To solve the problem, we need to determine the average rate of change of each function $f(x)$, $g(x)$, and $h(x)$ on the interval $[2, 5]$. Here's the step-by-step process:

1. Determine Average Rate of Change Formula:

The average rate of change of a function $y = f(x)$ on an interval $[a, b]$ is given by: $\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

2. Calculate for $g(x)$ using the given table:

From the table:

• $g(2) = 12$
• $g(5) = 12$

Substitute into the formula: $\text{Average rate of change for } g(x) = \frac{g(5) - g(2)}{5 - 2} = \frac{12 - 12}{3} = 0$

3. Calculate for $h(x) = -x^2 + 10x + 25$:

Evaluate $h(2)$ and $h(5)$: $h(2) = -(2)^2 + 10(2) + 25 = -4 + 20 + 25 = 41$ $h(5) = -(5)^2 + 10(5) + 25 = -25 + 50 + 25 = 50$

Substitute into the formula: $\text{Average rate of change for } h(x) = \frac{h(5) - h(2)}{5 - 2} = \frac{50 - 41}{3} = \frac{9}{3} = 3$

4. Calculate for $f(x)$ from the graph:

From the graph:

• $f(2) = 8$
• $f(5) = -4$

Substitute into the formula: $\text{Average rate of change for } f(x) = \frac{f(5) - f(2)}{5 - 2} = \frac{-4 - 8}{3} = \frac{-12}{3} = -4$

5. Order the Rates of Change:

Now, compare the average rates of change:

• $f(x): -4$
• $g(x): 0$
• $h(x): 3$

The order from least to greatest is: $f(x), g(x), h(x)$

Final Answer:

$\boxed{f(x), g(x), h(x)}$

Let me know if you'd like more clarification or a detailed breakdown of any step!

5 Relative Questions:

1. What happens to the average rate of change if the interval is expanded?
2. How does the behavior of the graph of $f(x)$ affect the rate of change on different intervals?
3. Why is $g(x)$'s rate of change zero in this example?
4. How would the rate of change change if $h(x)$ was a linear function instead of quadratic?
5. Can you determine instantaneous rates of change using this data?

Tip: Always interpret the graph carefully to determine exact values for computations!