average rate of change of a function over an interval

1. Identify the interval: The problem asks for the average rate of change over the interval $[1, 4]$. This means we need to consider the function's values at $x=1$ and $x=4$.

1. Determine the function's values at the endpoints of the interval: From the graph, we can see the coordinates of two points on the curve $f(x)$. The first point is at $(1, 1)$, so $f(1) = 1$. The second point is at $(4, 2)$, so $f(4) = 2$.

1. Apply the average rate of change formula: The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by the formula: $$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

4. Substitute the values into the formula: In this case, $a = 1$, $b = 4$, $f(a) = f(1) = 1$, and $f(b) = f(4) = 2$. Plugging these values into the formula, we get:
$$ \text{Average Rate of Change} = \frac{2 - 1}{4 - 1} $$

5. Calculate the average rate of change: Perform the subtraction in the numerator and the denominator:
$$ \text{Average Rate of Change} = \frac{1}{3} $$

The average rate of change of the graph of $f(x)$ over the interval $[1, 4]$ is $\frac{1}{3}$.