Question

Sketch the graphs of the following functions:

3. $y={\sin }^2\left(x-\frac{\mathrm{\pi }}{4}\right)$

Answer 1

$$\begin{gathered} y={\sin }^2\left(\mathrm{x}-{\displaystyle \frac{\mathrm{\pi }}{4}}\right) \\ y-0={\sin }^2\left(x-\frac{\mathrm{\pi }}{4}\right)..................\left(\mathrm{i}\right) \\ \mathrm{On}\mathrm{shifting}\mathrm{the}\mathrm{origin}\mathrm{at}\left(\frac{\mathrm{\pi }}{4},0\right),\mathrm{we}will\mathrm{have}: \\ \mathrm{x}=\mathrm{X}+\frac{\mathrm{\pi }}{4}\mathrm{and}\mathrm{y}=\mathrm{Y}+0 \\ \mathrm{On}\mathrm{subsitituting}\mathrm{the}\mathrm{values}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ \mathrm{Y}={\sin }^{2}X \\ \mathrm{Then},\mathrm{we}\mathrm{draw}\mathrm{the}\mathrm{graph}\mathrm{of}Y={\sin }^{2}X\mathrm{and}\mathrm{shift}\mathrm{it}\mathrm{by}\frac{\mathrm{\pi }}{4}\mathrm{to}\mathrm{the}\mathrm{right}. \end{gathered}$$
The obtained graph is given below.