Starting with the "basic cycle" for #tan(theta)# i.e. for #theta in [-pi/2,+pi/2]#
Then consider what values of #x# would place #(x+pi)# in this same range, i.e. #(x+pi) in [-pi/2,+pi/2]#
(Yes; I know: because #tan# has a cycle length of #pi# we could recognize that the cycle will repeat at exactly the same place, but let's do this for the more general case.)
#(x+pi)in[-pi/2,pi/2]color(white)("X"rarrcolor(white)("X")x in [-(3pi)/2,-pi/2]#
Giving us the "basic cycle" for #tan(x+pi)#:
Adding #5# to this to get #y=5+tan(x+pi)# simply shifts the points upward #5# units:
In the image below, I have added the "non-basic cycles" as well as the "basic cycle" used for analysis:
