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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!

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### 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!

The functions f(x), g(x), and h(x) are shown below. Select the option that represents the ordering of the functions according to their average rates of change on the interval -8 <= x <= 4, from least to greatest.

Learn how to order functions f(x), g(x), and h(x) based on their average rates of change in the interval -8 <= x <= 4. This problem involves interpreting graphs, quadratic functions, and calculating average rates of change, making it suitable for high school students.

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