Problem 9 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=3 \sin \frac{1}{2} x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = 3 \sin \frac{1}{2} x\) is 3 and its period is \(4\pi\).

Step 1: Identifying the amplitude

The amplitude of a sine function is given by the absolute value of its coefficient. In this case, the coefficient is 3. Therefore, the amplitude of given function \(y = 3 \sin \frac{1}{2} x\) is 3.

Step 2: Identifying the period

The period of the sine function can be calculated using the formula \(2\pi \div |B|\), where \(B\) is the coefficient of \(x\) inside the sine function. In this case, the coefficient of \(x\) is \frac{1}{2}. So, the period of the function is \(2\pi \div \frac{1}{2} = 4\pi\).

Step 3: Graphing one period of function

To graph one period of the function, first note that a typical sine function starts from the origin (0,0), reaches its highest point at \(\frac{1}{4}\) of its period, returns to the origin at half of its period, reaches its lowest point at \(\frac{3}{4}\) of its period, and returns to the origin at the end of its period. Given the amplitude and period of this function, the highest point becomes (0,3), the 'half period' point becomes \(2\pi , 0\), the lowest point becomes \(3\pi , -3\) and the end of the period is \(4\pi , 0\). Then, graph these points and join them smoothly in the shape of a sine curve, considering these key points.