Consider the general rational function $R(x) = \frac {P(x)}{Q(x)}$ where $P$ and $Q$ are polynomial functions ($Q \ne 0$).

As long as $Q(0) \ne 0$, the value of $R(0) = \frac {P(0)} {Q(0)}$ gives the second coordinate of the Y-intercept, $(0,R(0))$ for the graph of $R$.

The Factored Form:

There is one very useful form for a rational function $R$ that uses the fundamental theorem of algebra to understand $P(x)$ and $Q(X)$ as a products.

In this 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 :
$R(x) = \frac {P(x)} {Q(x)} =*\frac {A_P*P_1(x)*P_2(x)*...*P_m(x)*g_P(x)} {A_Q*Q_1(x)*Q_2(x)*...*Q_n(x)*g_Q(x)}$ .

This form is called the "factored" form of $R$. From the factor theorem of polynomial algebra, $x$ is an element of the domain of $R$ if and only if $x \ne s_i$ for any $i$, and $R(x) = 0$ if and only if $x$ is in the domain of $R$ and $x = r_j$ for some $j$.

Poles of $R$,

Considering the factored form expression of $Q(x) =A_Q*Q_1(x)*Q_2(x)*...*Q_n(x)*g_Q(x)$  we find the roots of $Q$, namely, $ s_1, s_2, ... , s_n$, are numbers where $R$ is not defined.

These numbers are sometimes described as the "poles" of $R$.

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.
In other subsections we explore how mapping diagrams for rational functions help us visualize and understand further the meaning of the shape of the graph of $R$.

Example OAF.RFF.1 : $R(x) = \frac {x-2}{x+1}$.

Example OAF.RFF.2 $R(x) =\frac {(x-2)^2} {(x+1)^3}$.

Example OAF.RFF.3 : $R(x) =\frac {x-2}{(x^2+1)(x+1)}$.

Example OAF.RFF.4 : $R(x) =\frac {(x-2)^2}{x^2(x+2)^3}$.

Example OAF.DRSF.0 : Dynamic Visualization of Standard Form for a Rational Function: Graphs and Mapping Diagrams