f(x)=cos(x-p/3)
The given function is \( f(x) = \cos(x - \frac{\pi}{3}) \). This is a cosine function that has been horizontally shifted.
Key Properties of the Function:
Period: The period of the cosine function is \( 2\pi \). Thus, the period of \( f(x) \) is also \( 2\pi \).
Amplitude: The amplitude of \( \cos(x) \) is 1. No vertical scaling is applied here, so the amplitude of \( f(x) \) is also 1.
Phase Shift: The expression \( x - \frac{\pi}{3} \) indicates a phase shift to the right by \( \frac{\pi}{3} \).
Vertical Shift: There is no vertical shift in this case as there is no constant added to the cosine function.
Important Points:
The cosine function starts at its maximum value. Since there is no vertical shift, the maximum value of \( f(x) \) is \( 1 \), occurring at \( x = \frac{\pi}{3} + 2k\pi \) (where \( k \) is any integer).
The minimum value of \( f(x) \) is \( -1 \), occurring at \( x = \frac{\pi}{3} + (2k + 1)\pi \).
Summary:
- Function: \( f(x) = \cos(x - \frac{\pi}{3}) \)
- Period: \( 2\pi \)
- Amplitude: 1
- Phase Shift: Right by \( \frac{\pi}{3} \)
- Max Value: 1 at \( x = \frac{\pi}{3} + 2k\pi \)
- Min Value: -1 at \( x = \frac{\pi}{3} + (2k + 1)\pi \)
If you need to analyze or visualize the graph of this function, let me know!
