Problem 52 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \( y=3 \cos (2 \pi x + 4 \pi) \) is 3, the period is 1, and the phase shift is -2 (meaning shifted 2 units to the right). The graph of one period of this function is a cosine wave with these characteristics.

Step 1: Identify the Parameters from Equation

From the function \( y=3 \cos (2 \pi x + 4 \pi) \), we can identify that \( A = 3 \), \( B = 2 \pi \), and \( C = 4 \pi \). There is no \( D \) term, so the function does not shift up or down.

Step 2: Calculate the Amplitude

The amplitude \( A \) of the function is the absolute value of the coefficient of the cosine function. Hence, the amplitude of the given function is 3.

Step 3: Calculate the Period

The period of the function is given by the formula \( \frac{2 \pi}{B} \). For the given function, \( B = 2 \pi \), so the period is \( \frac{2 \pi}{2 \pi} = 1 \).

Step 4: Calculate the Phase Shift

The phase shift of the function is calculated by the formula \( -\frac{C}{B} \). For the given function, \( C = 4 \pi \) and \( B = 2 \pi \), so the phase shift is \( -\frac{4 \pi}{2 \pi} = -2 \). The negative sign means that the graph will be shifted 2 units to the right.

Step 5: Graph the Function

To graph the function, the following values are represented on a graph: amplitude (3 units measured from the x-axis due to \( A = 3 \)), period (1 unit measured from left to right, since the period is 1), and phase shift (2 units to the right, as the phase shift is -2). The function should complete one full cycle over the span of one period. For an upward-opening cosine function, start at the amplitude, dip down to cross the x-axis at the halfway point of the period, reach the amplitude at the end of the period, and then repeat this for the next periods, considering the phase shift. Don't forget to label the graph with x and y units, and a title (e.g., \( y=3 \cos (2 \pi x + 4 \pi) \)).