Problem 27 - Algebra and Trigonometry

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \sin (\pi x+2)$$

Step-by-Step Solution

Answer

The amplitude is 3, the period is 2, and the phase shift is \(-2/\pi\).

Step 1: Determine the Amplitude

The amplitude is the coefficient of the sine function. In this case, the amplitude is 3. Hence, the amplitude is 3.

Step 2: Determine the Period

The period can be obtained with the formula \(2\pi / \omega\) where \(\omega\) is the coefficient of x inside the sine function, which in this case is \(\pi\). Hence, subsituting \(\pi\) into the formula we have \(2\pi / \pi\) which simplifies to 2. Thus, the period is 2.

Step 3: Determine the phase shift

The phase shift is determined by the constant being added or subtracted inside the function. Here, the constant is 2, so the phase shift is \(-2/ \pi\).

Step 4: Graphing the function

To graph one period of the function, plot the sine curve starting from the phase shift, with the maximum and minimum values being equal to the amplitude and -amplitude, respectively. The period determines the length along the x-axis that the graph repeats.