ANSWER
[tex]y = 3 \sin( \frac{2}{3}( x+ \frac{2}{\pi} )) - [/tex]
Or
[tex]y = - 3 \sin( \frac{2}{3}( x+ \frac{2}{\pi} )) - 1[/tex]
EXPLANATION
The general sine function is given by,
[tex]y = a \sin(bx \pm \: c) \pm d[/tex]
where
[tex] |a| = 2[/tex]
[tex]a =\pm2[/tex]
[tex] \frac{2\pi}{b} = 3\pi[/tex]
is the period.
This implies that,
[tex]b = \frac{2}{3} [/tex]
The phase shift is
[tex] \frac{b}{c} = \frac{\pi}{2} [/tex]
This implies that,
[tex]c = \frac{4}{3\pi} [/tex]
[tex]d = - 1[/tex]
is the vertical shift.
The required equation is:
[tex]y = \pm3 \sin( \frac{2}{3}( x+ \frac{2}{\pi} )) - 1[/tex]
