Rational Functions
where g(x)
and h(x) are
polynomials.
In the homework assignments, you will see some situations in which rational
functions arise.
A basic example of
a rational function is 
.
In fact, whenever g(x) and h(x) are linear functions, the graph of
f(x) is a translation of the graph of y = 
.
However, graphs of rational functions vary considerably.
Click below for a whirlwind tour of the graphs of some rational functions.
Our objective is to be able to make a rough sketch of the graph of a
rational function without actually plotting a large number of points.
Drawing a rational function can be complicated and we shall develop a systematic
procedure for obtaining enough information to sketch the graph. We relate
the discussion to a basic example
and to 
, 
and 
which illustrate various features found in the graphs of rational functions.
Their graphs are in table A.
 |
 |
|
|
 |
 |
 |
 |
 |
Preliminaries: The domain and x intercepts.
Unlike a polynomial function, the domain of a rational
function may not include all the real numbers. The zeros of the numerator
and denominator of a rational function provide
essential information about the graph. Click on this link for a discussion
of the domain and intercepts
We now assume that the numerator and denominator of f(x) have no common factor.
The sign of f(x)
Information describing where the graph
lies above and where below the x axis
is of great help in sketching the graph. In the discussion of
the graph of a polynomial function, we found the zeros of the polynomial, used these values to partition
the number line, and investigated the sign of the function in each of these regions. The Intermediate Value Theorem
ensured that the values of f(x) were either always positive or always negative in each of these regions.
A rational function is not defined (is discontinuous) at zeros of the denominator, and so we need to
partition the number line using zeros of both the numerator and the denominator.
Otherwise, the procedure is the same.
Click on this link for a discussion of the
sign of a function.
Vertical asymptotes
A rational function is not defined when the denominator is zero. For values of x close to a zero of the
denominator the graph is almost vertical, and zeros in the denominator give rise
to vertical asymptotes.
Click on this link for a discussion of
vertical asymptotes , and excercises relating to them.
Horizontal Asymptotes
A feature of the graph of f(x) =
that differs from those of graphs of polynomial functions is that
as the values of x grow in magnitude, both positively and negatively, the
value of f(x) approaches a finite value. In this
case f(x) approaches 0 and the graph becomes closer and closer to the line
y = 0. In this case, y = 0 is a horizontal
asymptote. The same is true of
the graph of many rational functions. Click
on this link for a discussion of
horizontal asymptotes , and excercises relating
to them.
* 1 *
* 2 *
* 3 *
* 4 *
and as x
You should now be ready to sketch the graph.