Question

Sketch the graph of the following pairs of function on the same axes:

(i) $y=\sin x,y=\sin \left(x+\frac{\mathrm{\pi }}{4}\right)$
(ii) y = sin x, y = sin 3x

Answer 1

(i)
First, we draw the graph of y = sin x.
Then, let us draw the graph of $y=\sin \left(x+\frac{\mathrm{\pi }}{4}\right)$.
$$\begin{gathered} y=\sin \left(\mathrm{x}+{\displaystyle \frac{\mathrm{\pi }}{4}}\right) \\ \Rightarrow y-0=\sin \left(x+\frac{\mathrm{\pi }}{4}\right)...\left(\mathrm{i}\right) \\ On\mathrm{shifting}\mathrm{the}\mathrm{origin}\mathrm{at}\left(-\frac{\mathrm{\pi }}{4},0\right),\mathrm{we}get: \\ \mathrm{x}=\mathrm{X}-\frac{\mathrm{\pi }}{4}\mathrm{and}\mathrm{y}=\mathrm{Y}+0 \\ On\mathrm{subsititut}ing\mathrm{the}\mathrm{values}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ \mathrm{Y}=\mathrm{sinX} \\ \mathrm{Then},\mathrm{we}\mathrm{draw}\mathrm{the}\mathrm{graph}\mathrm{of}Y=\sin X\mathrm{and}\mathrm{shift}\mathrm{it}\mathrm{by}\frac{\mathrm{\pi }}{4}\mathrm{to}\mathrm{the}\mathrm{left}. \end{gathered}$$
Then, we will obtain the following graph:


(ii)
First, we draw the graph of y = sin x.
Let us now draw the graph of y = sin 3x.
Step I- We find the value of c and a by comparing y = sin 3x with y = c sin ax, i.e. c = 1 and a =3.
Step II - Then, we draw the graph of y = sin x and mark the point where it crosses the x-axis.
Step III - Divide the x-coordinates of the points where y = sin x crosses x-axis by 3 (i.e. a =3) and mark the maximum value (i.e. c = 1) and minimum value (i.e.$-$c = $-$1).
Then, we obtain the following graph: