Precalculus (10th Edition)

$y=-3\sin{(3x+\frac{3\pi}{4})}$.
The general form for the sinusoidal function is: $y=A\sin{(\omega x-\phi)}+B$ where $A$ is the amplitude, $B$ is the vertical shift, we need $\omega$ can be computed from the period by the formula $\omega=\frac{2\pi}{T}.$ The phase shift is $\frac{\phi}{\omega}$, hence $\phi=\omega\cdot\text{phase shift}.$ Hence here: $A=-3$, $B=0$, $\omega=\frac{2\pi}{T}=\dfrac{2\pi}{\frac{2\pi}{3}}=3$ $\phi=\omega\cdot\text{phase shift}=3\cdot\frac{-\pi}{4}=\frac{-3\pi}{4}$. Hence our function is: $y=-3\sin{(3x+\frac{3\pi}{4})}$.