Question

Sketch the graph of the following curves on the same scale and the same axes:
\[y = \cos 2x\text{ and }y = \cos 2\left( x - \frac{\pi}{4} \right)\]

Answer 1

First, we draw the graph of y = cos 2x.
Let us now draw the graph of \[y = \cos2\left( x - \frac{\pi}{4} \right)\]

\[y = \cos2\left( x - \frac{\pi}{4} \right)\]

\[ \Rightarrow y - 0 = \cos2\left( x - \frac{\pi}{4} \right) . . . (i)\]

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

\[x = X + \frac{\pi}{4}\text{ and }y = Y + 0\]

On subsitituting the values in (i), we get: 

\[Y = \cos2X\]

\[\text{ Then, we draw the graph of Y }= \cos2X\text{ and shift it by }\frac{\pi}{4}\text{ to the right.}\]

 Then, we will obtain the following graph: