Question

what type of translation takes the graph of (f)x = |x| to the graph g(x) = |4 + x|?
A. vertical translation 4 units up
B. horizontal translation of 4 unit right
c. vertical translation of 4 units down
d. horizontal translation of 4 units left

Answer 1

if I had f(x)=x
then f(x+1)=x+1 this is a translation of -1 units parallel to the x axis
f(x)+4=x+4 this is a translation of 4 units parallel to the y axis

so in your case
f(x)=|x|
you not have |4+x|
so instead of a function of x, we now have 4+x=y so f(y)=|4+x|
since y is expressed in terms of x, we can call it a new function g(x) so g(x)=|4+x|

so what translation is this?

Answer 2

a more general term for a translation in x and y directions

f(x)=x
f(x+a)=x+a, translation -a parallel to x axis

f(x)=x
f(x)+b=x+b, translation b parallel to y axis