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Rational Functions - The Whole Thing

There are the three neighborhoods that this graph lives in:


There are three neighborhoods in our graph: left, middle, and right
 

Let's look at the left neighborhood:
* Remember your 2 sentences!

Possible graphs for the left section ... Since graphs hug asymptotes, our graph would need to do this ... or this
 

Since he's a function (and must pass the vertical line test), he can't do both.

What did we say about crossing the x-axis for this guy?  (Look back if you need to.)

Yep -- he's not allowed to cross the x-axis!  But, look at that bottom guy:

Possible graph for the left section ... if the graph has to hug the asymptotes y = 2 and x = -5, he's got to cross the x-axis!
 

But, he's not allowed to -- so, he's got to live upstairs!

We can use the same reasoning for the right neighborhood:


the left and right sections of the graph f( x ) = ( 2x^2 + 5 ) / ( x^2 - 25 )