Problem 9 - Precalculus Essentials

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 the period is \(4\pi\). For graphing, the wave will start from the origin (0,0), reach a peak of 3 at \(\pi\), go down to -3 at \(3\pi\), and then return to 0 at \(4\pi\).

Step 1: Identify the Type of Function and Parameters

The given function is \(y=3\sin\frac{1}{2}x\). This function follows the structure \(y=A\sin(Bx)\) which is a sine function where A is the amplitude and B affects the period.

Step 2: Determine the Amplitude

The amplitude is the value of A, which equals the absolute value of the number multiplied by \(\sin\). For this function, the amplitude is \(|3|\), which equals 3.

Step 3: Determine the Period

The period can be found through \(2\pi|B|\). Here B is equal to \(\frac{1}{2}\). Hence, the period here is \(2\pi|\frac{1}{2}|\) which simplifies to \(4\pi\).

Step 4: Graphing the Sine Function

For graphing, a basic understanding of the sine wave can be utilized. A sine function usually starts at zero, then reaches its peak at the mid-point of the period, which here is \(2\pi\) and then comes back to zero at the end of the period \(4\pi\). The amplitude tells us the maximum and minimum points, which in this case would be 3 and -3 respectively. By including these values, a complete wave from \(0\) to \(4\pi\) can be drawn.