To find the correct ordering of the functions according to their average rates of change over the interval $2 \leq x \leq 6$, we need to calculate the average rate of change for each function.

The formula for the average rate of change is:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$ where $f(x)$ is the function, and $[a, b]$ is the interval (in this case, $a = 2$ and $b = 6$).

1. Function $f(x)$:

From the table, we can see that for $f(x)$:

• $f(2) = 7$
• $f(6) = 31$

Average rate of change for $f(x)$: $\frac{f(6) - f(2)}{6 - 2} = \frac{31 - 7}{6 - 2} = \frac{24}{4} = 6$

2. Function $g(x)$:

From the table, we can see that for $g(x)$:

• $g(2) = 7$
• $g(6) = 31$

Average rate of change for $g(x)$: $\frac{g(6) - g(2)}{6 - 2} = \frac{31 - 7}{6 - 2} = \frac{24}{4} = 6$ So, both $f(x)$ and $g(x)$ have the same average rate of change.

3. Function $h(x) = x^2 - x + 28$:

Now, let's calculate $h(2)$ and $h(6)$:

• $h(2) = 2^2 - 2 + 28 = 4 - 2 + 28 = 30$
• $h(6) = 6^2 - 6 + 28 = 36 - 6 + 28 = 58$

Average rate of change for $h(x)$: $\frac{h(6) - h(2)}{6 - 2} = \frac{58 - 30}{6 - 2} = \frac{28}{4} = 7$

Ordering from least to greatest:

• $f(x)$ and $g(x)$ both have an average rate of change of 6.
• $h(x)$ has an average rate of change of 7.

Thus, the order is $f(x), g(x), h(x)$ from least to greatest.

Correct Answer:

The correct option is: $\boxed{f(x), g(x), h(x)}$ or written as: $f, g, h$.

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Do you want more details or have any questions?

Here are 5 related questions:

1. How do you calculate the average rate of change for a quadratic function?
2. What is the difference between the rate of change of linear and quadratic functions?
3. How does the average rate of change relate to the derivative of a function?
4. How can you use the average rate of change to determine if a function is increasing or decreasing?
5. How do you interpret the rate of change in real-world applications?

Tip: The average rate of change of a function over an interval is equivalent to the slope of the secant line connecting two points on the graph of the function within that interval.