Problem 8 - Precalculus Essentials

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=\sin 4 x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=\sin 4x\) is 1, and the period of the function is \(\frac{π}{2}\). A graph of one period of this function would start at (0,0), reach its peak at (\(\frac{π}{8}\), 1), return to the x-axis at (\(\frac{π}{4}\), 0), reach its minimum at (\(\frac{3π}{8}\), -1), and complete the period at (\(\frac{π}{2}\), 0).

Step 1: Identify the Amplitude

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.

Step 2: Identify the Period

The period of a sine or cosine function is given by dividing the standard period (2π) 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 \(T = \frac{2π}{|4|}= \frac{π}{2}\). Therefore, the period of this function is \(\frac{π}{2}\).

Step 3: Graph the Function

To graph the function \(y = \sin 4x\), plot the points corresponding to one period of the function. Since we have the period as \(\frac{π}{2}\), we start at x=0 and span x-values till \(\frac{π}{2}\). 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 \(\frac{π}{8}\) where y=1, at \(\frac{π}{4}\) y returns to 0, at \(\frac{3π}{8}\) y=-1 and finally y returns to 0 at \(\frac{π}{2}\). Connecting these dots will create a complete graph for one period of \(y = \sin 4x\).