1. The problem asks us to graph the function where the output is the greatest integer less than or equal to $\frac{1}{3}x$. 2. This function is written as $y=\left\lfloor \frac{1}{3}x \right\rfloor$, where $\left\lfloor \cdot \right\rfloor$ denotes the greatest integer (floor) function. 3. For each input \(x\), we calculate $\frac{1}{3}x$ and then take the greatest integer less than or equal to this value. 4. The function forms a step graph that jumps at multiples of 3, since $\frac{1}{3}x$ changes by 1 whenever $x$ increases by 3. 5. For example, when $x=0$, $y=\left\lfloor 0 \right\rfloor=0$; when $x=2$, $y=\left\lfloor \frac{2}{3} \right\rfloor=0$; when $x=3$, $y=\left\lfloor 1 \right\rfloor=1$; 6. Thus, the graph looks like a step function increasing by 1 at each integer multiple of 3. 7. This completes the explanation and characterization of the function $y=\left\lfloor \frac{1}{3}x \right\rfloor$.