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Magazine Chapter 5: Problem 34
Determine the amplitude and phase shift for each function, and sketch at least one cycle of the graph. Label five points as done in the examples. $$y=3 \cos (x+2 \pi / 3)-2$$
Short Answer
Expert verified
Amplitude: 3, Phase shift: \(-\frac{2\pi}{3}\) left
Step by step solution
01
- Identify the General Form
The general form of a cosine function is given by \[ y = A \cos(Bx + C) + D \] where: - \(A\) is the amplitude - \(\frac{2\pi}{B}\) is the period - \( - \frac{C}{B} \) is the phase shift - \(D\) is the vertical shift
02
- Identify the Amplitude
From the equation \( y = 3 \cos (x + \frac{2\pi}{3}) - 2 \), we see that \(A = 3\). Therefore, the amplitude is 3.
03
- Identify the Phase Shift
The phase shift is given by \( -\frac{C}{B} \). In the equation, \(C = \frac{2\pi}{3}\) and \(B = 1\). Thus, the phase shift is \( -\frac{\frac{2\pi}{3}}{1} = -\frac{2\pi}{3}\) units to the left.
04
- Identify the Vertical Shift
The vertical shift is determined by the value of \( D \). Here, \(D = -2\). Therefore, the graph is shifted downward by 2 units.
05
- Graphing the Function
To sketch one cycle of the graph: 1. Start with plotting the cosine function \( \cos(x) \) and adjust it for amplitude. 2. Shift it horizontally by \(-\frac{2\pi}{3}\) units.3. Lastly, shift it downward by 2 units. Label five key points on the graph (e.g., starting from the phase shift point, incrementing by the period divided by 4).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude
The amplitude of a trigonometric function describes the height of the wave from its central axis to its peak. For the cosine function, the general form is given by \[ y = A \, \text{cos} (Bx + C) + D \]Here, the amplitude is denoted by the absolute value of \( A \). In the provided exercise, the equation is \( y = 3 \, \text{cos} (x + \frac{2\pi}{3}) - 2 \). Thus, the amplitude is \( |3| = 3 \). This means the wave reaches 3 units above and below its central axis. Understanding this helps in accurately stretching or compressing the graph vertically.
Key points to remember about amplitude:
Key points to remember about amplitude:
- Height of the wave from the center line to the peak or trough
- Given by the absolute value of the coefficient A in front of the cosine function
Phase Shift
Phase shift refers to the horizontal displacement of the graph of a trigonometric function. In cosine functions, the phase shift is calculated by the formula: \[\text{Phase Shift} = -\frac{C}{B} \]In the given problem, \( C \) is \( \frac{2\pi}{3} \) and \( B \) is 1. Plugging these into the formula, we get: \( \text{Phase Shift} = -\frac{\frac{2\pi}{3}}{1} = -\frac{2\pi}{3} \). This means the graph of the function shifts \( \frac{2\pi}{3} \) units to the left. Understanding phase shifts helps determine where the wave starts on the horizontal axis.
Key points to remember about phase shift:
Key points to remember about phase shift:
- Horizontal displacement along the x-axis
- Calculated from \( -\frac{C}{B} \)
Cosine Function
The cosine function, written as \( \text{cos}(x) \), is one of the fundamental trigonometric functions. It is periodic and oscillates between -1 and 1. The general form of a cosine function includes amplitude, phase shift, and vertical shift: \[ y = A \, \text{cos} (Bx + C) + D \]In our example, the function is \( y = 3 \, \text{cos} (x + \frac{2\pi}{3}) - 2 \). This implies:
- Amplitude of 3
- Phase shift of \( \frac{2\pi}{3} \) units to the left
- Vertical shift downward by 2 units
Graphing Trigonometric Functions
Graphing trigonometric functions involves understanding and applying transformations to the basic function. For the cosine function \( y = 3 \, \text{cos} (x + \frac{2\pi}{3}) - 2 \) :
Label five key points starting from the phase shift point. These usually occur at \( 0, \frac{\text{Period}}{4}, \frac{\text{Period}}{2}, \frac{3\text{Period}}{4}, \text{and Period} \).
- Start with the basic cosine wave. This oscillates between -1 and 1.
- Apply the amplitude. Multiply the cosine values by 3 to stretch the wave vertically.
- Shift the wave \( \frac{2\pi}{3} \) units to the left for the phase shift.
- Finally, shift the entire graph 2 units down due to the vertical shift.
Label five key points starting from the phase shift point. These usually occur at \( 0, \frac{\text{Period}}{4}, \frac{\text{Period}}{2}, \frac{3\text{Period}}{4}, \text{and Period} \).
Vertical Shift
Vertical shift refers to moving the entire graph of the function up or down along the y-axis. This is determined by the value of \( D \) in the general cosine function: \[ y = A \, \text{cos} (Bx + C) + D \]In our example, \( D \) is -2. Hence, the entire graph of \( y = 3 \, \text{cos} (x + \frac{2\pi}{3}) - 2 \) is shifted downward by 2 units.
Key points to remember for vertical shift:
Key points to remember for vertical shift:
- Moves the wave up or down along the y-axis
- Determined by the constant \( D \) in the function
