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Graphing Rational Functions, n less than m - Problem 3

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Let’s graph a slightly harder rational function; y equals x minus 1 over x² minus x minus 6. Before I go into the actual graphing, I want to factor the denominator because it’s going to make it easier to find out what the vertical asymptotes are. The denominator factors into x plus or minus something and x plus of minus something. The factors of 6 that I probably need are 2 and 3, because I’m looking for -1 and so -3 and plus 2 will work. Because -3x plus 2x is –x and 2 times -3 is -6. So this will work.

Plot x intercepts and asymptotes, that’s usually the first thing I start with. So, the x intercepts will come from the zeros of the rational function, and they come from the zeros in the numerator, so x equals 1 is the zero. So 1 zero is going to be an x intercept and I can plot that right away. Now asymptotes, I’m going to get an asymptote at x equals -2 and x equals 3. Those will be my vertical asymptotes. And because the denominator has a degree larger than the numerator, I’m going to get y equals zero, as my horizontal asymptote. Let me make a note of that and draw those asymptotes on my graph. Y equals zero, x equals -2, and x equals 3. Those are my asymptotes.

The final thing to do is to plot some points to fill out our graph. I’ll plot some points x, y. I have an asymptote at x equals -2, why don’t I use -3 as a point? I’ve got -3 minus 1, -4, over -3 plus 2, -1, times -3 minus 3, -6. So I have 6 in the denominator and -4 in the numerator, -4 over 6, is -2/3. So (-3, -2/3) is about here.

What happens when x equals zero? This is between the asymptotes. When x equals zero I get -1, over 2 times -3, -6, so 1/6. (0, 1/6) this is one so, ½, 1/3, a third a way to a half is 1/6, right about there. And I guess I could plot x equals 1 or x equals 2 rather. At x equals 2, I get 2 minus 1 is 1, 2 plus 2 is 4, 2 minus 3, -1, -1/4. So this is ½, ¼, -¼ is right about here.

Finally let's plot one point over in this region. How about x equals 4? 4 minus 1, 3, 4 plus 2, 6, times 4 minus 3, 1. So ½. (4, ½). Now let’s keep in mind our asymptotes. Y equals zero is an asymptote, so we’re probably going to have something like this on this side. Here, this is kind of interesting. It’s below the x axis, from 1 to 3, so we’re probably going to get something like this. But it’s above the x axis between -2 and 1, so probably something like this. And then over here, we’re above the x axis, so probably it’s going to look like this. It’s a pretty interesting graph.

So just remember, plot x intercepts and asymptotes first, x intercepts come from the zeros in the numerator, the vertical asymptotes come from the zeros in the denominator. And whenever the numerator has degree less than the denominator, y equals zero is going to be your horizontal asymptote. Plot some points to fill out your graphs and you’ll get a pretty good graph most of the time.

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