91Ó°ÊÓ

The function to be plotted is :

$y=-4\sin \left(\frac{1}{8}x\right)$

Divide the interval $\left[0,16\mathrm{Ï€}\right]$into four subintervals,

each length:$\frac{16\mathrm{Ï€}}{4}=4\mathrm{Ï€}$

The x-coordinates of the five key points are :

The first x-coordinate is 0

second x-coordinate is $0+4\mathrm{Ï€}=4\mathrm{Ï€}$

Third x-coordinate is$4\mathrm{Ï€}+4\mathrm{Ï€}=8\mathrm{Ï€}$

Fourth x-coordinate is $8\mathrm{Ï€}+4\mathrm{Ï€}=12\mathrm{Ï€}$

Fifth x-coordinate is$12\mathrm{Ï€}+4\mathrm{Ï€}=16\mathrm{Ï€}$

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

Since $y=-4\sin \left(\frac{1}{8}x\right)$

Hence, multiply the y-coordinates of the five key points of $y=\sin x$by-4.

The five key points on the graph are :

$\left(0,0\right),\left(4\mathrm{Ï€},-4\right),\left(8\mathrm{Ï€},0\right),\left(12\mathrm{Ï€},4\right),\left(16\mathrm{Ï€},0\right)$

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 $4\mathrm{Ï€}$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 -4 to 4.

So range of the function is$\left[-4,4\right]$.