An asymptote of a function is a line that the function's graph approaches as the distance to the origin approaches infinity. It is very common for rational functions to have at least one asymptote. These can arise when extends to the right or left infinitely, and around -values making the denominator equal to zero. The rational function has two asymptotes, the -axis and the -axis. The shape of this function's graph is called a hyperbola, which means that it consists of two mirror images with bow-like appearances.
Comparing the function rule to the general form of a simple rational function makes it possible to identify the asymptotes and For these are and These can be drawn in a coordinate plane.
Plot some points both to the left and the right of the vertical asymptote. For the example, it would be appropriate to use, for instance, the -values and
The points can be added to the coordinate plane to begin to see the behavior of
The graph can now be drawn by connecting the points with a smooth curve. It must approach both the horizontal and vertical asymptotes.
Another common form of rational functions is where and are linear functions with different -intercepts. Just like simple rational functions, these are hyperbolas and have horizontal and vertical asymptotes. The horizontal asymptote is the line That is, the quotient of the leading coefficients of and The vertical asymptote is the linewhich is the -value that makes the denominator
Identify the asymptotes of the given rational function. Then, rewrite it in the form, and draw its graph.
The fraction can now be split into two, where one numerator is and the other is
Next, we can make a table to find points around the vertical asymptote.
Plotting the points gives the graph.
We can connect the points with two curves that tend toward the asymptote as and extend in both directions.
Below, two rational functions are graphed.
Match the graphs with their corresponding function rule.
We match the graphs with their corresponding rule by identifying the asymptotes. Observing the first graph, it looks as though the asymptotes are For the second graph, Let's now identify the asymptotes of the given rules. For rational functions in the form the vertical asymptote is and the horizontal asymptote is For the ones written as their vertical asymptote is and the horizontal asymptote is
|Function||Vertical asymptote||Horizontal asymptote|
The function has the same asymptotes as Graph I, and has the same asymptotes as Graph II. Thus, the graphs' corresponding function rules are