Problem 8 - Algebra and Trigonometry

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 is \(\pi/2\). The graph of one period is a sine wave starting at \(y=0\) for \(x=0\), reaching a maximum of \(y=1\) at \(x=\pi/4\), back to zero at \(x=\pi/2\), reaching a minimum of \(y=-1\) at \(x=3\pi/4\), and back to \(y=0\) for \(x=\pi\).

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function is the coefficient of the sinusoidal function, which in this case is \(1\). So, the amplitude of \(y = \sin 4x\) is \(1\).

Step 2: Identify the Period

The period of a sinusoid function can be determined by dividing \(2\pi\) by the absolute value of the frequency coefficient within the sinusoidal function. In this case, the frequency coefficient is \(4\). So the period is \(2\pi / 4 = \pi/2\).

Step 3: Graph the function

To graph one period of \(y = \sin 4x\), start by marking the x-axis at intervals of the period (\(\pi/2\)). Then, plot the points for \(x = 0, x = \pi/2, x = \pi, x = 3\pi/2, x = 2\pi\), and so on, to get the sinusoidal graph. Since the amplitude is \(1\), the maximum and minimum values on the \(y\) axis will be \(1\) and \(-1\), respectively.