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Graphing Rational Functions, n less than m - Problem 3
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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