which graph has a rate of change equal to $\frac{1}{3}$ in the interval between 0 and 3 on the x - axis?

Explanation:

Step1: Recall rate - of - change formula

The rate of change (slope) between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. We want to find the graph where for $x_1 = 0$ and $x_2=3$, $m=\frac{1}{3}$.

Step2: Check each graph

For the first graph: Let's assume two points on the line in the interval $x\in[0,3]$. If $x_1 = 0,y_1 = 1$ and $x_2=3,y_2 = 2$. Then $m=\frac{2 - 1}{3-0}=\frac{1}{3}$. For the second graph: It is not a straight - line in the interval $[0,3]$, and the rate of change is not constant in this interval. For the third graph: Let two points be $x_1 = 0,y_1 = 0$ and $x_2 = 3,y_2=1$. Then $m=\frac{1 - 0}{3-0}=\frac{1}{3}$. For the fourth graph: It is a non - linear graph in the interval $[0,3]$, and the rate of change is not constant in this interval.

Answer:

The first and the third graphs.