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

Solution Steps

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

1. Amplitude: The amplitude is the coefficient of the cosine function, which is 2.
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.
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 \( -\frac{c}{b} \).

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 \( 2 \).

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

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

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

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

Final Answer