Question

Sketch the graphs of the following pairs of functions on the same axes:
(i) $f\left(x\right)=\sin x,g\left(x\right)=\sin \left(x+\frac{\pi }{4}\right)$
(ii) f(x) = sin x, g(x) = sin 2x
(iii) f(x) = sin 2x, g(x) = 2 sin x
(iv) $f\left(x\right)=\sin \frac{x}{2},g\left(x\right)=\sin x$

Answer 1

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

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





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



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

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