Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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Transcribed Image Text:**Problem 2: Graphing a Sine Function**
Graph the function \( y = 1 + 2\sin\left(\frac{1}{2}x - \pi\right) \) over the interval \( 0 \leq x \leq 6\pi \).
**Function Analysis:**
1. **Transformation Basics:**
- The function is a transformed sine wave.
- The amplitude of the standard sine function is modified by a factor of 2, which means the graph will vertically stretch by 2.
- There is a vertical shift up by 1 unit, which affects the midline of the graph.
- The phase shift is determined by the \(- \pi\) inside the sine function, translating the graph to the right by \(\pi\) units.
- The horizontal dilation is caused by the \(\frac{1}{2}\) within the function, which increases the period of the sine wave.
2. **Period and Phase:**
- The standard period of \( \sin(x) \) is \( 2\pi \).
- For \( \frac{1}{2}x \), the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).
- Thus, every cycle of the sine wave completes over \( 4\pi \).
3. **Graph Details:**
- Start plotting at \( x = \pi \) due to the phase shift.
- The maximum point of the graph will be at \( y = 3 \) (1 + amplitude of 2), and the minimum point will be at \( y = -1 \) (1 - amplitude of 2).
- The midline of the wave is at \( y = 1 \).
These details will help in understanding the graphical representation of the given function over the specified interval.
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