Question

Sketch the graph of the following pair of functions on the same axes:
\[f\left( x \right) = \sin x, g\left( x \right) = \sin \left( x + \frac{\pi}{4} \right)\]

Answer 1

\[f\left( x \right) = \sin\frac{x}{2}, g\left( x \right) = \sin x\]
Clearly, sin x and \[\sin \left( x + \frac{\pi}{4} \right)\] is a periodic function with period 2π.
The graphs of \[f\left( x \right) = \sin x\text{ and }g\left( x \right) = \sin \left( x + \frac{\pi}{4} \right)\] on different axes are shown below:


If these two graphs are drawn on the same axes, then the graph is shown below.