Question

determine the amplitude, period, any vertical translation, and any phase shift of the given graph. 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.

Explanation:

Step1: Recall general cosine - function form

The general form of a cosine function is $y = A\cos(B(x - C))+D$, where $A$ is the amplitude, $\frac{2\pi}{|B|}$ is the period, $C$ is the phase - shift, and $D$ is the vertical translation.
For the function $y=-3\cos(x - \frac{\pi}{6})$, we have $A=-3$, $B = 1$, $C=\frac{\pi}{6}$, and $D = 0$.

Step2: Determine amplitude

The amplitude is $|A|$. Since $A=-3$, the amplitude $|A|=| - 3|=3$.

Step3: Determine period

The period is $\frac{2\pi}{|B|}$. Since $B = 1$, the period is $\frac{2\pi}{|1|}=2\pi$.

Step4: Determine vertical translation

Since $D = 0$ in the general form $y = A\cos(B(x - C))+D$, there is no vertical translation.

Step5: Determine phase - shift

The phase - shift is $C$. Here, $C=\frac{\pi}{6}$, so the phase - shift is $\frac{\pi}{6}$ units to the right.

Answer:

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