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.

To analyze the end behavior of the polynomial function \( f(x)=x^{4}+4 x^{3}+4 x^{2} \), we observe that it is a quartic polynomial (degree 4) with a positive leading coefficient (which is 1). According to the Leading Coefficient Test, since the degree is even and the leading coefficient is positive, the graph will rise to the left and rise to the right. Therefore, 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 extra fun facts to keep things interesting! Polynomial functions have been studied since ancient times, with notable advancements during the Renaissance. Mathematicians like Descartes developed new methods enabling us to analyze their behavior thoroughly, laying the groundwork for calculus and modern algebra. Those early thinkers would be amazed to see how we use their principles today! When working with polynomial functions, common mistakes include misidentifying intervals of increase or decrease and overlooking the influence of complex roots on the graph's shape. A good tip is to sketch a rough graph after identifying intercepts and local extrema to ensure you don’t overlook any crucial details before finalizing your analysis!