The end behavior of the polynomial function f(x) = (x+4)(x-1)^2(x-0)^2(x+7) is such that as x approaches positive infinity and negative infinity, f(x) also approaches positive infinity due to the leading term x^6 with a positive coefficient.
Step-by-step explanation:
End Behavior of Polynomial Functions
To determine the end behavior of the polynomial function f(x) = (x+4)(x-1)^2(x-0)^2(x+7), we need to look at the leading term when x is approaching infinity (positive or negative). Since the highest power of x in our function is the result of multiplying each term, which is x6 (as (x-1)^2 and (x-0)^2 both contribute x2), the leading term dictates the end behavior. Note that the leading coefficient is positive as it comes from the multiplication of the terms of the polynomial. Due to this, as x approaches positive infinity, f(x) will also approach positive infinity. Similarly, as x approaches negative infinity, f(x) will also approach positive infinity, as the highest power is even. So, the end behavior of the function can be described as:
As x approaches infinity, f(x) approaches infinity.
As x approaches negative infinity, f(x) approaches infinity.