1. **Problem Statement:** Given the graph of a polynomial function with local extrema at approximately $x=-9$, $x=-6$, $x=-2$, $x=3$, and $x=7$, answer the following: (a) Identify intervals where the function is increasing. (b) Find the $x$-values of local maxima. (c) Determine the sign of the leading coefficient. (d) Determine possible degrees of the polynomial. 2. **Key Concepts:** - A function is increasing where its graph goes upward as $x$ increases. - Local maxima occur at points where the function changes from increasing to decreasing. - The leading coefficient sign affects end behavior: positive means the right end goes to $+\infty$, negative means it goes to $-\infty$. - Degree parity and leading coefficient sign determine end behavior. 3. **Analyze intervals for increasing behavior:** - From $-\infty$ to $-9$: graph rises $\Rightarrow$ increasing. - From $-9$ to $-6$: graph falls $\Rightarrow$ decreasing. - From $-6$ to $-2$: graph rises $\Rightarrow$ increasing. - From $-2$ to $3$: graph falls $\Rightarrow$ decreasing. - From $3$ to $7$: graph rises $\Rightarrow$ increasing. - From $7$ to $\infty$: graph falls $\Rightarrow$ decreasing. So increasing intervals are $$(-\infty,-9), (-6,-2), (3,7)$$ 4. **Local maxima occur where function changes from increasing to decreasing:** - At $x=-9$ (rises then falls) - At $x=-2$ (rises then falls) - At $x=7$ (rises then falls) So local maxima at $$x=-9,-2,7$$ 5. **Sign of leading coefficient:** - The graph falls to bottom-left and falls to bottom-right. - Both ends go down $\Rightarrow$ leading coefficient is negative. 6. **Possible degree of the polynomial:** - Since both ends fall, degree must be even and leading coefficient negative. - Possible degrees from options: 4, 6, 8 (all even) - Odd degrees have opposite end behavior, so 5, 7, 9 are not possible. **Final answers:** (a) Increasing on $$(-\infty,-9), (-6,-2), (3,7)$$ (b) Local maxima at $$x=-9,-2,7$$ (c) Leading coefficient is negative. (d) Possible degrees: $$4,6,8$$