For the polynomial function \( f(x)=-4 x^{4}+8 x^{3} \), answer the parts a through e. a. Use the Leading Coefficient Test to determine the graph's end behavior. A. The graph of \( f(x) \) falls to the left and falls to the right. B. The graph of \( f(x) \) rises to the left and rises to the right. C. The graph of \( f(x) \) rises to the left and falls to the right. D. The graph of \( f(x) \) falls to the left and rises to the right.

To determine the end behavior of the polynomial function \( f(x) = -4x^{4} + 8x^{3} \), we observe that the leading term is \( -4x^{4} \). Since the leading coefficient is negative and the degree is even, we can conclude that the graph will fall on both ends. Therefore, the correct answer is: **A. The graph of \( f(x) \) falls to the left and falls to the right.** This means that as \( x \) approaches either positive or negative infinity, the value of \( f(x) \) decreases indefinitely. The idea that the end behavior is determined by the leading term is a key principle in polynomial functions. It allows for quick assessments without needing to analyze the entire function, which can seem daunting! Understanding this can save time and simplify your approach to graphing and analyzing polynomials.