91Ó°ÊÓ

The function to be plotted is

By comparing the given function $\sin \left(3x\right)$

with $A\sin \left(\mathrm{Ó\neg }x\right)$

we get amplitude: $A=1$

Time period :$$\begin{gathered} T=\frac{2\mathrm{Ï€}}{\mathrm{Ó\neg }} \\ =\frac{2\mathrm{Ï€}}{3} \end{gathered}$$

The graph will lie between -1 and 1 on the y-axis. One cycle begins at $x=0$and ends at$x=\frac{2\mathrm{Ï€}}{3}$

Divide the interval, $\frac{2\mathrm{π}}{3}÷4=\frac{\mathrm{π}}{6}$into four subintervals, each of length:$\frac{\mathrm{π}}{6}$

The x-coordinates of the five key points are :

First x-coordinate is 0

second x-coordinate is$0+\frac{\mathrm{Ï€}}{6}=\frac{\mathrm{Ï€}}{6}$

Third x-coordinate is$\frac{\mathrm{Ï€}}{6}+\frac{\mathrm{Ï€}}{6}=\frac{\mathrm{Ï€}}{3}$

Fourth x-coordinate is $\frac{\mathrm{Ï€}}{3}+\frac{\mathrm{Ï€}}{6}=\frac{\mathrm{Ï€}}{2}$

Fifth x-coordinate is $\frac{\mathrm{Ï€}}{2}+\frac{\mathrm{Ï€}}{6}=\frac{2\mathrm{Ï€}}{3}$

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

$0,\frac{\mathrm{Ï€}}{6},\frac{\mathrm{Ï€}}{3},\frac{\mathrm{Ï€}}{2},\frac{2\mathrm{Ï€}}{3}$

Since $y=\sin \left(3x\right)$

Multiply the y-coordinates of the five key points for $\sin x$by 1

The five key points on the graph are :

$\left(0,0\right),\left(\frac{\mathrm{Ï€}}{6},1\right)\left(\frac{\mathrm{Ï€}}{3},0\right),\left(\frac{\mathrm{Ï€}}{2},-1\right),\left(\frac{2\mathrm{Ï€}}{3},0\right)$

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 $\frac{\mathrm{Ï€}}{6}$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$\left[-1,1\right]$.