Choose the end behavior of the graph of each polynomial function. \( \square \) Falls to the left and rises to the right (a) \( f(x)=6 x^{3}-7 x^{2}-4 x+3 \) 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 Falls to the left and rises to the right (b) \( f(x)=x^{6}-x^{4}+4 x^{3}-5 x \) 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 Falls to the left and rises to the right (c) \( f(x)=-2 x(x-1)^{2}(x+3) \) 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

To determine the end behavior of a polynomial function, look at its degree and the sign of its leading coefficient. Here’s how to analyze each function: (a) f(x) = 6x³ – 7x² – 4x + 3 • The leading term is 6x³. • The degree is 3 (odd) and the coefficient is positive. • For an odd-degree polynomial with a positive leading coefficient, as x → –∞, f(x) → –∞ (falls to the left) and as x → +∞, f(x) → +∞ (rises to the right). Answer: Falls to the left and rises to the right (b) f(x) = x⁶ – x⁴ + 4x³ – 5x • The leading term is x⁶. • The degree is 6 (even) and the leading coefficient is positive. • For an even-degree polynomial with a positive leading coefficient, as x → ±∞, f(x) → +∞ (rises on both ends). Answer: Rises to the left and rises to the right (c) f(x) = –2x(x – 1)²(x + 3) • First, determine the overall degree: x (degree 1) multiplied by (x – 1)² (degree 2) multiplied by (x + 3) (degree 1) gives a total degree of 1 + 2 + 1 = 4. • The coefficient: The leading term from x, (x – 1)², and (x + 3) would be –2 · x · x² · x = –2x⁴. • The degree is 4 (even) and the leading coefficient is negative. • For an even-degree polynomial with a negative leading coefficient, as x → ±∞, f(x) → –∞ (falls on both ends). Answer: Falls to the left and falls to the right Thus, the correct end behaviors for the graphs are: (a) Falls to the left and rises to the right (b) Rises to the left and rises to the right (c) Falls to the left and falls to the right