Choose the end behavior of the graph of each polynomial function. \begin{tabular}{|l|l|} \hline (a) $f(x)=-3 x^{6}-4 x^{5}+5 x-5$ & Falls to the left and rises to the right \\ Rises to the left and falls to the right \\ Rises to the left and rises to the right \\ Falls to the left and falls to the right \\ (b) $f(x)=-3 x^{3}+x^{2}-2 x-7$ & Falls to the left and rises to the right \\ & Rises to the left and falls to the right \\ & Falls to the left and falls to the right \\ \hline (c) $f(x)=2 x(x-1)^{2}(x+3)$ & \begin{tabular}{l} Fills to the left and rises to the right \\ Rises to the left and falls to the right \end{tabular} \\ \hline \end{tabular}
Solution
Step 1 ：The end behavior of a polynomial function is determined by the degree and the leading coefficient of the polynomial. If the degree of the polynomial is even, the ends of the graph will point in the same direction. If the degree is odd, the ends will point in opposite directions. The sign of the leading coefficient will determine whether the graph rises or falls as x approaches positive or negative infinity.
Step 2 ：For the first function, the degree is 6 (which is even) and the leading coefficient is -3 (which is negative). Therefore, the graph falls to the left and falls to the right.
Step 3 ：For the second function, the degree is 3 (which is odd) and the leading coefficient is -3 (which is negative). Therefore, the graph rises to the left and falls to the right.
Step 4 ：For the third function, the degree is 4 (which is even) and the leading coefficient is 2 (which is positive). Therefore, the graph rises to the left and rises to the right.
Step 5 ：Final Answer: \n(a) \(\boxed{\text{Falls to the left and falls to the right}}\) \n(b) \(\boxed{\text{Rises to the left and falls to the right}}\) \n(c) \(\boxed{\text{Rises to the left and rises to the right}}\)
