The question asks to order the functions $f(x)$, $g(x)$, and $h(x)$ by their average rates of change on the interval $-8 \leq x \leq 4$, from least to greatest.

Step-by-Step Process:

1. Average Rate of Change Formula: The average rate of change of a function $y = f(x)$ over an interval $[a, b]$ is given by: $\frac{f(b) - f(a)}{b - a}$ Here, the interval is from $x = -8$ to $x = 4$.

For $f(x)$ (based on graph):

• From the graph, estimate the values of $f(-8)$ and $f(4)$.
• It appears that: $f(-8) \approx 25 \quad \text{and} \quad f(4) \approx -10$
• Average rate of change for $f(x)$: $\frac{f(4) - f(-8)}{4 - (-8)} = \frac{-10 - 25}{12} = \frac{-35}{12} \approx -2.92$

For $g(x)$ (from table):

• From the table, for $g(x)$: $g(-8) = 115 \quad \text{and} \quad g(4) = 7$
• Average rate of change for $g(x)$: $\frac{g(4) - g(-8)}{4 - (-8)} = \frac{7 - 115}{12} = \frac{-108}{12} = -9$

For $h(x)$ (given function):

• The function is $h(x) = -x^2 - 9x - 5$.
• Evaluate $h(-8)$ and $h(4)$: $h(-8) = -(-8)^2 - 9(-8) - 5 = -64 + 72 - 5 = 3$ $h(4) = -(4)^2 - 9(4) - 5 = -16 - 36 - 5 = -57$
• Average rate of change for $h(x)$: $\frac{h(4) - h(-8)}{4 - (-8)} = \frac{-57 - 3}{12} = \frac{-60}{12} = -5$

Conclusion:

• The average rates of change are: $f(x): -2.92, \quad g(x): -9, \quad h(x): -5$
• Ordering from least to greatest: $g(x) < h(x) < f(x)$

Let me know if you'd like further details, or if you have any other questions!

Follow-Up Questions:

1. How would the interval affect the average rate of change for $f(x)$ if it were different?
2. Can you determine the points where the instantaneous rate of change of $f(x)$ is zero based on the graph?
3. What is the physical interpretation of the average rate of change in terms of the graph of $f(x)$?
4. How does the quadratic nature of $h(x)$ affect its rate of change?
5. Could you use calculus to find the instantaneous rate of change at a specific point?

Tip:

Always verify that you're using the correct interval limits when calculating rates of change!