The perimeter of quadrilateral $PQRS$ equals $PQ+QR+RS+SP$. Since the dots are spaced 1 unit apart horizontally and vertically, then $PQ=4, QR=4$, and $PS=1$. Thus, the perimeter equals $4+4+RS+1$ which equals $RS+9$. We need to determine the length of $RS$. If we draw a horizontal line from $S$ to point $T$ on $QR$, we create a right-angled triangle $STR$ with $ST=4$ and $TR=3$. By the Pythagorean Theorem, $RS^{2}=ST^{2}+TR^{2}=4^{2}+3^{2}=25$. Since $RS>0$, then $RS=\sqrt{25}=5$. Thus, the perimeter of quadrilateral $PQRS$ is $5+9=14$.