Sketch the graphs of the following pairs of functions on the same axes:

(i) $f(x)=\sin x, g(x)=\sin \left(x+\frac{\pi}{4}\right)$

Clearly, $\sin x$ and $\sin \left(x+\frac{\pi}{4}\right)$ is a periodic function with period $2 \pi$.

The graphs of $f(x)=\sin x$ and $g(x)=\sin \left(x+\frac{\pi}{4}\right)$ on different axes are shown below:

 

 

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

 

(ii) f(x) = sin x, g(x) = sin 2x

Clearly, sin x and sin 2x is a periodic function with period 2π and π, respectively.

The graphs of f(x) = sin x and g(x) = sin 2x on different axes are shown below:

 

 

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

 

(iii) f(x) = sin 2x, g(x) = 2 sin x

Clearly, sin 2x and 2 sin x is a periodic function with period π and 2π, respectively.

The graphs of f(x) = sin 2x and g(x) = 2 sin x on different axes are shown below:

 

 

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

 

(iv) $f(x)=\sin \frac{x}{2}, g(x)=\sin x$

Clearly, $\sin \frac{x}{2}$ and $\sin x$ is a periodic function with period $4 \pi$ and $2 \pi$, respectively.

The graphs of $f(x)=\sin \frac{x}{2}$ and $g(x)=\sin x$ on different axes are shown below:

 

 

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