Final answer:
The end behavior for the function f(x) = -x⁴ + 4x² + x - 4 is such that as x approaches both positive and negative infinity, f(x) approaches negative infinity.The correct choice is Option A.
Step-by-step explanation:
To describe the end behavior of the function f(x) = -x⁴ + 4x² + x - 4, we look at the highest degree term since it has the most significant effect on the graph for large values of x. In this case, the highest degree term is -x⁴. The negative coefficient in front of the term implies that as x approaches positive infinity, f(x) approaches negative infinity. Similarly, as x approaches negative infinity, the even power of x ensures that the function will also approach negative infinity, leading to the behavior of the function mimicking an upside-down bowl.
Therefore, the correct description of end behavior for this function is: As x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches negative infinity. The correct choice is Option A.
The end behavior of a function can be determined by analyzing the leading term of the function. In the function f(x) = -x⁴ + 4x² + x - 4, the leading term is -x⁴. As the degree of the leading term is even and the coefficient is negative, the end behavior of the function is the same as that of the parent function y = -x⁴. As x approaches positive infinity, the function approaches negative infinity, and as x approaches negative infinity, the function approaches negative infinity. Therefore, the correct answer is A) As x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches negative infinity.