Answer
$y=-3 \ \sin (3x+\dfrac{3 \pi}{4})$
Work Step by Step
The general form for the sinusoidal function can be expressed as:
$y=A\sin{(\omega x-\phi)}+B ..(1)$
where $A$ is the amplitude and $B$ represents the vertical shift.
The $\omega$ can be found from the period by the formula as: $\omega=\dfrac{2\pi}{T}$ and the phase shift is $\dfrac{\phi}{\omega}$.
This means that $\phi=\omega \times \ Phase \ Shift$
We have: $A=-3$, $B=0$,
$\omega=\dfrac{2\pi}{T}=\dfrac{2\pi}{ 2\pi/3}=3$
$\phi=\omega \times \ Phase \ Shift=(3)(-\dfrac{\pi}{4})=-\dfrac{3 \pi}{4}$
Therefore, our sinusoidal function (1) becomes: $y=-3 \ \sin (3x+\dfrac{3 \pi}{4})$
