Problem 37 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=4 \cos 2 \pi x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=4 \cos 2 \pi x\) is 4. The period is 1.

Step 1: Find the Amplitude

The amplitude of a function, say \(y=A \cos(x)\) or \(y= A \sin(x)\) is given by the absolute value of the coefficient of the cosine or sine function. In our function \(y=4 \cos 2 \pi x\), the coefficient of cosine function is 4, so the amplitude is \(|4| = 4\).

Step 2: Determine the Period

The period of the function in a sinusoidal function \(y = A cos(bx)\) or \(y = A sin(bx)\) is calculated as \(2\pi/|b|\). For this function \(y=4 \cos 2 \pi x\), the coefficient of \(x\) is \(2\pi\). Therefore, the period will be \(2\pi/|2\pi| = 1\).

Step 3: Graph the Function

To graph one period of the function, set up an x-y coordinate system. On the x-axis represent one full period, and on the y-axis scale it to the amplitude. For \(y=4 \cos 2 \pi x\), the period is 1, and the amplitude is 4. So, plot the points over one period from 0 to 1 and join these points to form the graph. With a curve peaking at \(y=4\) and troughing at \(y=-4\). Note that the function will repeat this pattern over every period.