For the polynomial function \( f(x)=-4 x^{4}+8 x^{3} \), answer the parts a through e. 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.

The polynomial function \( f(x) = -4x^4 + 8x^3 \) is a degree 4 polynomial with a leading coefficient of -4. This means that as \( x \) approaches positive or negative infinity, the function will fall towards negative infinity. Therefore, the correct option is: A. The graph of \( f(x) \) falls to the left and falls to the right. Additionally, polynomial functions of even degree will always take the same direction on both ends, and since this function has a negative leading coefficient, that results in a downward trend on both sides. This property is consistent with other even-degree polynomial functions like \( x^2 \) or \( x^4 \).