[My tab crashed and I lost my edits. One more try.]
Graph of #2csc(2x - 1)#
graph{2 csc(2x - 1) [-10, 10, -5, 5]}
The trig functions like # csc x # all have period #2\pi.# By doubling the coefficient on #x#, that halves the period, so the function #csc(2x)# must have a period of #pi#, as must #2 csc(2x-1)#.
The phase shift for #csc(ax-b)# is given by #b/a.# Here we have a phase shift of #frac 1 2# radian, approximately #28.6^\circ#. The minus sign means #2csc(2x-1)# leads #2csc(2x)# so we call this a positive phase shift of #frac 1 2# radian.
#csc(x) = 1/sin(x)# so it diverges twice per period. The amplitude is infinite.
