91Ó°ÊÓ

The function to be plotted is :

$y=2\cos \left(\frac{1}{4}x\right)$

Divide the interval $\left[0,8\mathrm{Ï€}\right]$into four subintervals,

each of length:$\frac{8\mathrm{Ï€}}{4}=2\mathrm{Ï€}$

The x-coordinates of the five key points are :

First x-coordinate is 0

second x-coordinate is$0+2\mathrm{Ï€}=2\mathrm{Ï€}$

Third x-coordinate is$2\mathrm{Ï€}+2\mathrm{Ï€}=4\mathrm{Ï€}$

Fourth x-coordinate is $4\mathrm{Ï€}+2\mathrm{Ï€}=6\mathrm{Ï€}$

Fifth x-coordinate is$6\mathrm{Ï€}+2\mathrm{Ï€}=8\mathrm{Ï€}$

These values represent the x-coordinates of the five key points on the graph.

Since $y=2\cos \left(\frac{1}{4}x\right)$

Multiply the y-coordinates of the five key points for $y=\cos x$by 2.

The five key points on the graph are :

role="math" localid="1646929287738" $\left(0,2\right),\left(2\mathrm{Ï€},0\right),\left(4\mathrm{Ï€},-2\right),\left(6\mathrm{Ï€},0\right),\left(8\mathrm{Ï€},2\right)$

Plot the five key points obtained in Step 4 and fill in the graph. Extend the graph in each direction to obtain the complete graph. Notice that additional key points appear every $2\mathrm{Ï€}$radian.

As we can see that the value of $x$is set of all real number.

So domain is$\left(-∞,∞\right)$.

The y- value of the function in the graph lies from -1 to 1

So range of the function is$\left[-2,2\right]$