Question

Find the amplitude (if one exists), period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods. $$ y=-2 \cos \left(2 x-\frac{\pi}{2}\right) $$

Short answer

Amplitude: 2, Period: \(\pi\), Phase Shift: \(\frac{\pi}{4}\) units to the right.

Step-by-step solution

Identify the Amplitude

The amplitude of a function of the form \[ y = A \, \cos{(B x - C)} \ \, \text{or}\ \, y = A \, \sin{(B x - C)} \] is given by the absolute value of the coefficient in front of the cosine or sine function. For the given function \[ y = -2 \, \cos{(2 x - \frac{\pi}{2})} \] the amplitude is \[ \left| -2 \right| = 2. \]

Determine the Period

The period of a function of the form \[ y = A \, \cos{(B x - C)} \ \, \text{or}\ \, y = A \, \sin{(B x - C)} \] is given by \[ \frac{2\pi}{\left| B \right|}. \] For the given function \[ y = -2 \, \cos{(2 x - \frac{\pi}{2})} \] \[ B = 2, \ \, \text{so the period is}\ \frac{2\pi}{2} = \pi. \]

Find the Phase Shift

The phase shift of a function of the form \[ y = A \, \cos{(B x - C)} \ \, \text{or}\ \, y = A \, \sin{(B x - C)} \] is given by \[ \frac{C}{B}. \] For the given function \[ y = -2 \, \cos{(2 x - \frac{\pi}{2})} \] \[ C = \frac{\pi}{2}, \ \, \text{so the phase shift is} \ \, \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4} \, \, \text{units to the right.} \]

Sketch the Graph

To sketch the graph, follow these key steps:1. Since the amplitude is 2, the maximum and minimum values of the function will be 2 and -2, respectively.2. The period is \(\pi\), so the graph will repeat every \(\pi\) units.3. The phase shift is \(\frac{\pi}{4}\) units to the right, so start the cosine wave at \(x = \frac{\pi}{4}\).4. Label key points: one at each maximum, minimum, and x-intercept within two periods starting from \(\frac{\pi}{4}\).5. Reflect the graph of the cosine function because the coefficient is negative.The resulting graph will show a wave starting at \(\frac{\pi}{4}\) with peaks at 2 and troughs at -2, repeating every \(\pi\) units.

Key concepts

Amplitude of Trigonometric Functions

Amplitude refers to the height of a wave from the centerline to the peak (or trough). For functions of the form \( y = A \cos(Bx - C) \) or \( y = A \sin(Bx - C) \), the amplitude is given by the absolute value of A. In the given example, the function is \( y = -2 \cos(2x - \frac{\pi}{2}) \). Here, A is -2, so the amplitude is \( | -2 | = 2 \). This means that the graph will reach a maximum height of 2 and a minimum height of -2. Amplitude tells us how much the wave moves above and below the center line.

Period of Trigonometric Functions

The period of a trigonometric function represents the length of one complete cycle of the wave. For functions of the form \( y = A \cos(Bx - C) \) or \( y = A \sin(Bx - C) \), the period is calculated using the formula \( \frac{2\pi}{|B|} \). In our example, the function \( y = -2 \cos(2x - \frac{\pi}{2}) \) has a B value of 2. Plugging into our formula, the period is \( \frac{2\pi}{2} = \pi \). This tells us that the graph repeats every \( \pi \) units. Understanding the period helps you to anticipate the repeating pattern of the wave.

Phase Shift of Trigonometric Functions

Phase shift determines the horizontal displacement of the wave. For functions of the form \( y = A \cos(Bx - C) \) or \( y = A \sin(Bx - C) \), the phase shift is given by \( \frac{C}{B} \). For the given function \( y = -2 \cos(2x - \frac{\pi}{2}) \), the value of C is \( \frac{\pi}{2}\). With B being 2, we calculate the phase shift as \( \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4} \) units. Since this value is positive, it indicates a rightward shift. The phase shift helps us to know where our wave starts on the x-axis.

Graphing Techniques in Trigonometry

Graphing trigonometric functions involves plotting key points based on amplitude, period, and phase shift. Here’s how you can graph \( y = -2 \cos(2x - \frac{\pi}{2}) \):

• **Amplitude**: The graph ranges between 2 and -2.
• **Period**: The wave repeats every \( \pi \) units.
• **Phase Shift**: Start the wave at \( x = \frac{\pi}{4} \).
• **Key Points**: Identify maximum, minimum, and x-intercept points within two periods.
• **Reflection**: Because the coefficient is negative, reflect the wave.

With these steps, you can accurately graph trigonometric functions and understand their behavior.