To analyze the function \( y = 3\cos\left(\frac{1}{3}x\right) \) and find its amplitude, vertical shift, frequency \( B \), and phase shift, we can look at the general form of a cosine function:
\[ y = A \cos(Bx - C) + D \]
where:
- \( A \) is the amplitude,
- \( B \) affects the frequency,
- \( C \) is the phase shift,
- \( D \) is the vertical shift.
Now, we can identify the components from the given function:
-
Amplitude (A): The amplitude is the absolute value of \( A \). In this case, \( A = 3 \). So, the amplitude is: \[ \text{Amplitude} = |3| = 3 \]
-
Vertical Shift (D): The vertical shift is the value of \( D \). In this function, there is no additional constant added or subtracted, so \( D = 0 \). Thus, the vertical shift is: \[ \text{Vertical Shift} = 0 \]
-
Frequency (B): The frequency \( B \) can be extracted from the coefficient of \( x \) in the function. In this case, the function can be rewritten as: \[ B = \frac{1}{3} \] The frequency of the cosine function is related to the period. The period \( P \) is given by: \[ P = \frac{2\pi}{B} \] To find \( B \) in its typical interpretation around frequency, we recognize that the frequency (which refers to how many cycles occur in \( 2\pi \)), is thus the reciprocal due to the formula: \[ \text{Frequency} = \frac{1}{P} = \frac{B}{2\pi} = \frac{1/3}{2\pi} = \frac{1}{6\pi} \] However, keep in mind that commonly, \( B \) refers just to the coefficient of \( x \), so it's \( \frac{1}{3} \). You could clarify use based on context (period/actual frequency).
-
Phase Shift: The phase shift is given by the expression \( \frac{C}{B} \). Since there is no \( C \) (the function has no horizontal shift), let’s take \( C = 0 \). Therefore: \[ \text{Phase Shift} = \frac{C}{B} = \frac{0}{\frac{1}{3}} = 0 \]
In summary:
- Amplitude: \( 3 \)
- Vertical Shift: \( 0 \)
- Frequency \( B \): \( \frac{1}{3} \) (again this is the coefficient affecting period, not a usual frequency rate)
- Phase Shift: \( 0 \)