Question

determine the amplitude, period, any vertical translation, and any phase shift of y = - 3 cos(x - \frac{\pi}{6}). the amplitude is 3. the period is 2\pi. (type an exact answer, using \pi as needed.) select the correct choice below and fill in any answer boxes within your choice. a. the vertical translation is units. b. there is no vertical translation. select the correct choice below and fill in any answer boxes within your choice. a. the phase shift is units to the right. (type an exact answer, using \pi as needed.) b. there is no phase shift.

Explanation:

Step1: Recall amplitude formula

For $y = A\cos(Bx - C)+D$, amplitude is $|A|$. Here $A=- 3$, so $|A| = 3$.

Step2: Recall period formula

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

Step3: Check vertical translation

For $y = A\cos(Bx - C)+D$, vertical translation is given by $D$. In $y=-3\cos(x - \frac{\pi}{6})$, $D = 0$, so there is no vertical translation.

Step4: Check phase - shift

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

Answer:

The amplitude is 3.
The period is $2\pi$.
B. There is no vertical translation.
A. The phase shift is $\frac{\pi}{6}$ units to the right.