Rational Function Graphs

Another basic graph that you need to study is the basic rational function graph - that is, the graph of . Let's start by making a table of values:

Looking at this table you can see that the value of at x = 0 is undefined (since anything divided by zero is undefined.) This means that there is NO graph at x = 0. There is a vertical asymptote at this point.

An asymptote is an imaginary vertical or horizontal line on a graph that the graph approaches, but will never touch.

Let's put these points on a graph and see what an asymptote looks like. Plotting the points from the table of values we get:

I'd like to get a bit more detail about what is happening as the x-value of this graph approaches 0. I'm going to plot a few more points to help me see what is happening:

Now let's put these points on our graph:

Now let's connect these points with a smooth curve to get:

This is a much different graph than you are used to seeing! There is actually a vertical asymptote AND a horizontal asymptote on this graph. They exist on the x-axis (the function y = 0) and the y - axis (the function x = 0). It is important to remember that asymtotes are imaginary. We will sometimes draw them to help us graph, but they are NOT part of the final graph! (It is for this reason that if they are drawn at all, asymptotes will usually be drawn using dotted lines.)

Remember the shape of this graph. When we look at transformations, we will only examine the points (-1,-1) and (1,1) as our 'important' points. We will also move the asymptotes with the graph as needed in order to graph a transformation of .

Transformation Example

The standard form of the basic rational function looks like the following:

where
a is the vertical stretch
h is the horizontal shift and
k is the vertical shift

The a value might seem a little wierd sitting on the top of the fraction, but it comes from multiplying fractions. For example, if I perform the operation 2(), the result is 2/x. Let's work through an example:

Example

Sketch the graph of .

Solution

In this example, a is 2, h is POSITIVE 1 (to see why check the transformations lesson in the quadratic functions unit) and k is 3. First, start with the basic graph of :

Now since we are only drawing a sketch of the graph, it does not need to be perfect! (The graph you see on this page will be perfect, as it was generated by a computer, but yours does not have to be quite that good.) We will focus on the points (-1,-1), (1,1), and the asymptotes to help us draw our new graph. Start with the first transformation:

1. a is 2. Multiply all y-coordinates by 2. In our two important points ((-1,-1) and (1,1)) this gives us (-1,-2) and (1,2). Redraw the graph using those new points. (Hint: Just put those 2 points on the graph, and then draw the basic shape of the graph passing though the points and approaching (BUT NOT TOUCHING) the asymptotes:

This is the graph of . I have included the asymptotes (as dotted green lines) but it is VERY hard to see them as they sit right on the axes of the graph.

2. For this graph, h is 1. This means that all points (and the asymptotes!) need to be moved one unit to the left. Below I have shown the graph with the important points and the asymptotes shifted, BEFORE drawing in the whole graph:

As with the last step, draw the graph running through those 2 points, and bounded by (approaching but not touching) the asymptotes. Your next step should look similar to the following:

This is the graph of .

3. Last, we have to deal with the k value. In this example, k is 3. This means we must shift all points (and asymptotes) up three units. As with the last step I have shown you the graph with the points and asymptotes only, and then the final graph.

Now draw the graph through the points and bounded by the asymptotes to get:

This is the final graph of .

Now that you have seen the basic graph of and its transformations, you can head back to the Polynomial and Rational Functions main page.