For the polynomial function \( f(x)=x^{4}, 4 x^{3}+4 x^{2} \), 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 falls to the right. C. The graph of \( f(x) \) rises to the left and rises to the right. D. The graph of \( f(x) \) falls to the left and rises to the right.

The leading term of the polynomial \( f(x) = x^4 + 4x^3 + 4x^2 \) is \( x^4 \), which has a positive leading coefficient and an even degree. According to the Leading Coefficient Test, this means that as \( x \) approaches positive or negative infinity, the function \( f(x) \) will rise in both directions. So, the correct answer is: C. The graph of \( f(x) \) rises to the left and rises to the right. Now, let’s sprinkle in some fun facts! The fourth-degree polynomial functions, also known as quartics, can have up to four real roots and can display a wide variety of shapes. They are visually similar to their close relatives, cubic functions, but with an added level of complexity that can lead to fascinating behaviors in their graphs! For real-world applications, these quartic polynomials can model various phenomena, from optimizing areas and volumes in geometry to predicting the trajectory of objects in physics. Whenever you dive into motion equations or surface area calculations, quartic functions might just be hiding in those calculations!