1. **Problem Statement:** Determine the end behavior of the graphs of the given polynomial functions. 2. **Key Concept:** The end behavior of a polynomial function depends on the degree and the leading coefficient. - If the degree is even and the leading coefficient is positive, the graph rises on both ends. - If the degree is even and the leading coefficient is negative, the graph falls on both ends. - If the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right. - If the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right. 3. **(a) For** $f(x) = -x^4 + 2x^3 - 7x^2 + 3x$: - Degree is 4 (even). - Leading coefficient is $-1$ (negative). - Therefore, the graph falls to the left and falls to the right. 4. **(b) For** $f(x) = 2x^5 - 8x^3 + 5x + 4$: - Degree is 5 (odd). - Leading coefficient is $2$ (positive). - Therefore, the graph falls to the left and rises to the right. 5. **(c) For** $f(x) = -3(x - 1)(x + 5)^2$: - Expand the factors to find degree and leading coefficient: - $(x + 5)^2 = x^2 + 10x + 25$ - Multiply by $(x - 1)$: $x^3 + 10x^2 + 25x - x^2 - 10x - 25 = x^3 + 9x^2 + 15x - 25$ - Multiply by $-3$: $-3x^3 - 27x^2 - 45x + 75$ - Degree is 3 (odd). - Leading coefficient is $-3$ (negative). - Therefore, the graph rises to the left and falls to the right. **Final answers:** - (a) The graph falls to the left and falls to the right. - (b) The graph falls to the left and rises to the right. - (c) The graph rises to the left and falls to the right.