91Ó°ÊÓ

The amplitude of a function is the maximum or minimum value the function can reach. For the given function \(y = \sin 4x\), the coefficient in front of the sine is 1 (as there is no explicit number). Therefore, the amplitude of this function is 1.

The period of a sine or cosine function is given by dividing the standard period (2Ï€) by the absolute value of the coefficient in front of x, inside the function. For the given function \(y = \sin 4x\), we have a coefficient of 4 in front of x. The standard period would be \(T = \frac{2Ï€}{|4|}= \frac{Ï€}{2}\). Therefore, the period of this function is \(\frac{Ï€}{2}\).

To graph the function \(y = \sin 4x\), plot the points corresponding to one period of the function. Since we have the period as \(\frac{Ï€}{2}\), we start at x=0 and span x-values till \(\frac{Ï€}{2}\). The sine function begins with (0, 0), reaches its maximum at \(\frac{1}{4}\) of its period, returns to zero at half of its period, reaches its minimum at \(\frac{3}{4}\) of its period, and then returns back to zero completing its period. Applying this understanding to the current function, plot (0,0), then move to \(\frac{Ï€}{8}\) where y=1, at \(\frac{Ï€}{4}\) y returns to 0, at \(\frac{3Ï€}{8}\) y=-1 and finally y returns to 0 at \(\frac{Ï€}{2}\). Connecting these dots will create a complete graph for one period of \(y = \sin 4x\).