2.7 Graphs of Rational Functions

Guidelines for Graphing Rational Functions

Let , where N(x) and D(x) are polynomials that have no common factors. To graph this function,

1. Find the y-intercept by solving for x = 0

2. Solve for N(x) = 0. The real solutions are the x-intercepts.

3. Solve for D(x) = 0. The real solutions are the vertical asymptotes.

4. Find the horizontal asymptote

5. Plot each of these on the graph, representing the intercepts as points, and the asymptotes as dashed lines.

6. Plot a point between and a point beyond each x-intercept and vertical asymptote, if necessary. Connect the points with smooth curves between and beyond the vertical asymptotes

Example

1. Y-intercept(s): f(0) gives no y-intercepts

2. X-intercept(s): N(x)=0 gives no x-intercepts

3. Vertical asymptotes: x=0

4. Horizontal asymptotes: y=0

5 and 6. Graph. Both asymptotes are along the axes in this example, so they are not easily seen.

A basic rule of thumb for graphs of this type is that around the vertical asymptotes, the graph often does opposite things on either side (for example, the graph above falls on the left of the vertical asymptote, and rises on the right). This is not always true, but we will learn about the predictable exceptions to this rule later.

Hole in a graph

Sometimes, holes can be found in graphs. This occurs when the numerator and denominator of the rational function have exactly the same factor.

For example, the function can be written as , in which the factor (x-1) can be found in the numerator and the denominator. If this is cancelled out in the numerator and the denominator, we are left with the function , just like in the graph above. But what is different about these two graphs?

The answer is that there is a hole in the graph of at x = 1, the location of the eliminated factor (x-1). When looking at this graph in a graphing calculator, it is useful to view the zoom decimal window, in order to best see this hole. When graphing this type of function by hand, the graph can be drawn with an open circle in the line at x = 1.

It is easy to tell that there will be a hole in a graph, if, when graphing the function, there is an x-intercept and a vertical asymptote at the same place. At this point in the graph, the intercept and asymptote should be removed, and at this location, there should be a hole.