Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 3E
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Question
![### Analysis of Sine Curve Characteristics
#### Graph Analysis
The graph displays one complete period of a sine curve, represented in red. The x-axis and y-axis intersect at the origin.
Key Points on the Graph:
- The sine curve reaches its maximum value at \((0, 3)\).
- It crosses the x-axis at \(\left(-\frac{1}{4}, 0\right)\) and \(\left(\frac{3}{4}, 0\right)\).
- The sine curve reaches its minimum value at \(\left(\frac{1}{2}, -3\right)\).
- The y-values range from -3 (minimum) to 3 (maximum).
#### Task
Identify the amplitude, period, and horizontal shift of the given sine curve. Assume the absolute value of the horizontal shift is less than the period.
#### Given:
- **Amplitude**: 3 (This is already marked correct with a green check).
- **Period**: [Input field]
- **Horizontal shift**: [Input field]
#### Solution Explanation
To find the amplitude, period, and horizontal shift:
1. **Amplitude (A)**: The amplitude is the distance from the centerline (median value) of the sine curve to its peak. Here the amplitude is given as 3.
2. **Period (T)**: The period is the length of one complete cycle of the sine curve. By observation, since the curve completes one cycle from \(\left(-\frac{1}{4}, 0\right)\) to \(\left(\frac{3}{4}, 0\right)\), the period appears to be 1.
3. **Horizontal Shift (C)**: This is how far the entire graph is shifted horizontally from the usual sine curve. From the graph given, the curve appears to start at \(x = -\frac{1}{4}\), suggesting a horizontal shift.
Place appropriate values into the input fields for period and horizontal shift based on this analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedc716be-e066-4234-bd1a-1a86837575c3%2F24c3f640-3e4f-4b9f-a894-e545f4a4113f%2Fp93vn76_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Analysis of Sine Curve Characteristics
#### Graph Analysis
The graph displays one complete period of a sine curve, represented in red. The x-axis and y-axis intersect at the origin.
Key Points on the Graph:
- The sine curve reaches its maximum value at \((0, 3)\).
- It crosses the x-axis at \(\left(-\frac{1}{4}, 0\right)\) and \(\left(\frac{3}{4}, 0\right)\).
- The sine curve reaches its minimum value at \(\left(\frac{1}{2}, -3\right)\).
- The y-values range from -3 (minimum) to 3 (maximum).
#### Task
Identify the amplitude, period, and horizontal shift of the given sine curve. Assume the absolute value of the horizontal shift is less than the period.
#### Given:
- **Amplitude**: 3 (This is already marked correct with a green check).
- **Period**: [Input field]
- **Horizontal shift**: [Input field]
#### Solution Explanation
To find the amplitude, period, and horizontal shift:
1. **Amplitude (A)**: The amplitude is the distance from the centerline (median value) of the sine curve to its peak. Here the amplitude is given as 3.
2. **Period (T)**: The period is the length of one complete cycle of the sine curve. By observation, since the curve completes one cycle from \(\left(-\frac{1}{4}, 0\right)\) to \(\left(\frac{3}{4}, 0\right)\), the period appears to be 1.
3. **Horizontal Shift (C)**: This is how far the entire graph is shifted horizontally from the usual sine curve. From the graph given, the curve appears to start at \(x = -\frac{1}{4}\), suggesting a horizontal shift.
Place appropriate values into the input fields for period and horizontal shift based on this analysis.
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