Question 917246
"Middle point"  would be the local maximum, which on inspecting the graph appears to be x=1/2, and y=1/2.


{{{f(x)=(4x(2-x))/((x+2)^2)}}}, factored, your function.
If you want to exactly find that "middle point" on the graph, that local maximum, you can use first derivative, and equate to zero.  This should be easy IF you have enough derivative Calculus skill; otherwise, you will learn this in first semester of Calculus.


You want to know about the end-behavior.  Notice the degree of numerator and denominator both 2.  The ratio of the leading terms will directly give you the horizontal asymptote, which is the end-behavior.  You already found this asymptote pre-assumed to be done correctly.  You would have compared {{{-4x^2}}} to {{{x^2}}}; unbounded in either direction, the other terms in the rational expression become increasingly LESS significant, and the end behavior  (a limit) approaches {{{-4x^2/x^2}}} which simplifies to {{{-4}}}.


Quadrant 3 Question:
Just test the signs of the intervals that are part of x<0.  If f is negative, then that is in quadrant 3.