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To estimate the average rate of change from \( x = 1 \) to \( x = 4 \), we use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

In this case:
- \( x_1 = 1 \) and \( x_2 = 4 \)
- From the graph:
  - At \( x = 1 \), \( f(1) \approx 4 \)
  - At \( x = 4 \), \( f(4) \approx 0 \)

Now, applying these values into the formula:

\[
\text{Average Rate of Change} = \frac{f(4) - f(1)}{4 - 1} = \frac{0 - 4}{3} = \frac{-4}{3} \approx -1.3
\]

Thus, the average rate of change from \( x = 1 \) to \( x = 4 \) is approximately \( -1.3 \).

Would you like more details or further clarification?

Here are 5 related questions:
1. How would the average rate of change differ if we chose different intervals?
2. What is the meaning of a negative average rate of change in this context?
3. How can you determine the instantaneous rate of change at a specific point from a graph?
4. What is the difference between average and instantaneous rates of change?
5. How does the shape of the graph influence the average rate of change over various intervals?

**Tip**: The steeper the slope between two points, the greater the magnitude of the rate of change.

Based on the graph above, estimate (to one decimal place) the average rate of change from x = 1 to x = 4.

Learn how to estimate the average rate of change from x = 1 to x = 4 using a graph. This problem covers key mathematical concepts like functions, graph interpretation, and the average rate of change formula. Suitable for students in Grades 9-11.

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