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Graphing Rational Functions, n less than m - Problem 1
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We’re graphing rational functions starting out with the case, n is less than m and the degree of the numerator is less than m the degree of the denominator. Now here, we have y equals 5 over 4 minus x². The first thing I like to do when I’m graphing rational functions, is plot the x intercepts and the asymptotes. Now x intercepts come from the zeros of the function, which come from the zeros of the numerator. Now this numerator is never going to equal zero, so there are no x intercepts.
What about asymptotes? In this case, the vertical asymptotes will come from zeros in the denominator. The denominator equals zero when 4 minus x² equals zero, x equals 2 or x equals -2. The horizontal asymptotes, because the degree of the denominator is bigger than the degree of the numerator, y equals zero is automatically the horizontal asymptote. I want to make note of that on my graph. Y equals zero is my horizontal asymptote and I’ll draw my vertical asymptotes; x equals negative 2 and x equals 2.
What I usually like to do next is plot a few points. I’ll plot at least one point in each these regions. Let’s start with a point in this left region, -3. I have 5 over 4 minus -3². 4 minus 9, that’s 5 over -5, which is -1. So at -3, we have -1. That’s going to be a point. Let’s try, how about at zero, what happens? At zero we have 5 over 4 minus zero, 5 over 4. 5/4, is about here. And, you may recognize that this is an even function. If you replace x by negative x, you get this exact same expression, and that means that there is going to be symmetry at the y axis. This point will have a reflection over here. There’ll be a point that’s the mirror image of this one across the y axis. I don’t really need to plot 3, I know I’m going to get -1.
How about x equals 1, let’s see what happens in here a little bit more. X equals 1, I get 5 over 4 minus 1², 4 minus 1, 5 over 3. At x equals 1, I get 5/3, which is 1 and 2/3, about here and then again, using symmetry when I have a point here. And keeping in mind the asymptotic behavior, you know that the graph is going to look something like this. It will be symmetrical about the y axis, and, keeping in mind that y equals zero is an asymptote, the function’s probably going to look like this and like this. And it will be symmetric.
That’s it, that’s my graph of y equals 5 over 4 minus x². Now just remember the steps.
First plot x intercepts and asymptotes, then make a table of values. I would say at least one value per region, the regions that the asymptotes divide the plane into. That’s it, that’s your graph.
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