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The problem asks for ordering the functions \( f(x) \), \( g(x) \), and \( h(x) \) according to their average rates of change over the interval \( 4 \leq x \leq 6 \), from least to greatest.

### Step 1: Average rate of change for \( h(x) \)
The function for \( h(x) \) is given as:

\[
h(x) = x^2 - x + 32
\]

The average rate of change is calculated as:

\[
\text{Average rate of change of } h(x) = \frac{h(6) - h(4)}{6 - 4}
\]

- \( h(6) = 6^2 - 6 + 32 = 36 - 6 + 32 = 62 \)
- \( h(4) = 4^2 - 4 + 32 = 16 - 4 + 32 = 44 \)

\[
\text{Average rate of change of } h(x) = \frac{62 - 44}{6 - 4} = \frac{18}{2} = 9
\]

### Step 2: Average rate of change for \( g(x) \)
For \( g(x) \), the values are given in a table:

\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
4 & 19 \\
6 & 35 \\
\hline
\end{array}
\]

The average rate of change of \( g(x) \) over the interval is:

\[
\text{Average rate of change of } g(x) = \frac{g(6) - g(4)}{6 - 4} = \frac{35 - 19}{2} = \frac{16}{2} = 8
\]

### Step 3: Average rate of change for \( f(x) \)
From the graph of \( f(x) \), estimate the values of \( f(6) \) and \( f(4) \):

- \( f(6) \approx 25 \)
- \( f(4) \approx 20 \)

Thus, the average rate of change of \( f(x) \) is:

\[
\text{Average rate of change of } f(x) = \frac{f(6) - f(4)}{6 - 4} = \frac{25 - 20}{2} = \frac{5}{2} = 2.5
\]

### Step 4: Ordering the functions
The average rates of change are:

- \( f(x): 2.5 \)
- \( g(x): 8 \)
- \( h(x): 9 \)

Thus, the correct order from least to greatest is:

\[
f(x), g(x), h(x)
\]

The correct answer is: **\( f(x), g(x), h(x) \)**.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. What is the geometric meaning of the average rate of change for a function over an interval?
2. How can you calculate the average rate of change from a graph without precise points?
3. How does the average rate of change relate to the slope of a secant line?
4. Can the average rate of change be negative, and what would that indicate about the function?
5. How would the rate of change differ over different intervals for \( f(x) \), \( g(x) \), and \( h(x) \)?

**Tip:** When calculating the average rate of change from a graph, always approximate values carefully to avoid errors.

The functions f(x), g(x), and h(x) are shown. Order them according to their average rates of change over the interval 4 ≤ x ≤ 6 from least to greatest.

Learn how to order functions based on their average rate of change over a given interval. This problem includes a quadratic function, a tabulated function, and a function presented on a graph. Suitable for Grades 9-11.

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