Question

Sketch the graph of the following functions on the same scale.
(i) y=cos x and y =cos $(x-\frac{\pi }{4})$
(ii) y=cos 2 x and y = cos2 $(x-\frac{\pi }{4})$
(iii) y =cos x and y = cos $\left(\frac{x}{2}\right)$

Answer 1

(i)

(ii) We have,
$y=cos2(x-\frac{pi}{4})$
$\Rightarrow y-0=cos2(x-\frac{\pi }{4})$ ...(i)
Shifting the origin at $(\frac{\pi }{4},0)$, we obtain
$x=X+\frac{\pi }{4},y=Y+0$
Substituting these values in (i), we get Y=cos 2X.
Thus we draw the graph of Y = cos 2x and shift it by $\frac{\pi }{4}$ to the right to get the required graph.

(iii) To obtain the graph of y= cos $\frac{x}{2}$ we first draw the graph of y= cos x in the interval $[0,2\pi ]$ and then divide the x - coordiantes of the points where it crosses x - axis by $\frac{1}{2}$.