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We are tasked with finding the interval on which the average rate of change of the function \( f(x) \) equals \( \frac{3}{2} \). The graph of \( f(x) \) is provided, and the average rate of change is defined as:

\[
\text{Average rate of change on } [a, b] = \frac{f(b) - f(a)}{b - a}.
\]

We will now calculate the average rate of change for each given interval in the options and see if any match the value \( \frac{3}{2} \).

### 1. Interval: \( [-1, 1] \) (Option A)
For this interval:
- \( f(-1) = 1 \)
- \( f(1) = 2 \)

The average rate of change is:
\[
\frac{f(1) - f(-1)}{1 - (-1)} = \frac{2 - 1}{2} = \frac{1}{2}.
\]
This is not equal to \( \frac{3}{2} \).

### 2. Interval: \( [-1, 2] \) (Option B)
For this interval:
- \( f(-1) = 1 \)
- \( f(2) = 3 \)

The average rate of change is:
\[
\frac{f(2) - f(-1)}{2 - (-1)} = \frac{3 - 1}{3} = \frac{2}{3}.
\]
This is also not equal to \( \frac{3}{2} \).

### 3. Interval: \( [-2, 1] \) (Option C)
For this interval:
- \( f(-2) = 0 \)
- \( f(1) = 2 \)

The average rate of change is:
\[
\frac{f(1) - f(-2)}{1 - (-2)} = \frac{2 - 0}{3} = \frac{2}{3}.
\]
This is also not equal to \( \frac{3}{2} \).

### 4. Interval: \( [-2, -1] \) (Option D)
For this interval:
- \( f(-2) = 0 \)
- \( f(-1) = 1 \)

The average rate of change is:
\[
\frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{1 - 0}{1} = 1.
\]
This is not equal to \( \frac{3}{2} \).

---

After checking all options, none of the intervals result in an average rate of change equal to \( \frac{3}{2} \), which suggests that the intervals provided in the problem may need to be reconsidered or there's a mismatch in values between the question and graph.

Do you want to go over any of the steps again or look for other potential intervals?

Here are five questions for you:
1. How is the average rate of change of a function similar to the slope of a line?
2. Can the average rate of change be negative?
3. What is the geometric interpretation of the average rate of change on a graph?
4. How would you find the average rate of change over a smaller interval?
5. How does the average rate of change differ from the instantaneous rate of change?

**Tip**: The average rate of change formula can be understood as calculating the slope between two points on the graph of a function.

On which of the intervals is the average rate of change of f(x) equal to 3/2?

Discover how to find the interval on a graph of f(x) where the average rate of change is equal to 3/2. This problem includes calculating the slope using the average rate of change formula and interpreting the graph. Suitable for Grades 9-12.

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