Problem 28 - Algebra and Trigonometry

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \sin (2 \pi x+4)$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=3 \sin (2 \pi x+4)\) is 3, the period is 1, and the phase shift is approximately -0.64 or -64% of the period.

Step 1: Determine the Amplitude

The amplitude of the function is the absolute value of the coefficient A of the sine function. In this case, given the function \(y=3 \sin (2 \pi x+4)\), the amplitude is \(|3|\) which equals 3.

Step 2: Determine the Period

The period of the function is the reciprocal of the coefficient B in front of x inside the sine function. For a sine function, the period is usually \(2pi\) divided by the absolute value of B. Thus, for the given function \(y=3 \sin (2 \pi x+4)\), the period is \(2\pi / |2\pi|\) which is 1.

Step 3: Determine the Phase Shift

The phase shift of the function is the value of C divided by B, then flip the sign. Given the function \(y=3 \sin (2 \pi x+4)\), the phase shift is \(-4 / (2\pi)\), which equals approximately -0.64 or -64% of the period.

Step 4: Graph the Function

The graph of one period of the function can now be sketched based on the amplitude, period, and phase shift. Start by drawing a sine wave that goes up to 3 and down to -3 (amplitude), over a distance of 1 (period). Then shift the wave to the right by the phase shift of approximately -0.64. It's recommended to use a graphing tool for a precise representation.