Problem 35 - Algebra and Trigonometry

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\frac{1}{2}-\frac{1}{2} \cos \left(2 x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{2}\), Period: \(\pi\), Phase Shift: \(\frac{\pi}{6}\) right.

Step 1: Identify the Cosine Function Form

The given function is \( y = \frac{1}{2} - \frac{1}{2} \cos \left(2x - \frac{\pi}{3}\right) \). We need to rewrite it in the standard form of a cosine function, which is \( y = C + A \cos(Bx - D) \). Comparing, we find \( C = \frac{1}{2} \), \( A = -\frac{1}{2} \), \( B = 2 \), and \( D = \frac{\pi}{3} \).

Step 2: Calculate the Amplitude

The amplitude of a cosine function is the absolute value of \( A \). Here, \( A = -\frac{1}{2} \), so the amplitude is \( |-\frac{1}{2}| = \frac{1}{2} \).

Step 3: Calculate the Period

The period of a cosine function is given by \( \frac{2\pi}{|B|} \). For \( B = 2 \), the period becomes \( \frac{2\pi}{2} = \pi \).

Step 4: Determine the Phase Shift

The phase shift \( \phi \) is given by \( \frac{D}{B} \). So with \( D = \frac{\pi}{3} \) and \( B = 2 \), the phase shift is \( \frac{\frac{\pi}{3}}{2} = \frac{\pi}{6} \). Since \( D \) is positive, the phase shift is to the right.

Step 5: Graph One Complete Period

To graph one complete period, start at the phase shift \( x = \frac{\pi}{6} \) and extend to \( x = \frac{\pi}{6} + \pi = \frac{7\pi}{6} \). At \( x = \frac{\pi}{6} \), the cosine function starts at its maximum or minimum depending on the sign of \( A \). The function \( y = \frac{1}{2} - \frac{1}{2} \cos \left(2x - \frac{\pi}{3}\right) \) starts at its maximum of 1 and completes one cycle at \( x = \frac{7\pi}{6} \).

Key Concepts

Amplitude

The amplitude of a cosine function is one of the most straightforward concepts, yet it plays a crucial role in understanding how the function behaves. To put it simply, the amplitude is the measure of how much a wave oscillates above and below its central position. When looking at any cosine function in the form of \( y = A \cos(Bx - D) + C \), the amplitude is the absolute value of \( A \). This means you always make sure not to consider the sign of \( A \) but only its magnitude.

• In our function \( y = \frac{1}{2} - \frac{1}{2} \cos(2x - \frac{\pi}{3}) \), we found \( A = -\frac{1}{2} \).
• The amplitude is \(|A| = | -\frac{1}{2} | = \frac{1}{2} \).

This amplitude tells us that the graph of the cosine function will reach 0.5 units above and below the midline \( y = \frac{1}{2} \). In essence, amplitude is all about the height of the waves from the centerline.

Period

The period of a cosine function is an exciting element that controls how fast the function repeats its pattern. The period determines the horizontal length it takes for the cosine wave to start repeating itself. For the standard cosine function \( y = A\cos(Bx - D) + C \), the period can be found using the formula:\[ \text{Period} = \frac{2\pi}{|B|} \]Here's how we calculate it for our function:

• In \( y = \frac{1}{2} - \frac{1}{2} \cos(2x - \frac{\pi}{3}) \), \( B = 2 \).
• Thus, the period is \( \frac{2\pi}{2} = \pi \).

This period \( \pi \) indicates how frequently the graph completes one full cycle. When the period is shorter, the cycles repeat more quickly, leading to more oscillations over the same length of the x-axis. For our example, after every \( \pi \) units along the x-axis, the cosine function returns to its starting point and begins a new cycle.

Phase Shift

Phase shift is a concept that depicts how much a graph has been horizontally shifted from its usual position. A positive phase shift moves the graph to the right, while a negative one shifts it to the left. Here's how phase shift is calculated:In the function \( y = A\cos(Bx - D) + C \), the phase shift \( \phi \) is determined by the formula:\[ \text{Phase Shift} = \frac{D}{B} \]Applying this to our function:

• With \( D = \frac{\pi}{3} \) and \( B = 2 \), the phase shift is \( \frac{\frac{\pi}{3}}{2} = \frac{\pi}{6} \).
• This positive result tells us that the graph of the function moves \( \frac{\pi}{6} \) units to the right.

Understanding phase shift helps predict where the wave starts on the x-axis. It essentially tells us that the graph is no longer centered at the origin but instead begins its pattern a bit to the right for our equation.