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College Algebra: Basic Rational Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 9:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 100 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: Rational Functions (2 lessons, $2.97)

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 at 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 UT-Austin 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, p-adic 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

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So let's just take a look at some very basic, very simple rational functions and get a sense of how they visually look. Because what I want to have happen is like with the polynomial stuff, with like quadratics and cubics and stuff, when someone just says to you a very simple rational function, it would be great if in your mind you had just even a very rough visual sense of what's going on, what it looks like.
So let's just start off with the very, very basic people, and then we'll build from there. The first person I want to look at is the following: f(x) = . So notice that really is a rational function because I've got a polynomial, a very simply one, namely the polynomial 1, it's constant, divided by a simple polynomial just x. So what does that look like? Well, we could make a little chart and have sort of x and f(x) and plot some points. So let's put in -2, -1, 0, 1, and 2. If I plug in -2 I just take the reciprocal, so I see -1/2. If I plug in -1 I just see -1. If I plug in zero, that's undefined. In fact, zero is not a point in the domain, so, in fact, this is undefined. Not allowed. If I plug in 1 I get 1, if I plug in 2 I get . So if I plot this what would this look like? Well, I have -2, -1/2, so that would come in way over here. And then I have (-1, -1), so that's sort of over here. You can see it sort of looks like this. And then I can't cross that line, but I have (1, 1), and then I have (2, ). And if you think about it, if I put bigger and bigger x values in, those reciprocals will get smaller and smaller. Like if I put in 1000 I'll have 1/1000. If I put in 1,000,000, I'll have 1/1,000,000, so as I go further out, this curve is getting closer and closer to the line y = zero. So we have a graph that looks like this. And that's sort of a standard-looking hyperbola. You can see it. It has as its vertical asymptote the y-axis, and as its horizontal asymptote, the x-axis, and just hugs up against both those things. That is the standard function. So when I say you should have a picture that looks just like that. Just got that from plotting a few points.
Let's try another example. How about f(x) = . Let's put another more exotic polynomial down there. Let's see what that would look like. Well, we plot some points. We still see that zero is not a good thing to plot in there, so let's avoid the zero point. But I'll put in -2, I'll put in -1, I'll put in 1, and I'll put in 2. If I put in -2 here, I see , which is . If I put a -1 and square it, downstairs I still get 1. 1 is 1 and 2 is . So now what does this look like? Well, now what I see is the following. At -2-1/4, so at -2 I'm really low. At -1 I'm at 1, so now I'm just going to be climbing a little bit. And then we know we don't cross that line because, of course, I can't have x = 0 anywhere, but then at 1 I'm at 1, and at 2 I'm at . So what I see is I'm going to be heading up to this, and again, as I put bigger and bigger values in, whether they're negative or positive, I'm going to be getting smaller and smaller positive numbers. So what I see is this kind of function. It has a vertical asymptote, that's the y-axis and the horizontal asymptote is the x-axis, but it's different than the one we just saw, because this one has both of its wings on the top because I'm squaring. I can never have a -y value, but since I'm squaring those values will always be positive. So, in fact, it's sort of this picture flipped up, but also the squared means it's a little bit sharper. It's a little teeny bit sharper function. Anyway, that's what this looks like.
Let's try another one. How about the negative of the first one? So let's let f(x) = -. What would that look like? Well, if you think about it, it would be exactly this picture, but all the negative values would become positive and all the positive values would become negative. You can make a little table of them, but you see that in fact, we just would get this picture flipped over the x-axis. And so what we'd see here would just look like this. You get the exact same picture, but what used to be negative will now be positive, and what used to be positive will now be negative. So it's the exact same picture as before, but just flip-flop the roles. So it's the same function, just flipped over the x-axis.
How about this one? I'll use green. How about f(x) = . If we just remember the shifting and the translating thing that we talked about a while back, we can actually see, instead of just plotting points, that this is going to be the same thing as this, but I'm changing the y values. I'm adding 1. So remember, if you add to y, go high. So this is just going to raise everything up one unit. So I should just take this picture right here and just shift it up one unit. And so if I do that, I think what I'd see is this. Let's try this right now. The exact same picture as this, but just move everything up, including the asymptote, so that asymptote is going to be now at 1. So let's see how that's going to play out. And now what I see here is at 1 we have this asymptote and we have the exact same function as we saw before. And notice, by the way, that if I set this equal to zero and solve this, I see it crosses the x-axis now at -1. So it's literally the exact same picture as what we had before, but I just moved the whole thing up one unit, and you can see that it actually looks like that. I just took the whole picture and slid it up once. So this piece went to there, and this piece went to there, and this line that used to be the asymptote went up one unit, so just a shift up one.
I thought I'd try one last one with you guys, just for fun. How about this? How about f(x) = ? Well, now I'm shifting the x's and so I'm going to shift either to the right or to the left, and how does that work again? Well, this is a classic mistake--classic mistake number 8 on my top 10 list. Number 8 - the shifting function mistake. And remember, add to y, go high, add to x, go west. Since I'm subtracting, I should go east. So that means I should be moving in this direction three units. Another way of thinking about that is asking where would this line go. This is where things are undefined. Where are things undefined here? When x = 3. So instead of going here, I'm going to go one, two, three. So I'm shifting everything over to the east by three units, and if you graph that what you'd see is the following: one, two, three units, and there's the vertical asymptote, and this picture just looks like this and then looks like this. So again, exact same picture as this, but I just shifted now to the right by three units, because that's what this thing tells me to do.
So, in fact, you can see that a lot of these, for example, this, this, and this, these three just follow immediately from the first one by using the shifting techniques we talked about earlier. And this one, if you plot some points, you'll actually see you've got something sort of interesting, you get this function where both wings are sort of up. In fact, that's one of the ones I showed just a moment ago here--another version of that there.
Anyway, those are some very basic type of rational functions where I hope when you see those things you can get a sense as to how things are moving, whether it's like this or like this, or shifted, up, or down, around, sideways, so forth. Get a sense of basic rational functions.
Polynomial and Rational Functions
Rational Functions
Basic Rational Functions Page [1 of 2]

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