graph{sqrt(x) [-10, 10, -5, 5]}
graph{(sqrtx) - 9 [-8.04, 11.96, -12.84, -2.84]}
translation by the vector #(""_-9 ^ 0)# means that there is no shift parallel to (along) the #x-#axis, but there is a shift parallel to the #y-# axis.
the shift is #-9#, meaning that the graph is shifted downwards by #9# units.
for example:
the #y#-intercept of #y = sqrtx# is #(0,0)#, since #sqrt0 = 0#.
meanwhile, when #x=0# for the graph #y = sqrtx - 9#, #y# is #-9#.
the #x#-intercept of #y = sqrtx - 9# is found at #(81,0)#.
#sqrt81 = 9#
when #y = sqrtx - 9#, #y = 0# while #x# is #81#.
when#y = sqrtx#, #y = 9# while #x# is #81#.
this shows that, to translate from the graph of #y = sqrtx# to the graph of #y = sqrtx - 9#, you do not need to change #x#, but you need to subtract #9# from the #y-#value of all points on the graph.
in vector notation, this is #(""_-9 ^ 0)#.
the top number shows the number of units that need to be added to the #x#-value of each point on the graph. in other words, it shows how far right the graph needs to shift. (if it is negative, then the graph shifts to the left).
here, the top number is #0#, so the graph does not shift left or right.
the bottom number shows the number of units that need to be added to the #y#-value of each point on the graph. therefore, it shows how far upwards the graph needs to shift. (if it is negative, then the graph shifts to downwards).
here, the bottom number is #-9#, so the graph shifts downwards by #9# units.
