Problem 10 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=\frac{1}{2} \sin \frac{\pi x}{3} $$

Step-by-Step Solution

Answer

The amplitude of the function \(y=\frac{1}{2} \sin \frac{\pi x}{3}\) is \(\frac{1}{2}\) and the period is 6.

Step 1: Identify the Amplitude

To find the amplitude, identify the absolute value of the coefficient of the sine function. In this case, the value before \(\sin\) function is \( \frac{1}{2}\). The amplitude is therefore \(\frac{1}{2}\).

Step 2: Identify the Period

To find the period, take the coefficient B of the variable inside the sinusoidal function and apply it to the expression \(\frac{2\pi}{B}\). Here, B is \(\frac{\pi}{3}\). Therefore, substitute B into the formula to find the period: \[\frac{2\pi}{\frac{\pi}{3}}=6\].

Step 3: Statement of the solution

After finding both the amplitude and period of the given function, we are able to state that the amplitude of the function \(y=\frac{1}{2} \sin \frac{\pi x}{3}\) is \(\frac{1}{2}\) and the period is 6.