Problem 13 - Precalculus Essentials

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

Step 1: Determine Amplitude

The amplitude A is the absolute value of the coefficient of the \(\sin\) term. For \(y=-3 \sin 2\pi x\), the amplitude \(A\) is \(|-3|\), or \(3\).

Step 2: Determine Period

The period P of the sine function is found by dividing \(2\pi\) by the absolute value of the coefficient of \(x\). In this case, the period is \(P = \frac{2\pi}{|2\pi|} = 1\).

Step 3: Plot the function

To plot the function \(y=-3 \sin 2\pi x\), start by marking the period along the x-axis. The period for this function is 1, so mark this on the x-axis. Then find the midline (which is y=0 in this case), and plot points at the maximum, minimum, and both zero crossings (one at the start of the period and one at the end). The maximum point will be at (0.25,3), the zero crossings will be at (0,0) and (1,0), and the minimum point will be at (0.75,-3). Draw a smooth curve through these points to complete one period of the function.