Solution:

Starting at $J$, Tess’s distance from home increases as she runs from $J$ to $K$.

From $K$ to $L$, her distance continues to increase, but with a different shape, reaching its maximum at the opposite corner $L$.

From $L$ to $M$ and then from $M$ back to $J$, her distance decreases back to $0$, again changing behavior at $M$.

Graph $D$ is the only graph with this increase, changed-rate increase, changed-rate decrease, and final decrease to $0$.

Thus, D is the correct answer.