How do you find the end behavior of #y = -x^4+3x^3-3x^2+6x+8#?

Question

Sadie Ackerman · Accepted Answer

The end behavior of a function is the behavior of the function as x approaches positive infinity or negative infinity.

So we have to do these two limits:

#lim_(xrarr-oo)f(x)#

and

#lim_(xrarr+oo)f(x)#.

Than:

#lim_(xrarr-oo)(-x^4+3x^3-3x^2+6x+8)=lim_(xrarr-oo)(-x^4)=-oo#

and

#lim_(xrarr+oo)(-x^4+3x^3-3x^2+6x+8)=lim_(xrarr+oo)(-x^4)=-oo#.

This is because the power #-x^4# is the highest power and its behavior is the same of the whole function.

Greyson Brown · Answer

undefined To find the end behavior of the polynomial function $y = -x^4 + 3x^3 - 3x^2 + 6x + 8$, we examine the leading term as $x$ approaches positive and negative infinity.

The leading term of the polynomial is $ -x^4 $. As $x$ approaches positive infinity, the term $x^4$ becomes very large, and since it's multiplied by -1, $y$ approaches negative infinity.

Similarly, as $x$ approaches negative infinity, $x^4$ still becomes very large, but since it's multiplied by -1, $y$ approaches negative infinity.

Therefore, the end behavior of the function is that as $x$ approaches positive or negative infinity, $y$ approaches negative infinity.