To find the average rate of change of the function $g(x)$ from $x = -5$ to $x = -2$, we use the formula:

$\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}$

where $x_1 = -5$ and $x_2 = -2$.

Step-by-Step Solution:

1. Identify the Points: From the graph:

   • At $x = -5$, the corresponding $y$-value $g(-5)$ appears to be approximately $-3$.
   • At $x = -2$, the corresponding $y$-value $g(-2)$ appears to be approximately $3$.

2. Calculate the Difference in $y$-values: $g(-2) - g(-5) = 3 - (-3) = 3 + 3 = 6$

3. Calculate the Difference in $x$-values: $x_2 - x_1 = -2 - (-5) = -2 + 5 = 3$

4. Calculate the Average Rate of Change: $\text{Average Rate of Change} = \frac{6}{3} = 2$

Answer:

The average rate of change of the function $g(x)$ from $x = -5$ to $x = -2$ is $2$.

Would you like further clarification or have any additional questions?

Here are some related questions that might interest you:

1. How does the average rate of change differ from the instantaneous rate of change?
2. Can we determine the behavior of the function based on the average rate of change?
3. How would the average rate of change change if we looked at a different interval?
4. What is the significance of a positive or negative average rate of change?
5. How would you calculate the rate of change if given a different function, like a piecewise function?

Tip: When finding the average rate of change, always ensure to check the function values accurately from the graph to avoid estimation errors.