https://file.aiquickdraw.com/mathbot/file/1725597384186gcfgm4iw.jpg

The image shows a graph of a function and a question asking for the average rate of change from \( x = 1 \) to \( x = 4 \). To find the average rate of change, we need to compute the difference in the function's values at these points and divide by the difference in \( x \).

The formula for the average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is:

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

Let's find the values of \( f(1) \) and \( f(4) \) from the graph.

Looking at the graph:
- At \( x = 1 \), the function appears to be around \( f(1) \approx 1 \).
- At \( x = 4 \), the function appears to be around \( f(4) \approx 16 \).

Now, let's plug in these values:

\[
\text{Average rate of change} = \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = \frac{15}{3} = 5
\]

So, the average rate of change from \( x = 1 \) to \( x = 4 \) is **5**.

Would you like to know more details about this calculation or ask any questions? Here are some related questions:

1. What is the difference between average rate of change and instantaneous rate of change?
2. How do you find the derivative of a function graphically?
3. How do you determine the function's equation given its graph?
4. Can the average rate of change of a function be negative? When does this occur?
5. What are some applications of the average rate of change in real-world scenarios?

**Tip:** The average rate of change is essentially the slope of the line connecting two points on a graph, which gives insight into how fast the function is increasing or decreasing between those points.

Use the graph of the function to find its average rate of change from x = 1 to x = 4. Simplify your answer as much as possible.

Learn how to calculate the average rate of change of a function between x = 1 and x = 4 using a graph. The solution applies key concepts in calculus, including the formula for average rate of change and the slope of a line. Ideal for students in Grades 9-12.

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