Problem 13 - 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 its period is 1.

Step 1: Determine the amplitude

Amplitude is given by the absolute value of the coefficient of the sine function. In this case, the coefficient of the \(\sin\) function is -3, so the amplitude is \(\|-3\|=3\).

Step 2: Determine the period

The period is determined by \(P=2\pi/B\), where B is the coefficient of x in the argument of the sine function. Here, \(B=2\pi\), thus \(P=2\pi/2\pi=1\).

Step 3: Graph the function

Now that we know the amplitude (3) and the period (1), we can draw the graph of the function. Remember that this is a negative sine function, so it will be reflected over the x-axis. The graph starts from the origin (0,0), reaches down to the minimum point (-3), goes back to zero (completing the half period), reaches the maximum point (3) and returns to zero (completing one full period). The graph then repeats this pattern.