MATH 133 -- Intensive Calculus for Science

September 21, 2005

An "interesting" rational function:

     f(x) = (x^2-3*x+2)/(x^2-9)

The numerator factors as  ( x - 1)( x   - 2),  so the

graph will have x- axis intercepts  at   x = 1,2.  

The denominator factors as ( x  + 3)( x  - 3) ,   so the

graph will have vertical asymptotes at   x  = -3, 3.

The ``end behavior'' (as   proc (x) options operator, arrow; infinity end proc  and   proc (x) options operator, arrow; -infinity end proc ) can

be seen by multiplying top and bottom by  1/ x^2

(1-3/x+2/(x^2))/(1-9/(x^2))   ->  1.  

Here's a Maple plot where we specify the ranges

of   x-   and   y- values to plot, and tell Maple

to watch out for the vertical asymptotes:

>    plot((x^2-3*x+2)/(x^2-9),x=-4..4,y=-10..10,discont=true);

[Maple Plot]

Kind of hard to tell there are   x- axis intercepts, isn't it.  Let's try ``zooming in'':

>    plot((x^2-3*x+2)/(x^2-9),x=0..3,y=-1..1,discont=true);

[Maple Plot]

That's better!