Return to Assignments Fractional functions and graphs
Assignment 10
Investigate the function:
and sketch its graph.
Solution
For we have a vertical asymptote (for
the function becomes
). When we try to find a horizontal asymptote we encounter some problems, because for
or
the graph approaches
. In order to get more information of the behavior of the function for
or
, we rewrite the function as follows:
We see that the graph approaches:
when or
. In fact, this parabola is a special asmptote.
It is clear that the graph has no intersection points with the -axis (
). The intersection point with the
-axis can be found by solving the equation:
We notice immediately that this equation has the solution and thus we can factorize the function
:
The quadratic function has a discriminant which is less than is and thus has no solutions. Thus the only intersection point with the
-as is
.
Because of the special nature of the function we look at it in more detail near . We notice that the second term
near
has either the largest (positive or negative) value:
and
Based on this investigation we can sketch the graph. We also depicted the parabola.