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Find the average rate of change in g on the intervals [-4,4] and [0,9]. Let g(x) be the graph shown below

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!

Find the average rate of change in g on the intervals [-4, 4] and [0, 9]. Let g(x) be the graph shown below.

Learn how to calculate the average rate of change of a graph on intervals [-4, 4] and [0, 9] using the average rate of change formula. This involves interpreting the values of the function g(x) at specific points and applying the formula. Suitable for high school students in Grades 9-12.

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