Graphing a Translation of a Rational Function

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

We are going to take a look at rational functions and their graphs. Specifically, we will study the translations of rational functions and two different ways to graph these translations by getting certain information from the function itself.

Translations of Rational Functions

Suppose that your boss just walked into your office at work to give you the model y = 3/x that fits some data that she is analyzing. She asks you to graph it for her. Mathematically speaking, the function is an example of a rational function. A rational function is a function with the variable in the denominator. They are sometimes called butterfly functions because graphs of simple rational functions have two parts that look somewhat like butterfly wings.


graphtransrat1


You enter the function your boss gave you into your computer system and print out the graph.


graphtransrat2


Notice the graph approaches, but does not touch, the x and y axes (the lines x = 0 and y = 0). When a graph approaches a line in this way, we call the line an asymptote of the function.

You print off the graph and realize that in doing so, you've used up the last of the printer paper. As soon as the graph is printed, your boss comes running in saying she messed up, and that the model should have been y = 3/(x - 2) + 4. All the printer paper is used up, so you can't use your computer system to print a new graph! How are you going to produce a graph of the correct model?

As it turns out, the correct model is actually what is called a translation of the rational function y = 3/x, where a translation is a sliding of a graph along a straight line. Luckily, we have a nice trick for graphing these when we know the graph of the original function! Let's take a look at how to determine the graph of the correct model from the graph of the old model.

Graphing Translations from Known Graphs

In general, this rational function takes the form:


y = a/(x - h) + k


This is a translation of the function y = a/x, and h and k give us all of the information we need to perform the translation and graph y = a/(x - h) + k. You see the graph of y = a/(x - h) + k is the graph of y = a/x translated h units horizontally and k units vertically.

In the case of your work example, y = 3/(x - 2) + 4, h = 2 and k = 4. Therefore, the correct model is the graph of y = 3/x translated 2 units to the right and 4 units up. Great! All we have to do is take the graph of y = 3/x that we already have and shift it 2 units right and 4 units up.


graphtransrat3


Problem solved! You deserve a raise!

This is a great way to graph translations of rational functions when we have the original graph. But, what if we don't have the original graph? Let's look at how we can graph y = a/(x - h) + k if we don't have the graph of y = a/x.

Graphing Translations without the Original Graph

When we have a rational function of the form y = a/(x - h) + k, h and k not only give us the horizontal and vertical translations of y = a/x, but they also tell us the asymptotes of the function y = a/(x - h) + k. You see, x = h and y = k are the vertical and horizontal asymptotes, respectively, of y = a/(x - h) + k. Therefore, we can use the following steps to graph y = a/(x - h) + k when we don't have the graph of y = a/x.

  1. Draw in the lines x = h and y = k. These are your asymptotes.
  2. Use the function to find points to the left and right of x = h, and plot them.
  3. Connect the points in the shape of a rational function. The two wings should approach the asymptotes, but never touch them.

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