Question

Describe how the graph of the function
F (x) = sin (x - pu/6) differs from the graph of sin x.

Answer 1

The function \( F(x) = \sin\left(x - \frac{\pi}{6}\right) \) is a horizontally shifted version of the basic sine function \( \sin(x) \). Here are the key differences between the graphs of \( F(x) \) and \( \sin(x) \):

1. **Horizontal Shift**: The function \( F(x) \) has a horizontal shift to the right by \( \frac{\pi}{6} \). This means every point on the graph of \( \sin(x) \) is moved \( \frac{\pi}{6} \) units to the right. For example, where \( \sin(x) \) has a maximum at \( x = \frac{\pi}{2} \), \( F(x) \) will have a maximum at \( x = \frac{\pi}{2} + \frac{\pi}{6} = \frac{2\pi}{3} \).

2. **Same Amplitude and Period**: Both functions have the same amplitude of 1 and the same period of \( 2\pi \). This means the height of the waves remains unchanged and they still complete a full cycle in the same horizontal span.

3. **Key Points**: The key points (such as intercepts, maxima, and minima) of \( \sin(x) \) will occur at different \( x \)-coordinates for \( F(x) \). For instance:
- The y-intercept of \( \sin(x) \) is \( \sin(0) = 0 \), while for \( F(x) \), the y-intercept is \( F(0) = \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \).

Overall, the primary difference between the graphs of \( F(x) = \sin\left(x - \frac{\pi}{6}\right) \) and \( \sin(x) \) is the horizontal shift of the entire sine wave to the right by \( \frac{\pi}{6} \) units, while maintaining the same shape, amplitude, and period.