To find the interval of $x$ where the average rate of change is the smallest, we can calculate the rate of change between each pair of points on the graph and compare them.

The rate of change (slope) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1}$

Let's calculate the rate of change for each interval:

1. From $x = 4$ to $x = 10$:

   • Points: $(4, 6)$ and $(10, 20)$ $\frac{20 - 6}{10 - 4} = \frac{14}{6} \approx 2.33$

2. From $x = 10$ to $x = 27$:

   • Points: $(10, 20)$ and $(27, 28)$ $\frac{28 - 20}{27 - 10} = \frac{8}{17} \approx 0.47$

3. From $x = 27$ to $x = 37$:

   • Points: $(27, 28)$ and $(37, 42)$ $\frac{42 - 28}{37 - 27} = \frac{14}{10} = 1.4$

4. From $x = 37$ to $x = 67$:

   • Points: $(37, 42)$ and $(67, 58)$ $\frac{58 - 42}{67 - 37} = \frac{16}{30} \approx 0.53$

Conclusion:

The smallest rate of change occurs in the interval from $x = 10$ to $x = 27$, with a rate of approximately 0.47.

Would you like more details or explanations?

Here are some related questions to consider:

1. What does the rate of change tell us about the graph?
2. How can we interpret negative rates of change in a graph?
3. What would a rate of change of zero signify?
4. How can the steepness of the graph be related to the rate of change?
5. Can the average rate of change differ if we select a larger interval?

Tip: The rate of change gives you the slope of the line connecting two points. A smaller rate of change means a less steep line.