Question

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

Answer 1

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]}