Graphing rational functions
         Vertical asymptotes of rational functions
         Horizontal asymptotes of rational functions
         The oblique or slant asymptote of rational functions
         Graphs of rational functions examples
The graphs of the rational functions
  Rational functions - a ratio of two polynomials  
- Reciprocal function
  - Translation of the reciprocal function,     called linear rational function.  
The functions that most likely have vertical, horizontal and/or slant asymptotes are rational functions.
Vertical asymptotes of rational functions
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.
Horizontal asymptotes of rational functions
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.
The oblique or slant asymptote of rational functions
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.
Example:  Given the rational function   sketch its graph.
Solution:  The vertical asymptote can be found by finding the root of the denominator, 
                      x + 2 = 0       =>       x -2  is the vertical asymptote.
Since the degree of the numerator is exactly one more than the degree of the denominator the given rational function has the slant asymptote.
By dividing the numerator by the denominator
obtained is the slant asymptote  y = x 
and the remainder  3/(x + 2) that vanishes as x approaches positive or negative infinity.
 
Calculus contents A
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