Rational functions are basically just giant fractions with one polynomial divided by another, but they're a bummer to graph because there's so much to keep track of: horizontal asymptotes, vertical asymptotes, slant asymptotes (oblique asymptotes), x-intercepts, y-intercepts, zeros... In other words, these are a grab bag of everything you've ever seen this whole year. You even have to factor! And making them scarier is that on the test teachers often put full-page problems with 5 parts asking you to break these down every which way! Not fun, certainly, and very time-consuming; but it really is five separate questions, and I'll show you how to get through them one at a time.
Vertical Asymptotes & Holes
The most important thing about rational functions is that you're never ever allowed to divide by zero (those values aren't in domain); hence, the first thing we'll do in every problem is set the denominator equal to zero. Which zero is a vertical asymptote and which is a hole? What's a hole? And how can an asymptote be friendly or unfriendly? Stay tuned.
Horizontal asymptotes come down to three simple rules that you'll just have to memorize, but most students don't have too much trouble with that. Key is that equations of horizontal lines are always y = number, so don't get confused and use x!
Slant Asymptotes (a.k.a. Oblique Asymptotes)
On the down side, these involve polynomial long division. On the up side, many teachers don't even cover this sub-topic, so check your syllabus! (If you don't remember polynomial long division, check out my polynomial division chapter.)
Rational Function Graphing Examples
In this video I work a couple of full-length rational function problems like the worst you might have on the test. We'll find: vertical asymptotes, holes, horizontal asymptotes, X- & Y-intercepts, domain, range, and we'll even draw the graphs!
If you do not have an account, you should get one, because it is awesome! You can save a playlist for each test or each chapter, and save your "greatest hits" into a "watch right before the final" list (not that we recommend cramming, but when in Rome...)