determine the amplitude, period, any vertical translation, and any phase shift of the given graph. y = - 2…

Explanation:

Step1: Recall amplitude formula

For $y = A\sin(Bx - C)+D$, amplitude is $|A|$. Here $A=-2$, so amplitude $= | - 2|$.

Step2: Calculate amplitude

$| - 2|=2$.

Step3: Recall period formula

The period of $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Here $B = 1$, so $T=\frac{2\pi}{|1|}=2\pi$.

Step4: Determine vertical translation

For $y = A\sin(Bx - C)+D$, vertical translation is given by $D$. Here $D = 0$, so no vertical translation.

Step5: Determine phase - shift

The phase - shift of $y = A\sin(Bx - C)+D$ is $\frac{C}{B}$. Here $C=\frac{\pi}{6}$ and $B = 1$, so phase - shift is $\frac{\frac{\pi}{6}}{1}=\frac{\pi}{6}$ to the right.

Answer:

Amplitude: 2 Period: $2\pi$ Vertical translation: None Phase - shift: $\frac{\pi}{6}$ to the right