Problem 106 - Algebra and Trigonometry

Question

Determine the amplitude, period, and phase shift of \(y=-2 \cos \left(2 x-\frac{\pi}{2}\right) .\) Then graph one period of the function. (Section 5.5, Example 6)

Step-by-Step Solution

Answer

The amplitude is 2, the period is \(\pi\), and the phase shift is \(\frac{\pi}{4}\). The graph starts at \(\frac{\pi}{4}\), reaches 2 at \(\frac{\pi}{4} + \frac{\pi}{8}\), returns to 0 at \(\frac{\pi}{4} + \frac{\pi}{4}\), goes to -2 at \(\frac{\pi}{4} + \frac{3\pi}{8}\), and then returns to 0 at \(\frac{\pi}{4} + \frac{\pi}{2}\), which is one full period of the graph.

Step 1: Identify the amplitude

The amplitude is the absolute value of the coefficient of the cosine function. In this case, A = |-2| = 2.

Step 2: Identify the period

The period can be determined using the formula \(T = \frac{2\pi}{B}\), where B is the coefficient of x in the argument of the cosine function. In this case, B = 2, so the period T = \(\frac{2\pi}{2}\) = \(\pi\).

Step 3: Identify the phase shift

The phase shift can be determined by the term -C in the argument of the cosine function. Here, the shift is given by \(2x - \frac{\pi}{2}\). Solving \(2x - \frac{\pi}{2} = 0\) for x gives the phase shift as \(\frac{\pi}{4}\).

Step 4: Plot the function

Plot the graph using the identified amplitude, period and phase shift. Start the graph at the phase shift, let it reach the amplitude at a quarter of the period, come back to the x-axis at half of the period, reach the negative of the amplitude at three quarters of the period, and then come back to the x-axis at the end of the period. Note that the function is negated, meaning the graph is a reflection about the x-axis compared to a usual cosine function.

Key Concepts

Amplitude

The amplitude of a trigonometric function refers to how much it varies from its central position, often the middle of its oscillation.
For the cosine function, the amplitude is represented as the maximum vertical distance from the center of the wave to its peak.

Specifically, the amplitude is determined by the absolute value of the coefficient in front of the cosine term. In our given function, which is \(y = -2 \cos (2x - \frac{\pi}{2})\), the coefficient is -2.

To find the amplitude, we take the absolute value:

• Coefficient = -2
• Amplitude = |-2| = 2

This tells us that the waves of our cosine function reach 2 units above and below its middle line.

Period

The period of a trigonometric function describes how long it takes for the function to complete one full cycle before it starts repeating.
When looking at the cosine function, this is influenced by the coefficient of the variable inside the cosine, next to the \(x\).

The formula to calculate the period (\(T\)) for a cosine function is:\[T = \frac{2\pi}{B}\]where \(B\) is the coefficient of \(x\).

For our function, \(y = -2 \cos (2x - \frac{\pi}{2})\), the coefficient \(B\) is 2, so:

• \(T = \frac{2\pi}{2} = \pi\)

This means the graph of the function will complete its cycle every \(\pi\) units of \(x\). Understanding the period is crucial for knowing how frequently the wave pattern repeats.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric function from its usual position. It shows how far along the \(x\)-axis the wave begins.To find the phase shift in a function like \(y = -2 \cos (2x - \frac{\pi}{2})\), you solve the inside function for zero:\(2x - \frac{\pi}{2} = 0\).Solving for \(x\), you add \(\frac{\pi}{2}\) to both sides:

• \(2x = \frac{\pi}{2}\)
• \(x = \frac{\pi}{4}\)

This result, \(\frac{\pi}{4}\), is the phase shift, meaning the entire wave has been shifted to the right by \(\frac{\pi}{4}\) units. By understanding phase shifts, you can accurately determine where the cosine wave starts along the \(x\)-axis.

Cosine Function

The cosine function is a fundamental part of trigonometry, often used to model periodic behavior like sound waves, tides, and more.
The basic form is \(y = A \cos(Bx + C) + D\).
Each letter represents:

• \(A\) for amplitude, dictating wave height
• \(B\) affecting the period
• \(C\) resulting in a phase shift
• \(D\) moving the graph vertically up or down

For the function \(y = -2 \cos (2x - \frac{\pi}{2})\),

• Amplitude \(= 2\)
• Period \(= \pi\)
• Phase shift \(= \frac{\pi}{4}\)

Notice the negative sign in front, which flips the graph across the \(x\)-axis, making it reflect the usual cosine pattern. This classic function is essential in understanding wave-related concepts and mathematical modeling.