Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the period and amplitude. $$y=\frac{3}{2} \cos \frac{\pi x}{2}$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\) is \(\frac{3}{2}\) and the period is \(4\).

Step 1: Identify the Amplitude

The amplitude \(A\) is the multiplier of the cosine function, which determines its maximum magnitude or height. In the function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\), the amplitude is the coefficient in front of the cosine, which is \(\frac{3}{2}\).

Step 2: Identify the Period

The period \(T\) is found by the logic \(B=\frac{2\pi}{T}\). In this function, the coefficient in front of \(x\) inside the cosine function is \(\frac{\pi}{2}\). So, equating it to \(\frac{2\pi}{T}\), we solve for \(T\): \(\frac{2\pi}{T} = \frac{\pi}{2} \implies T = \frac{2\pi}{(\pi/2)} = 4\). So, the period of the function is \(4\).