The graphs of the elementary functions The graphs of algebraic functions The graphs of rational functions
Basic properties of rational functions
The graph of the reciprocal function, equilateral or rectangular hyperbola
Translation of the reciprocal function, linear rational function
The graphs of rational functions
 · Rational functions - a ratio of two polynomials - Reciprocal function - Translation of the reciprocal function,     called linear rational function. Basic properties of rational functions
The functions that most likely have vertical, horizontal and/or slant asymptotes are rational functions.
So, vertical asymptotes occur when the denominator of the simplified rational function is equal to 0. Note that the simplified rational function has cancelled all factors common to both the numerator and denominator.
The existence of the horizontal asymptote is related to the degrees of both polynomials in the numerator and the denominator of the given rational function.
Horizontal asymptotes occur when either, the degree of the numerator is less then or equal to the degree of the denominator.
In the case when the degree (n) of the numerator is less then the degree (m) of the denominator, the x-axis
y = 0 is the asymptote.
If the degrees of both polynomials, in the numerator and the denominator, are equal then,  y = an/bm  is the horizontal asymptote, written as the ratio of their highest degree term coefficients respectively.
When the degree of the numerator of a rational function is greater than the degree of the denominator, the function has no horizontal asymptote.
A rational function will have a slant (oblique) asymptote if the degree (n) of the numerator is exactly one more than the degree (m) of the denominator that is if  n = m + 1.
Dividing the two polynomials that form a rational function, of which the degree of the numerator pn(x) is exactly one more than the degree of the denominator qm(x), then
pn(x) = Q(x) · qm(x) + R     =>      pn(x)/qm(x) = Q(x) + R/qm(x)
where, Q(x) = ax + b is the quotient and R/qm(x) is the remainder with constant R.
The quotient Q(x) = ax + b represents the equation of the slant asymptote.
As x approaches infinity (or negative infinity), the remainder R/qm(x) vanishes (tends to zero).
Thus, to find the equation of the slant asymptote, perform the long division and discard the remainder.
The graph of a rational function will never cross its vertical asymptote, but may cross its horizontal or slant asymptote.
The graph of the reciprocal function, equilateral or rectangular hyperbola
The graph of the reciprocal function y = 1/x or  y = k/x is a rectangular (or right) hyperbola of which asymptotes are the coordinate axes.
 If k > 0 then, the function is decreasing from zero to negative infinity and from positive infinity to zero, i.e., the graph of the rectangular hyperbola opening in the first and third quadrants as is shown in the right figure. The vertices,  Translation of the reciprocal function, linear rational function
 The rational function by dividing the numerator by denominator,
 can be rewritten into where, is the constant, are the vertical and the horizontal asymptote respectively.
Therefore, the values of the vertical and the horizontal asymptotes correspond to the coordinates of the horizontal and the vertical translation of the reciprocal function  y = k/x as is shown in the figure below.    Intermediate algebra contents 