The function $f$ is defined by $f(x)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants. The graph of $y=f(x)$ in the $xy$-plane passes through the point $(4,0)$, and the $x$-coordinate of its vertex is $1$. If $a=3$, what is the value of $b$?