Problem 15 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\sin \frac{2}{3} x$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Amplitude

The amplitude is given by the absolute value of the coefficient attached with sine function, which is \(a\). In the equation, the value of \(a\) is -1. Therefore, the amplitude is \(|-1|=1\).

Step 2: Identify the Period

The period of the graph can be calculated using the formula for the period of a sin function, which is \(\frac{2\pi}{|b|}\). In the equation, the value of \(b\) is \(\frac{2}{3}\). So, the period is \(\frac{2\pi}{|\frac{2}{3}|} = 3\pi\).

Step 3: Plotting the function

First, draw the x and y axes. Next, draw a sine wave, starting from the origin point (0,0). Reflect the wave over the x-axis because of the negative sign. The wave should pass through the points (0, 0), \(\frac{3\pi}{2}\), (0,0), (3\pi), (0,0). The wave should have a maximum height of 1 and minimum of -1 as the amplitude is 1.