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Graphing a Rational Expression - Concept

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!

: The graph of a basic rational function (1/x) is easy to do by plotting key points. When graphing rational functions, the functions are asymptotic to either the x-axis and y-axis or to certain lines if there are shifts in the graphs. More complex graphs of rational functions include functions with graph shifts.

The graph of a basic rational expression 1 over x so for this little lesson we want to look at f of x is equal to 1 over x the function 1 over x and we're going to graph this by just plotting some key points. Okay so behind me I have a table we're just going to plug in some values and see what values come out so if x is equal to negative 5 we end up with negative one fifth, x is negative 1 1 over -1 just cancels out to negative 1 okay negative one fifth, this one is little bit more complicated if we plug in negative one fifth we get one over negative one fifth this is a complex fraction so we actually multiply by the opposite so this is one over time one times negative 5 over 1 which turns into -5. We plug in 0 one over 0 what is one over 0? We can't divide by 0 so this is undefined and then we're going to go to the same numbers without the negative sign the numeric values are going to come out exactly the same as up here but they'll be positive instead so 1 over 1 fifth is just going to be 5 this is 1 and then finishing up with one fifth.
Okay so let's go over to the graph and plot these points. This particular example I'm not terribly concerned with complete accuracy but we are going to sure to get the rough idea of what this graph looks like. Okay so -5, negative one fifth we go up back 5 units negative one fifth is a fairly small number okay negative 1 up [IB] positive it should be negative let's drop that down -1, -1 is right over here, negative one fifth is pretty small number pretty close to 0 so it's going to be just a pretty close to x axis and we go down 5. Okay plotting the same points when we're going positive we can't plot anything when x is zero and so we continue our pattern x is one fifth we go up to 5, 0.1, 1 and lastly the 0.5, one fifth. okay, connecting the dots, what we end up with is a graph that comes down round like this kind of missed my point but we get that idea of the general shape and another piece that comes down and around like this okay? Not exact but you get an idea what this graph looks like.
What you notice is we have 2 pieces okay it's divided by the x axis and the y axis. And what this is actually called are called asymptotes I'm going to write up there because it's a pretty hard word to say and spell so asymptotes I don't know really know why the p is in there you don't really say it out but it's how you spell it. So what we have here is a horizontal asymptote which means that the graph is basically going to get very very close to 0 but it's never actually going to touch it so think about what happened when we put in really large numbers of x 1000, 1 million things like that, we're going to get 1 over 1000, 1 over a million, get very very small numbers but they're still positive numbers they're getting closer and closer to 0, so what a horizontal asymptote is saying is the graph is going to approach it and approach but never actually touch it, so here we have a horizontal asymptote at y=0 okay by the same logic we have a vertical asymptote at x=0 okay again we can plug plug in very very small numbers for x one of a very small number turns into a very big number and so what's going to happen is the graph is going to shoot up to infinity or shoot down to negative infinity. Okay so that a vertical asymptote it's pretty easy to remember a vertical is up and down horizontal is side to side.
So what we've done is plot some key points to create our graph and then talked about some language that is involved in the graph of 1 over x.

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