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This image shows a graph with an oscillating wave pattern, likely representing a sinusoidal function. Here’s a detailed description of the graph: - **Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). - Both axes have an arrow indicating positive directions. - **Scales and Units:** - The \( x \)-axis is marked with values 0 and 1, indicating units for measurement. - The \( y \)-axis is marked with values 0 and 1, also for measurement purposes. - **Wave Pattern:** - The wave starts slightly above the origin, peaks before reaching \( x = 1 \), crosses the \( x \)-axis, dips below it, and then rises back up forming two oscillations. - The pattern is symmetric around the \( y \)-axis. This kind of graph is typical for demonstrating properties of sine or cosine functions, such as wavelength, amplitude, and frequency. ### Key Concepts for Trigonometric Functions - **Amplitude:** - The height from the centerline to the peak of the curve (positive or negative). - **Formula:** Amplitude = A - **Period:** - The length of one complete cycle of the curve. - **Formula:** Period = B - **Phase Shift:** - The horizontal shift of the curve along the x-axis. - **Vertical Shift:** - The movement of the curve up or down on the y-axis. - **Equation for Each Curve in Terms of Cosine:** - The general form of a cosine function is: \[ y = A \cos(B(x - C)) + D \] - Where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. Use these parameters to analyze and graph trigonometric functions effectively.

This image shows a graph with an oscillating wave pattern, likely representing a sinusoidal function. Here’s a detailed description of the graph: - **Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). - Both axes have an arrow indicating positive directions. - **Scales and Units:** - The \( x \)-axis is marked with values 0 and 1, indicating units for measurement. - The \( y \)-axis is marked with values 0 and 1, also for measurement purposes. - **Wave Pattern:** - The wave starts slightly above the origin, peaks before reaching \( x = 1 \), crosses the \( x \)-axis, dips below it, and then rises back up forming two oscillations. - The pattern is symmetric around the \( y \)-axis. This kind of graph is typical for demonstrating properties of sine or cosine functions, such as wavelength, amplitude, and frequency. ### Key Concepts for Trigonometric Functions - **Amplitude:** - The height from the centerline to the peak of the curve (positive or negative). - **Formula:** Amplitude = A - **Period:** - The length of one complete cycle of the curve. - **Formula:** Period = B - **Phase Shift:** - The horizontal shift of the curve along the x-axis. - **Vertical Shift:** - The movement of the curve up or down on the y-axis. - **Equation for Each Curve in Terms of Cosine:** - The general form of a cosine function is: \[ y = A \cos(B(x - C)) + D \] - Where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. Use these parameters to analyze and graph trigonometric functions effectively.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This image shows a graph with an oscillating wave pattern, likely representing a sinusoidal function. Here’s a detailed description of the graph: - **Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). - Both axes have an arrow indicating positive directions. - **Scales and Units:** - The \( x \)-axis is marked with values 0 and 1, indicating units for measurement. - The \( y \)-axis is marked with values 0 and 1, also for measurement purposes. - **Wave Pattern:** - The wave starts slightly above the origin, peaks before reaching \( x = 1 \), crosses the \( x \)-axis, dips below it, and then rises back up forming two oscillations. - The pattern is symmetric around the \( y \)-axis. This kind of graph is typical for demonstrating properties of sine or cosine functions, such as wavelength, amplitude, and frequency.
Transcribed Image Text:This image shows a graph with an oscillating wave pattern, likely representing a sinusoidal function. Here’s a detailed description of the graph: - **Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). - Both axes have an arrow indicating positive directions. - **Scales and Units:** - The \( x \)-axis is marked with values 0 and 1, indicating units for measurement. - The \( y \)-axis is marked with values 0 and 1, also for measurement purposes. - **Wave Pattern:** - The wave starts slightly above the origin, peaks before reaching \( x = 1 \), crosses the \( x \)-axis, dips below it, and then rises back up forming two oscillations. - The pattern is symmetric around the \( y \)-axis. This kind of graph is typical for demonstrating properties of sine or cosine functions, such as wavelength, amplitude, and frequency.
### Key Concepts for Trigonometric Functions - **Amplitude:** - The height from the centerline to the peak of the curve (positive or negative). - **Formula:** Amplitude = A - **Period:** - The length of one complete cycle of the curve. - **Formula:** Period = B - **Phase Shift:** - The horizontal shift of the curve along the x-axis. - **Vertical Shift:** - The movement of the curve up or down on the y-axis. - **Equation for Each Curve in Terms of Cosine:** - The general form of a cosine function is: \[ y = A \cos(B(x - C)) + D \] - Where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. Use these parameters to analyze and graph trigonometric functions effectively.
Transcribed Image Text:### Key Concepts for Trigonometric Functions - **Amplitude:** - The height from the centerline to the peak of the curve (positive or negative). - **Formula:** Amplitude = A - **Period:** - The length of one complete cycle of the curve. - **Formula:** Period = B - **Phase Shift:** - The horizontal shift of the curve along the x-axis. - **Vertical Shift:** - The movement of the curve up or down on the y-axis. - **Equation for Each Curve in Terms of Cosine:** - The general form of a cosine function is: \[ y = A \cos(B(x - C)) + D \] - Where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. Use these parameters to analyze and graph trigonometric functions effectively.
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