Use the given function to complete parts (a) through (e) below \( f(x)=x^{4}-9 x^{2} \) a) Use the Leading Coefficient Test to determine the graph's end behavior A. The graph of \( f(x) \) rises left and falls right 2 B. The graph of \( f(x) \) rises left and rises night C. The graph of \( f(x) \) falls left and falls night D. The graph of \( f(x) \) falls left and rises right

To determine the end behavior of the polynomial function \( f(x) = x^4 - 9x^2 \) using the Leading Coefficient Test, we focus on the leading term \( x^4 \). Since the degree (4) is even and the leading coefficient (1) is positive, the graph will rise on both ends. This means that the correct answer is: B. The graph of \( f(x) \) rises left and rises right. For additional insight, it's essential to know that even-degree polynomials with a positive leading coefficient will always have both ends of the graph rising. This characteristic can be particularly useful when sketching graphs or analyzing their behavior without relying on full computations. When analyzing polynomial functions, remember that critical points and inflection points can provide significant insights into a function's overall shape. However, be wary of common mistakes, like misidentifying the leading coefficient or overlooking the degree's even or odd nature, which can led to confusion regarding end behavior.