Question

How do you graph `y=6csc(3x+(2pi)/3)-2`?

Answer 1

See Socratic graph and explanation.

Explanation:

y is of period `(2pi)/3`.

As `|csc(theta)|>=1, |y+2|>=6`, giving `y >=4 and <=--8`

The graph has vertical asymptotes, when `csc(3x+2pi/3)=oo`, giving

`x= 1/3(kpi-2pi/3), k =0, +-1, +-2, +-3, ...`

For that matter, half period `1/3pi` is the difference between two

consecutive asymptote x values.

The graph for one period #x in(-2/9pi, 4/9pi) is also included.

graph{(y+2)sin(3x+2pi/3)-6=0 [-40, 40, -21, 19]}

graph{(y+2)sin(3x+2pi/3)-6=0 [-.7, 1.4, -14.2, 12]}

Not-to-scale graph to reveal spacing `pi/3` = 1.05, for the three

asymptotes at `x = -.7, .35 and 1.4`