Problem 12 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Calculate Amplitude

In the given function \(y=3 \sin 2 \pi x\), the coefficient of the sine function is 3. Therefore, the amplitude of the function is 3.

Step 2: Calculate Period

In the given function \(y=3 \sin 2 \pi x\), the coefficient of x inside the sine function is \(2 \pi\). Therefore, the period of the function is \(\frac{2 \pi}{2 \pi}\), which simplifies to 1.

Step 3: Draw the Graph

Now graph the function paying close attention to the amplitude and period. Start at \(x = 0\) and end at \(x = 1\) since the period is 1. The sine function starts at 0, reaches a maximum at the \(\frac{1}{4}\) of the period, comes back to 0 at half the period, reaches a minimum at \(\frac{3}{4}\) of the period, and finally comes back to 0 at the end of the period. The maximum and minimum values will be equal to the amplitude and negative amplitude respectively (here, 3 and -3).