y = -2 sin (3x + π/3) + 1

Explanation:

Step1: Identify the amplitude

The general form of a sinusoidal function is $y = A\sin(Bx - C)+D$. For the function $y=-2\sin(3x + \frac{\pi}{3})+1$, the amplitude is given by $|A|$. Here, $A=-2$, so the amplitude $|A| = 2$.

Step2: Identify the period

The period of a sinusoidal function $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Here, $B = 3$, so the period $T=\frac{2\pi}{3}$.

Step3: Identify the phase - shift

The phase - shift of the function $y = A\sin(Bx - C)+D$ is $\frac{C}{B}$. For the function $y=-2\sin(3x+\frac{\pi}{3})+1$, we can rewrite it as $y=-2\sin(3(x+\frac{\pi}{9})) + 1$. So the phase - shift is $-\frac{\pi}{9}$ (a shift to the left).

Step4: Identify the vertical shift

The vertical shift of the function $y = A\sin(Bx - C)+D$ is $D$. Here, $D = 1$.

Answer:

Amplitude: 2; Period: $\frac{2\pi}{3}$; Phase - shift: $-\frac{\pi}{9}$; Vertical shift: 1