Describe how the graph of the function

Question

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

Answer 1

The function $F(x)=\sin (x-\frac{\pi }{6})$ 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 $F(0)=\sin (-\frac{\pi }{6})=-\frac{1}{2}$.

Overall, the primary difference between the graphs of $F(x)=\sin (x-\frac{\pi }{6})$ 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.