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Graph of one complete cycle of the equation y = -2- cos(2x- x). Identify the period, amplitude and phase shift for this equation.

Graph of one complete cycle of the equation y = -2- cos(2x- x). Identify the period, amplitude and phase shift for this equation.

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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### Graph of one complete cycle of the equation \( y = -2 - \cos(2x - \pi) \). --- **Identify the period, amplitude and phase shift for this equation.** --- #### Explanation of the Graph: The graph shown is a Cartesian plane with the x-axis and y-axis marked. There are no specific plots or curves visible, but it is mentioned that it relates to the trigonometric equation \( y = -2 - \cos(2x - \pi) \). #### Understanding the Given Equation: 1. **Equation:** \( y = -2 - \cos(2x - \pi) \) 2. **Cosine Function Analysis:** - The equation is in the form \( y = A + B \cdot \cos(Cx + D) \), where: - \( A = -2 \) - \( B = -1 \) - \( C = 2 \) - \( D = -\pi \) 3. **Amplitude:** - Amplitude \( = |B| \) - Here, the amplitude \( = |-1| = 1 \). 4. **Period:** - The period \( T \) of the function \( \cos(Cx) \) is given by \( \frac{2\pi}{C} \). - In this case, \( C = 2 \). - Therefore, \( T = \frac{2\pi}{2} = \pi \). 5. **Phase Shift:** - The phase shift \( \phi \) is calculated by solving \( Cx + D = 0 \). - Here, \( 2x - \pi = 0 \implies x = \frac{\pi}{2} \). - Therefore, the phase shift is \( \frac{\pi}{2} \). #### Conclusion: - **Amplitude:** 1 - **Period:** \( \pi \) - **Phase Shift:** \( \frac{\pi}{2} \) The graph would typically plot one full cycle of the cosine function, considering the period and phase shift calculated above. The vertical shift is down 2 units due to the -2 term in the equation.
Transcribed Image Text:### Graph of one complete cycle of the equation \( y = -2 - \cos(2x - \pi) \). --- **Identify the period, amplitude and phase shift for this equation.** --- #### Explanation of the Graph: The graph shown is a Cartesian plane with the x-axis and y-axis marked. There are no specific plots or curves visible, but it is mentioned that it relates to the trigonometric equation \( y = -2 - \cos(2x - \pi) \). #### Understanding the Given Equation: 1. **Equation:** \( y = -2 - \cos(2x - \pi) \) 2. **Cosine Function Analysis:** - The equation is in the form \( y = A + B \cdot \cos(Cx + D) \), where: - \( A = -2 \) - \( B = -1 \) - \( C = 2 \) - \( D = -\pi \) 3. **Amplitude:** - Amplitude \( = |B| \) - Here, the amplitude \( = |-1| = 1 \). 4. **Period:** - The period \( T \) of the function \( \cos(Cx) \) is given by \( \frac{2\pi}{C} \). - In this case, \( C = 2 \). - Therefore, \( T = \frac{2\pi}{2} = \pi \). 5. **Phase Shift:** - The phase shift \( \phi \) is calculated by solving \( Cx + D = 0 \). - Here, \( 2x - \pi = 0 \implies x = \frac{\pi}{2} \). - Therefore, the phase shift is \( \frac{\pi}{2} \). #### Conclusion: - **Amplitude:** 1 - **Period:** \( \pi \) - **Phase Shift:** \( \frac{\pi}{2} \) The graph would typically plot one full cycle of the cosine function, considering the period and phase shift calculated above. The vertical shift is down 2 units due to the -2 term in the equation.
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