How do you graph, identify the domain, range, and asymptotes for #y=csc(3theta+pi/2)+3#?

1 Answer
Aug 3, 2018

Answer:

See explanation and graph.

Explanation:

#y = csc ( 3theta + pi/2 ) +3, 3theta + pi/2 ne# asymptotic #kpi#,

#k = 0, +-1, +-2, +-3, ...#

#rArr theta ne# asymptotic #( 2k - 1 ) pi/6#

csc value #notin { -1, 1 )#. So,

#y notin ( -1 +3, 1 + 3 ) = ( 2, 4 )

Period is the period of #sin ( 3theta + pi/2 ) = (2pi)/3#

Phase shift #= ( - pi/2)/3 = - pi/6#

Vertical shift = 3

See graph, depicting all these aspects.
graph{((y-3)sin (3x+pi/2) -1)(y-3+0x)(y-2+0x)(y-4+0x)(x+5/6pi -0.0001y)(x+pi/6 -0.0001y)(x-pi/6 +0.0001y)(x+pi/2 - 0.001y)=0[ -3 3 -3 12]}