Problem 38 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=5 \cos 2 \pi x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=5 \cos 2 \pi x\) is 5 and the period is 1.

Step 1: Determine the Amplitude

The amplitude(A) of the function \(y=5 \cos 2 \pi x\) is the absolute value of the coefficient of cos, which is |5| = 5.

Step 2: Calculate the Period

The period(P) of the function \(y=5 \cos 2 \pi x\) is determined by \(P= \frac {2\pi}{|B|}\), where B is the coefficient of x inside the cos. Here, B = \(2\pi\), so the Period \(P= \frac {2\pi}{2\pi}=1\)

Step 3: Plot the Graph

To plot the function \(y=5 \cos 2 \pi x\), mark a point at every interval of the period (in this case, 1) on the x-axis. The amplitude indicates the highest and lowest points the function will reach. Since the amplitude is 5, the function will reach the points 5 and -5 on the y-axis. The cos function starts at its peak, so at x=0, y=5. Then it continues to the next points at -5 (half the period) and ends back at 5 (full period). Continue this to get one full period of the function.