To find the average rate of change of the function $g(x)$ on the given intervals $[-4,4]$ and $[0,9]$, we use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}$ where $a$ and $b$ are the endpoints of the interval.

1. Interval $[-4, 4]$:

• From the graph, at $x = -4$, the value of $g(-4)$ is approximately 6.
• At $x = 4$, the value of $g(4)$ is approximately -8.

Now, apply the formula:

$\text{Average Rate of Change on } [-4, 4] = \frac{g(4) - g(-4)}{4 - (-4)} = \frac{-8 - 6}{4 - (-4)} = \frac{-14}{8} = -\frac{7}{4}$

Thus, the average rate of change on $[-4, 4]$ is $-\frac{7}{4}$.

2. Interval $[0, 9]$:

• From the graph, at $x = 0$, the value of $g(0)$ is approximately -6.
• At $x = 9$, the value of $g(9)$ is approximately 3.

Now, apply the formula:

$\text{Average Rate of Change on } [0, 9] = \frac{g(9) - g(0)}{9 - 0} = \frac{3 - (-6)}{9 - 0} = \frac{3 + 6}{9} = \frac{9}{9} = 1$

Thus, the average rate of change on $[0, 9]$ is $1$.

Summary:

• The average rate of change on $[-4, 4]$ is $-\frac{7}{4}$.
• The average rate of change on $[0, 9]$ is $1$.

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How would you interpret a negative average rate of change?
2. Can the average rate of change be the same as the instantaneous rate of change at a point?
3. How do you find the average rate of change over a non-symmetric interval like $[1, 5]$?
4. What is the significance of the slope of the secant line between two points on a curve?
5. How would you calculate the rate of change if $g(x)$ were given as a function rather than a graph?

Tip: The average rate of change is the slope of the secant line between two points on a graph!