find the amplitude, period, and horizontal shift of the function. y = -2 sin(6(x + π/6)) amplitude period…

Explanation:

Step1: Recall amplitude formula

For $y = A\sin(B(x - C))$, amplitude is $|A|$. Here $A=-2$, so amplitude $= |-2| = 2$.

Step2: Recall period formula

The period of $y = A\sin(B(x - C))$ is $T=\frac{2\pi}{|B|}$. Here $B = 6$, so $T=\frac{2\pi}{6}=\frac{\pi}{3}$.

Step3: Recall horizontal - shift formula

For $y = A\sin(B(x - C))$, the horizontal shift is $C$. The function is $y=-2\sin(6(x+\frac{\pi}{6}))$, which can be written as $y=-2\sin(6(x-(-\frac{\pi}{6})))$. So the horizontal shift is $-\frac{\pi}{6}$.

Answer:

amplitude: 2 period: $\frac{\pi}{3}$ horizontal shift: $-\frac{\pi}{6}$