Consider the general rational function $R(x) = \frac {P(x)}{Q(x)}$
where $P$ and $Q$ are polynomial functions ($Q \ne 0$).
Understanding the behavior of $R$ involves examining both its zeroes
and poles which are finite in number and connected to linear factors
raised to powers.
In its factored form,
$R$ is expressed using powers of linear functions, $P_j(x) =(x-
r_j)^{p_j}$, with $ j = 1,2, ... , m$, and a monic
polynomial $g_P$ which has no real roots in the numerator and
$Q_i(x) =(x- s_i)^{q_i}$, with $ i = 1,2, ... , n$, and a
monic polynomial $g_Q$ which has no real roots :
$R(x) = \frac {P(x)} {Q(x)} =\frac
{A_P*P_1(x)*P_2(x)*...*P_m(x)*g_P(x)}
{A_Q*Q_1(x)*Q_2(x)*...*Q_n(x)*g_Q(x)}$
The importance of the roots and poles of
$R$ to the general behavior of $R$ near the roots and poles is
covered in most 2nd and advanced courses in algebra.
Consider numbers $x_-$ and $x_+$ close to $t$, a root or a
pole of $R$, with $x_+ >t$ and $x_-<t$. One of two things can
happen:
- $R(x_+)*R(x_-) >0$ or
- $R(x_+)*R(x_-) <0$.
These cases determine the behavior of $R$ close to $t$.
When $t$ is a root of $R$ , $R(x_+)*R(x_-) >0$ if and
only if $t$ is a
local extreme point of $R$.
When $t$ is a pole of $R$ and $t$ is not a root of $P$,
$R(x_+)*R(x_-) >0$ if and only if $R$ is unbounded
positive close to $t$ or $R$ is unbounded negative close to $t$. [A
"vertical line" asymptote for the graph of $R$.]
When $x$ is very large in magnitude - either positive or
negative- the relative degrees of $P$ and $Q$ determine the
behavior of $R$.
Suppose the degree of $P$ is $d_P$ and the degree of $Q$ is $d_Q$.
There are three cases:
- If $d_P > d_Q$ then the values of $R$ are unbounded when
$x$ is very large in magnitude. In this case the long division
of polynomials determines a polynomial $P_Q$ where when $x$ is
very large in magnitude, $R(x) \approx P_Q(x)$.
- If $d_P < d_Q$ then $R(x) \approx 0$, when $x$ is very
large in magnitude.
- If $d_P = d_Q$ then $R(x) \approx \frac {A_P} {A_Q}$ when $x$
is very large in magnitude.
In case 1 the graph of $R$ will appear very close to the graph of
the polynomial $P_Q$. This is described by saying "the function $R$
has a polynomial asymptote."
In cases 2 and 3 the graph of $R$ appears very close to the graph of
the horizontal line $y = 0$ or $ y = \frac {A_P} {A_Q}$. This is
described by saying "the function $R$ has a horizontal asymptote."
The following examples of mapping diagrams for rational functions
visualize these behaviors and help with understanding further the
meaning of the shape of the graph of $R$.
Example
OAF.DRF.0 :
Dynamic Visualization for a Rational Function $f(x) =\frac
{P(x)}{Q(x)}$: Graphs and Mapping Diagrams