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

Feb 19, 2017

See Socratic graph and explanation.

#### Explanation:

y is of period $\frac{2 \pi}{3}$.

As $| \csc \left(\theta\right) | \ge 1 , | y + 2 | \ge 6$, giving $y \ge 4 \mathmr{and} \le - - 8$

The graph has vertical asymptotes, when $\csc \left(3 x + 2 \frac{\pi}{3}\right) = \infty$, giving

$x = \frac{1}{3} \left(k \pi - 2 \frac{\pi}{3}\right) , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

For that matter, half period $\frac{1}{3} \pi$ 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 $\frac{\pi}{3}$ = 1.05, for the three

asymptotes at $x = - .7 , .35 \mathmr{and} 1.4$