Question 849547
the solution is:


(2x) / (x+1)


the graph that is generated looks like this:


{{{graph(600,600,-5,5,-5,5,(2x)/(x+1))}}}


the table that could be generated looks like this:


<pre>
        x         y = (2x) / (x+1)

        -2             4
        -1             undefined
         0             0
         1             1
         2             4/3
</pre>


since the vertical asymptote was at -1, the denominator of the graph had to be some for of x + 1 because then -1 + 1 = 0 and the graph is undefined at that point.


the numerator of the graph had to be the same level as the denominator, because then the graph has a horizontal asymptote at whatever level of y the numerator was at when the value of x goes to infinity.


a recommended method that i learned is that you divide the numerator and the denominator by the highest level exponent.   This makes everything that is not important cancel out.


For example:


Assuming the numerator is 2x and the denominator is (x+1), and you want to find the horizontal asymptote, factor out the greatest exponent term.


2x factors out an x.


(x+1) factors out an x.


you are left with:


x * (2) / (x * (1 + 1/x)


now, as x goes to infinity, the numerator stays at 2 and the denominator stays at 1, so the limit of the expression as x goes to infinity is equal to 2.


the solution is part luck and part detective work in finding what the expression could be.   


not sure if i would be successful every time, but it worked this time.


some tutorials on the web that may be helpful are:


<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm</a>


<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut40_ratgraph.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut40_ratgraph.htm</a>