Problem 7 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determine the Amplitude

The amplitude of the function \(y=\sin 2x\) is determined by the coefficient A in front of the sine function. In this case, A=1, so the amplitude is \(|1|=1\).

Step 2: Determine the Period

The period of the function \(y=\sin 2x\) is determined by \(\frac{2\pi}{|B|}\), where B is the coefficient in front of x within the sine function. Here, B=2, so the period is \(\frac{2\pi}{|2|}=\pi\).

Step 3: Graph the Function

Now we can graph one period of the function. Draw the x-axis and y-axis. Mark the x-axis from 0 to \(\pi\). For the y-axis, mark from -1 to 1. The graph of \(y=\sin 2x\) starts at (0,0), rises to (π/4,1), falls back to (π/2,0), falls to 3π/4,-1, and ends at (π,0). Connect these points smoothly to get the graph of one period of the function.