You want to know what transformation maps [tex]3 \sqrt{x} [/tex] onto [tex]3 \sqrt{x-4} +3[/tex].
Note [tex]f(x)=3\sqrt{x}[/tex] so [tex]f(x-4)=3 \sqrt{x-4} [/tex] so [tex]f(x-4)+3= \sqrt{x-4}+3 [/tex].
We just analyse the function to see the translations. f(x-4) corresponds to a horizontal translation 4 units in the positive x direction. f(x)+3 corresponds to a vertical translation 3 units in the positive y direction.
So overall, the parent function moves 4 units right and 3 units up. This is a translation by the vector [tex] \left[\begin{array}{r}4&3\end{array}\right] [/tex].
