Problem 30 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The amplitude of the function is 3, the period is 1, and the phase shift is -2.

Step 1: Identify the amplitude

The amplitude of a sine function is the absolute value of the coefficient of the sine term. In this case, the function is \(y=-3 \sin (2 \pi x+4 \pi)\), therefore, the amplitude (A) would be \(|-3|\), which is 3.

Step 2: Identify the period

The period of the sine function can be grabbed by equating \(2\pi/B\) to the period. Here, B is the coefficient of x, which is \(2\pi\). Thus, the period is \(2\pi /(2\pi) = 1\).

Step 3: Identify the phase shift

The phase shift may be determined by seeing what value of x would cause the argument of the sine function to be zero. Thus, we set \(2\pi x+4\pi = 0\) and solve for x. The solution, x = -2, is the phase shift (C).

Step 4: Plotting the function

Plotting a function consists of a sine wave with an amplitude of 3 (peaks and troughs will reach 3 and -3 respectively), a period of 1 (the pattern will repeat every 1 unit along the x-axis), and a phase shift of -2 (the entire graph will be shifted 2 units in the negative x direction).