The problem asks to find the ordering of the average rates of change of three functions $f(x)$, $g(x)$, and $h(x)$ on the interval $[-3, -1]$, from least to greatest.

Given:

1. Function $f(x)$: Represented by a graph.
2. Function $g(x)$: Provided as a table.
3. Function $h(x)$: Given as the equation $h(x) = x^2 + 2x + 4$.

The average rate of change of a function over an interval $[a, b]$ is calculated as: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$ where $a = -3$ and $b = -1$ for this problem.

1. Rate of change for $g(x)$ from the table:

For $g(x)$, use the values at $x = -3$ and $x = -1$:

• $g(-3) = 31$
• $g(-1) = 13$

The average rate of change for $g(x)$ is: $\text{Rate of Change for } g(x) = \frac{g(-1) - g(-3)}{-1 - (-3)} = \frac{13 - 31}{-1 + 3} = \frac{-18}{2} = -9$

2. Rate of change for $h(x)$ from the equation:

The equation for $h(x)$ is $h(x) = x^2 + 2x + 4$.

Evaluate $h(x)$ at $x = -3$ and $x = -1$: $h(-3) = (-3)^2 + 2(-3) + 4 = 9 - 6 + 4 = 7$ $h(-1) = (-1)^2 + 2(-1) + 4 = 1 - 2 + 4 = 3$

The average rate of change for $h(x)$ is: $\text{Rate of Change for } h(x) = \frac{h(-1) - h(-3)}{-1 - (-3)} = \frac{3 - 7}{-1 + 3} = \frac{-4}{2} = -2$

3. Rate of change for $f(x)$ from the graph:

For $f(x)$, read the values from the graph at $x = -3$ and $x = -1$:

• $f(-3) = -2$
• $f(-1) = 2$

The average rate of change for $f(x)$ is: $\text{Rate of Change for } f(x) = \frac{f(-1) - f(-3)}{-1 - (-3)} = \frac{2 - (-2)}{-1 + 3} = \frac{2 + 2}{2} = \frac{4}{2} = 2$

Summary of Rates of Change:

• $g(x)$: $-9$
• $h(x)$: $-2$
• $f(x)$: $2$

Ordering from least to greatest:

$g(x) < h(x) < f(x)$

Let me know if you want any further clarifications or additional details. Here are some related questions to expand your understanding:

1. How do you interpret the average rate of change graphically?
2. How can you derive the formula for the average rate of change for any function?
3. What role does the sign of the average rate of change play in determining the behavior of a function?
4. Can you find the instantaneous rate of change for any of these functions at $x = -2$?
5. How would the ordering change if the interval were different, say from $[-4, -2]$?

Tip: Remember, the average rate of change over an interval is the slope of the secant line connecting the two points on the graph corresponding to the endpoints of the interval.