Sketch the graphs of the following curves on the same scale and the same axes:

Solution:

(i) First, we draw the graph of y = cos x.

Let us now draw the graph of $y=\cos \left(x-\frac{\pi}{4}\right)$.

$y=\cos \left(\mathrm{x}-\frac{\pi}{4}\right)$

$\Rightarrow y-0=\cos \left(x-\frac{\pi}{4}\right)$                   ...(i)

On shifting the origin at $\left(\frac{\pi}{4}, 0\right)$, we get $:$

$\mathrm{x}=\mathrm{X}+\frac{\pi}{4}$ and $\mathrm{y}=\mathrm{Y}+0$

On subsitituting the values in $(\mathrm{i})$, we get:

$\mathrm{Y}=\cos X$

Then, we draw the graph of $Y=\cos X$ and shift it by $\frac{\pi}{4}$ to the right.

 Then, we will obtain the following graph:



(ii) First, we draw the graph of y = cos 2x.

Let us now draw the graph of $y=\cos 2\left(x-\frac{\pi}{4}\right)$.

$y=\cos 2\left(\mathrm{x}-\frac{\pi}{4}\right)$

$\Rightarrow y-0=\cos 2\left(x-\frac{\pi}{4}\right)$            ...(i)

On shifting the origin at $\left(\frac{\pi}{4}, 0\right)$, we get:

$\mathrm{x}=\mathrm{X}+\frac{\pi}{4}$ and $\mathrm{y}=\mathrm{Y}+0$

On subsititut ing the values in (i), we get:

$\mathrm{Y}=\cos 2 X$

Then, we draw the graph of $Y=\cos 2 X$ and shift it by $\frac{\pi}{4}$ to the right.

 Then, we will obtain the following graph:



(iii) First, we draw the graph of y = cos x.

Let us now draw the graph of $y=\cos \left(\frac{x}{2}\right)$.

$\Rightarrow y=\cos \frac{1}{2}(x)$

 Then, we will obtain the following graph:


(iv) First, we draw the graph of y = cos2 x.
Let us now draw the graph of y = cos x.
 
Then, we will obtain the following graph: