1. **Problem Statement:** Given the function $f$ with the following characteristics: - Domain: $[-9, \infty)$ - $x$-intercept(s): $0$ - $y$-intercept: $0$ - Increasing on $(-2, \infty)$ - Decreasing on $(-3, -2)$ - Constant on $(\infty, -3)$ (likely a typo, interpreted as no constant interval) - Relative minimum at $x = -2$ with value $f(-2) = -9$ - No relative maximum - $f(-8) = -8$ - $f(x) = -9$ at $x = -2$ - $f$ is neither even nor odd 2. **Goal:** Understand the behavior and sketch the function based on the given information. 3. **Key Concepts:** - $x$-intercept means $f(x) = 0$ at $x=0$. - $y$-intercept means $f(0) = 0$. - Increasing means $f'(x) > 0$. - Decreasing means $f'(x) < 0$. - Relative minimum at $x = -2$ means $f'(-2) = 0$ and $f''(-2) > 0$. - No relative maximum means no point where $f'(x) = 0$ and $f''(x) < 0$. 4. **Analysis:** - The function decreases from $x = -3$ to $x = -2$. - At $x = -2$, the function has a relative minimum with value $-9$. - The function increases from $x = -2$ onward. - The function passes through the origin $(0,0)$. - At $x = -8$, $f(-8) = -8$. 5. **Summary:** - The function starts at $x = -9$ (domain start), likely near $f(-9)$ (not given). - It is constant or undefined before $-3$ (given constant interval is likely a typo). - Decreases on $(-3, -2)$ to reach minimum $-9$ at $x = -2$. - Increases on $(-2, \infty)$ passing through $(0,0)$. 6. **No explicit formula is given, so the function is described qualitatively.** 7. **Desmos LaTeX:** Since no explicit formula is provided, we leave it minimal. Final answer: The function $f$ has a relative minimum at $x = -2$ with value $-9$, passes through the origin, decreases on $(-3,-2)$, and increases on $(-2, \infty)$ with domain $[-9, \infty)$ and is neither even nor odd.