To determine the end behavior, let's take the limit as #xtooo# and #xto-oo#.
In our polynomial, #f(x)#, the first term is what will dominate the end behavior, because it has the highest degree. So we can find the limit of that:
#lim_(xtooo) color(red)(-2)color(blue)(x^4)=-oo#
As #x# gets very large, the blue term will always be positive, but the #-2# (red) will turn it negative. This is why our limit evaluates to #-oo#.
#lim_(xto-oo) color(red)(-2)color(blue)(x^4)=-oo#
As #x# gets very negative, the even exponent will make the term positive, but the red #-2# on the outside will make it negative. Thus, this limit will also evaluate to #-oo#.
In general, the function is downward opening because of the negative coefficient on the #x^4# term.
Hope this helps!
