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.