To identify the amplitude, period, vertical shift, and phase shift of the function \( y = 3 \cos\left(\frac{1}{3}x\right) \), we can analyze it in the context of the general cosine function form:
\[ y = A \cos(B(x - C)) + D \]
Where:
- \( A \) is the amplitude,
- \( B \) affects the period,
- \( C \) is the phase shift,
- \( D \) is the vertical shift.
From the given function, we can identify the parameters:
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Amplitude: The amplitude \( A \) is the absolute value of the coefficient in front of the cosine function. Here, \( A = 3 \), so the amplitude is \( 3 \).
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Period: The period \( P \) of a cosine function is given by the formula:
\[ P = \frac{2\pi}{|B|} \]
In our function, \( B = \frac{1}{3} \). Thus, the period is:
\[ P = \frac{2\pi}{\frac{1}{3}} = 2\pi \cdot 3 = 6\pi \]
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Vertical Shift: The vertical shift \( D \) refers to any constant added to the function. In this equation, there is no \( + D \) term, which means \( D = 0 \). Therefore, there is no vertical shift, and the vertical shift is \( 0 \).
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Phase Shift: The phase shift \( C \) can be identified from the term \( (x - C) \). In our case, it is written as \( \frac{1}{3}x \), which can be thought of as \( \cos\left(\frac{1}{3}(x - 0)\right) \). Hence, \( C = 0 \), meaning there is no phase shift.
In summary:
- Amplitude: \( 3 \)
- Period: \( 6\pi \)
- Vertical Shift: \( 0 \)
- Phase Shift: \( 0 \)
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