12th Grade Math
posted by plato .
How do you graph
f(x)= ((x+5)(x4)^(2))/((x2)(x^(4))

First of all some critical points.
From the single factor of (x+5) at the top, we can say it crosses at (5,0)
from the double factor of (x4)^2, the graph will "touch" the xaxis and then reverse direction.
From the x^4 at the bottom, the yaxis will be a vertical asymptote,
From the (x2)at the bottom, there will be a vertical asymptote at x=2
The highest power at the top will be +x^3 and the highest power at the bottom will be +x^5, so as x approaches infinity in either the positive or negatives, it will approach zero and the graph will approach the xaxis from the top.
I tried some positive and negative values of x close to zero and the function value was negative. Also a value of x = 1.9 gave me a negative value. So the "loop between 0 and 2 lies below the xaxis, suggesting that there is a maximum value for that part of the graph, but still below the xaxis
Also after (4,0), the graph will rise ever so slightly for a maximum just to the right of (4,0) and then approach the xaxis.
The same thing will happen on the left at (5,). The graph will come up from its yaxis asymptote, cross at (5,0), rise just ever so slightly and then drop down to approach the xaxis
If you have a graphing calculator, you can zoom in on these critical areas, but on a large scale the small changes near (4,0) and (5,0) will be hardly noticeable. 
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