To determine the interval of $x$ with the smallest average rate of change for the given function, we calculate the rate of change for each interval on the graph. The rate of change (slope) for an interval $[x_1, x_2]$ is given by:

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

Here are the calculations for each interval based on the points provided:

1. Interval $[3, 7]$: Points (3, 76) and (7, 213) $\frac{213 - 76}{7 - 3} = \frac{137}{4} = 34.25$

2. Interval $[7, 19]$: Points (7, 213) and (19, 656) $\frac{656 - 213}{19 - 7} = \frac{443}{12} \approx 36.92$

3. Interval $[19, 22]$: Points (19, 656) and (22, 1081) $\frac{1081 - 656}{22 - 19} = \frac{425}{3} \approx 141.67$

4. Interval $[22, 38]$: Points (22, 1081) and (38, 1218) $\frac{1218 - 1081}{38 - 22} = \frac{137}{16} \approx 8.56$

The smallest average rate of change occurs on the interval $[22, 38]$ with a rate of approximately $8.56$.

Would you like further details on these calculations or additional help?

Here are five questions that could expand your understanding:

1. How does calculating the rate of change help in understanding the behavior of a function?
2. Can the smallest rate of change always be considered the slowest increase or decrease in values?
3. What would the rate of change look like for intervals with a decrease in values?
4. How would we interpret the average rate of change if the function were not linear?
5. How does the concept of average rate of change apply to real-world scenarios?

Tip: Remember that the rate of change is essentially the slope between two points, representing how steeply or gradually a function changes between them.