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Graphing a Rational Expression - Problem 1

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Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Graphing a simple rational expression with a transformation so for this example we are going to look at f(x) is equal to one 1 over x plus 2. And what I want to do is draw some parallels to some other functions that we are doing now.

So say we are talking about f(x) is equal to x². We know that graph to be a basic parabola and if I said g(x) is equal to x² minus 4 that’s going to shift the graph down 4. So basically it’s going to keep the same shape and instead of going through the origin, it’s now going to go down 4 units and look pretty much the same. Just a rough sketch of what’s going on with this.

With rational expressions the idea is exactly the same. We know what 1 over x looks like, we know that 1 over x looks like this so all we do with the plus 2 is it shifts everything up 2 spaces. So the graph of this gets moved up 2. We have vertical asymptote at the y axis if we move our vertical line up and down it stays the same. But what is going to change is we have a horizontal, normally we have a horizontal asymptote at 0. That gets moved up 2 so what actually happens is we know how our horizontal axis asymptote right there. I always draw it in just to sort of keep my point of reference on what’s going on, and then our graph is going to keep the exact same shape.

Something like this something like that so basically all we’ve done is we’ve taken our normal 1 over x graph and just moved it up two units vertical asymptote stays the same, horizontal asymptote gets moved up 2, so horizontal asymptote is now at y equals 2.

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