The function f(x) = -2(x-1)^3(x+3) exhibits end behavior where, as x approaches positive infinity, f(x) goes to negative infinity and as x approaches negative infinity, f(x) goes to positive infinity. This is a consequence of the leading term having a negative coefficient and the highest power being odd.
Step-by-step explanation:
The end behavior of the function f(x) = -2(x-1)^3(x+3) pertains to how the function behaves as x approaches positive or negative infinity. This function is a polynomial, and its end behavior is determined by the leading term, which in this case is dictated by the highest power of x. Since the highest power of x is odd and the leading coefficient is negative, we know the following about its end behavior:
As x approaches positive infinity (right-end behavior), f(x) goes to negative infinity.
As x approaches negative infinity (left-end behavior), f(x) goes to positive infinity.
This is due to the fact that negative coefficients flip the direction of the end behavior in comparison to positive coefficients and odd powers dictate opposite behaviors for each end.