Use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function.
\[
f(x)=-5 x^{4}+2 x^{3}-x+9
\]
Choose the correct answer below.
A. The graph of $f(x)$ falls to the left and falls to the right.
B. The graph of $f(x)$ falls to the left and rises to the right.
C. The graph of $f(x)$ rises to the left and rises to the right.
D. The graph of $f(x)$ rises to the left and falls to the right.

Step 1 ：The given polynomial function is \(f(x)=-5 x^{4}+2 x^{3}-x+9\).

Step 2 ：The Leading Coefficient Test is used to determine the end behavior of the graph of a polynomial function.

Step 3 ：The Leading Coefficient Test states that if the degree of the polynomial is even, and the leading coefficient is positive, the end behavior of the graph will be: Up to the left and up to the right.

Step 4 ：If the degree of the polynomial is even, and the leading coefficient is negative, the end behavior of the graph will be: Down to the left and down to the right.

Step 5 ：If the degree of the polynomial is odd, and the leading coefficient is positive, the end behavior of the graph will be: Down to the left and up to the right.

Step 6 ：If the degree of the polynomial is odd, and the leading coefficient is negative, the end behavior of the graph will be: Up to the left and down to the right.

Step 7 ：In this case, the degree of the polynomial is 4 (which is even) and the leading coefficient is -5 (which is negative).

Step 8 ：Therefore, according to the Leading Coefficient Test, the end behavior of the graph will be: Down to the left and down to the right.

Step 9 ：\(\boxed{\text{The correct answer is A. The graph of } f(x) \text{ falls to the left and falls to the right.}}\)