Question

Identify the amplitude, vertical shift, frequency B, and phase shift for the function y = 3cos(1/3x)

Answer 1

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:

1. **Amplitude (A)**: The amplitude is the absolute value of \( A \). In this case, \( A = 3 \). So, the amplitude is:
\[
\text{Amplitude} = |3| = 3
\]

2. **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:
\[
\text{Vertical Shift} = 0
\]

3. **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).

4. **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: \( 0 \)