, the bottom is negative, the top is negative. The whole thing becomes positive. So, I must be getting up. Therefore, I see I must be going up like this. Somehow or another, I come down like that. Look how exotic that looks. I'm asymptotic down that way.
You can see where I should cross the axis somewhere. Where do I cross the axis? That's where this whole thing equals 0 and if you solve that, you'll see the top equals 0, when x = ^5/[3]. So, ^5/[3]. So, this is a very exotic function. How did I graph it? I put in the asymptotes and very carefully looked at the intervals and saw how the curve should go. We'll take a look at a whole bunch more examples coming up next.
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