https://file.aiquickdraw.com/mathbot/file/1727395624542n2zy5c6h.jpg

To calculate the average rate of change between \( x = 1 \) and \( x = 4 \), we can use the formula for the average rate of change of a function \( f(x) \) over the interval \( [a, b] \):

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

Here, \( a = 1 \) and \( b = 4 \). From the graph:

- \( f(1) \approx 7 \)
- \( f(4) \approx 10 \)

Now, applying these values into the formula:

\[
\text{Average rate of change} = \frac{10 - 7}{4 - 1} = \frac{3}{3} = 1.0
\]

So, the average rate of change from \( x = 1 \) to \( x = 4 \) is approximately **1.0**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the average rate of change compare to the instantaneous rate of change?
2. What is the significance of a positive average rate of change?
3. Can you determine where the slope of the curve is steepest between \( x = 1 \) and \( x = 4 \)?
4. How would the result change if we used different points, such as \( x = 2 \) and \( x = 5 \)?
5. What real-world phenomena could this graph represent?

**Tip:** The average rate of change over an interval gives the slope of the secant line connecting two points, providing a general idea of the behavior of the function over that interval.

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 solution explains key calculus concepts like rate of change and the slope of the secant line. Suitable for high school students.

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