Problem 11 - Precalculus Essentials

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=4 \sin \pi x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=4 \sin \pi x\) is 4, while its period is 2. The graph of one period of this function is a sine curve starting at the origin (0,0), rising to the maximum value at (0.5,4), descending back down to (1,0), reaching a minimum at (1.5,-4), and returning back to the x-axis at (2,0).

Step 1: Identify the Amplitude

The amplitude is defined as the absolute value of the coefficient of the sine function. In this case the function is \(y=4 \sin \pi x\) so the amplitude is \(|4|\) which equals 4.

Step 2: Identify the Period

The period of a function is the horizontal length of one complete cycle. For any sine function, the default period is \(2\pi\). However, inside the sine function, we have \(\pi x\). This value of \(\pi\) is essentially the scaling factor or 'frequency' of the function. So the period of this function is going to be \(\frac{2\pi}{|\pi|}\) which equals 2.

Step 3: Graphing the Function

When graphing the function, begin by identifying key values from the function. The amplitude is 4, which tells us the maximum and minimum value of the function (4 and -4). The period is 2, which tells us the distance over which the function completes one cycle. Start at x=0 and rise to the maximum value at x=0.5, descend back down to zero at x=1, continue to the minimum value at x=1.5, and finally back up to zero at x=2. Repeat this pattern for more cycles.