1. **State the problem:** Analyze the graph of the function $y = g(x)$ to determine its symmetry, intervals of increase/decrease, local extrema, absolute extrema on the interval $[-2.5, 4.5]$, and the average rate of change from $x = -1$ to $x = 2$. 2. **Determine if $g(x)$ is even, odd, or neither:** - A function is even if $g(-x) = g(x)$ for all $x$. - It is odd if $g(-x) = -g(x)$. - From the description, $g(x)$ is not symmetric about the y-axis or origin. Thus, $g(x)$ is neither even nor odd. 3. **Identify intervals where $g(x)$ is increasing:** - From the graph, $g(x)$ increases from approximately $x = -2$ to $x = 0$ (rising to local max at $(0,3)$). - Then after the local min at $x=1.5$, it increases again from $x hicksim 1.5$ to $x hicksim 2.75$. 4. **Identify intervals where $g(x)$ is decreasing:** - From the local max at $x = 0$ to the local min at $x = 1.5$, $g(x)$ decreases. - From the local max at $x = 2.75$ onward to $x = 4$, $g(x)$ decreases. 5. **Find local maxima and minima:** - Local maxima at approximately $(0, 3)$ and $(2.75, 2)$. - Local minimum at approximately $(1.5, 1)$. 6. **Find absolute maximum and minimum on $[-2.5, 4.5]$:** - From graph and described endpoints approximate values: - Endpoints roughly near $(-2.5, $ hicksim$-1)$ and $(4.5, $ hicksim$0)$. - Absolute max is local max at $(0, 3)$. - Absolute min is near endpoint $(−2.5, -1)$. 7. **Find average rate of change from $x = -1$ to $x = 2$:** - Average rate of change formula: $$\frac{g(2) - g(-1)}{2- (-1)} = \frac{g(2)-g(-1)}{3}$$ - Approximate $g(2)$ from graph is about 1.8 (between local min and max). - Approximate $g(-1)$ from graph is about 1 (between start and local max). - Compute: $$\frac{1.8 - 1}{3} = \frac{0.8}{3} \approx 0.267$$ **Final answers:** - $g(x)$ is neither even nor odd. - Increasing on $(-2, 0)$ and $(1.5, 2.75)$. - Decreasing on $(0, 1.5)$ and $(2.75, 4)$. - Local maxima: $(0, 3)$ and $(2.75, 2)$. - Local minimum: $(1.5, 1)$. - Absolute max on $[-2.5, 4.5]$ is $(0, 3)$. - Absolute min near $(-2.5, -1)$. - Average rate of change from $-1$ to $2$ is approximately $0.267$.