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College Algebra: Rational Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 4:13
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 45 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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Now I want us to start thinking about what are referred to as rational functions. Now what's a rational function? Well, a rational function is just a function that has a numerator and a denominator, and both the numerator and the denominator are actually polynomials. So basically what we're going to do is we've looked at polynomials for a while, now we're going to take quotients of polynomials and take a look at what those functions look like. Let me show you what some of those functions look like. These are all graphs, by the way, of actual, real, honest-to-goodness rational functions. So you can see they look a lot different than just polynomials. For one thing you'll notice there's this line. In this case it's at -2, x = -2, where the curve does not cross. The curve just sort of heads up to it, and on this side he heads down to it, but in fact, never crosses. In fact, never actually touches. This line is called an asymptote, in particular, this is a vertical asymptote because it's a vertical line and the function sort of inches up to it and heads toward it and wants to snuggle up against it, but in fact, never, never touches it. And you can see, as the function comes down here, it gets closer and closer to this. That's called a vertical asymptote.
There's also horizontal asymptotes, and you can see that right here. This x-axis is actually a horizontal line, and notice that again, the function wants to head up and nestle right against it, but never actually touches it. Similarly, down here. And this is sort of the general shape of a rational function. There may be these asymptotes, both horizontal and vertical, where the function wants to come up to and touch, kiss, but actually never, in fact, touches.
Now let me show you another picture of one. Here's one where you see the same kind of activity. You see again, a horizontal asymptote. Now the horizontal asymptote is at the height y = 3, and you'll see a vertical asymptote here at x = 4, and you can see that the curve wants to sort of shape around those asymptotes, never touches them, but in fact, heads up and wants to sort of bump up against them, even though they actually never touch. Again, a general sense of what these rational functions look like. By the way, what do you think is happening here at x = 4? Well, since the function's not defined, my guess is that's probably a point, x = 4, where the denominator equals zero. Because where the denominator equals zero, that's where the function is undefined, and it turns out that it may be the case that if the denominator's equal to zero, we may have one of these vertical asymptotes, so that's a good place to sort of look out for what's going on here.
Let's look at another example. This is even more exotic. Look at this. Here we have three wings. We have one wing that goes up like this, and then we have a wing that sort of looks a little kinky. It sort of comes down here, it hugs up against the line x = -2, and then sort of brushes up against the line x = 3, and again we have this horizontal asymptote at the x-axis. So you can see, for example, there may, in fact, be more than one vertical asymptote. We may, in fact, have many vertical asymptotes where in different regions the function wants to hug up against those asymptotes. That's completely okay. In fact, here, my guess is that maybe in the denominator we see a -2 makes the bottom zero, but also a 3 makes the bottom zero. Maybe, in fact, the denominator is a quadratic, and it had two roots to make that zero. So that's possible.
And as one last example, it may be the case that, in fact, we have a rational function where it always stays on top. In fact, they don't always have to be opposite, like one down here and one--it's not like a yin and a yang kind of thing. You might actually have part of the thing up here, and the other part of the thing up here. Notice, there's still an asymptote at the y-axis--there's a vertical asymptote there, and there's also a horizontal asymptote. So we don't always have to have that activity of coming up against one side and going up the other. These are all examples of rational functions, and those are just quotients of polynomials. What we're going to start to look at is, how do we actually, first of all, find these different asymptotes, and how do we determine how the function sort of hugs around, and that hugging and so forth. So we're going to start getting into some romantic sessions about hugging, asymptotes, and the like. I'll see you there.
Polynomial and Rational Functions
Rational Functions
Understanding Rational Functions Page [1 of 1]

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