Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The amplitude of the function is 4, the period is \(\pi\), and there is a phase shift of \(\pi/2\) to the right.

Step 1: Determine the Amplitude

The amplitude is the coefficient in front of the cosine function. For the function \(y=4 \cos (2 x-\pi)\), the amplitude is 4.

Step 2: Determine the Period

The period, usually represented as \(2\pi\) in the standard cosine function, is affected by the inner coefficient. For the function \(y=4 \cos (2 x-\pi)\), the inner coefficient is 2. The period is therefore \(\pi\). This is calculated by dividing \(2\pi\) by the absolute value of the inner coefficient; therefore \(2\pi/|2| = \pi\).

Step 3: Determine the Phase Shift

The phase shift is the horizontal movement of the function from the origin. In the given function \(y=4 \cos (2 x-\pi)\), the term -\(\pi\) causes a phase shift. The phase shift can be calculated by dividing the term by the negative of the inner coefficient (2); therefore \(-\pi/-2 = \pi/2\). This means there's a phase shift of \(\pi/2\) to the right.

Step 4: Draw the Graph

To create one complete period of the function, start by sketching a horizontal axis. Mark a point halfway along the X-axis that will represent the phase shift of \(\pi/2\). Then mark points to indicate a complete period (\(\pi\)) from the phase shift. Draw a cosine curve that peaks at 4 (the amplitude) and falls to -4, showing one complete wave cycle from maximum to minimum and back to maximum. The graph rises and falls around the X-axis accordingly.