Questions: Find the phase shift, period, and amplitude of the function. y=2 cos(2 π x + (3 π)/2) + 2 Give the exact values, not decimal approximations. Phase shift: II π Period: 5 Amplitude: I Don't Know

Find the phase shift, period, and amplitude of the function.
y=2 cos(2 π x + (3 π)/2) + 2

Give the exact values, not decimal approximations.

Phase shift: II π

Period: 5

Amplitude: I Don't Know

Transcript text: Unproctored Placement Assessment Question 19 Find the phase shift, period, and amplitude of the function. \[ y=2 \cos \left(2 \pi x+\frac{3 \pi}{2}\right)+2 \] Give the exact values, not decimal approximations. Phase shift: II $\square$ $\square$ $\pi$ Period: $\square$ $\times$ 5 Amplitude: $\square$ I Don't Know Submit

Solution

Solution Steps

To find the phase shift, period, and amplitude of the function $ y = 2 \cos \left(2 \pi x + \frac{3 \pi}{2}\right) + 2 $:

Amplitude: The amplitude is the coefficient of the cosine function, which is 2.
Period: The period of a cosine function $ \cos(bx) $ is given by $ \frac{2\pi}{b} $. Here, $ b = 2\pi $, so the period is 1.
Phase Shift: The phase shift is calculated by solving $ bx + c = 0 $ for $ x $, where $ c = \frac{3\pi}{2} $. The phase shift is $ -\frac{c}{b} $.

Step 1: Amplitude

The amplitude of the function $ y = 2 \cos \left(2 \pi x + \frac{3 \pi}{2}\right) + 2 $ is the coefficient of the cosine term. Therefore, the amplitude is $ 2 $.

Step 2: Period

The period of a cosine function $ \cos(bx) $ is given by $ \frac{2\pi}{b} $. Here, $ b = 2\pi $, so the period is:

\[ \frac{2\pi}{2\pi} = 1 \]

Step 3: Phase Shift

The phase shift is calculated by solving $ bx + c = 0 $ for $ x $, where $ c = \frac{3\pi}{2} $. The phase shift is:

\[ -\frac{c}{b} = -\frac{\frac{3\pi}{2}}{2\pi} = -\frac{3}{4} \]