3) Determine the period, horizontal shift, vertical shift, domain, range, and vertical asymptotes. The chart has the five key points of the basic function. Complete the chart with the corresponding 5 key points of the transformed functions. Draw your own axis and properly scale and label them (make sure it's BIG enough for me to see everything). Sketch one cycle of only the final graph (do not need the basic graph) (5pts) y=2csd(x+7)-1 period is domain is range is horizontal shift is vertical shift is vertical asymptotes are
3) Determine the period, horizontal shift, vertical shift, domain, range, and vertical asymptotes. The chart has the five key points of the basic function. Complete the chart with the corresponding 5 key points of the transformed functions. Draw your own axis and properly scale and label them (make sure it's BIG enough for me to see everything). Sketch one cycle of only the final graph (do not need the basic graph) (5pts) y=2csd(x+7)-1 period is domain is range is horizontal shift is vertical shift is vertical asymptotes are
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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Help please with Q.3
Transcribed Image Text:**Determining Transformations and Key Characteristics of Trigonometric Functions**
This educational exercise focuses on transforming the basic trigonometric functions and identifying important properties such as period, shifts, domain, range, and vertical asymptotes.
**Task:**
1. **Functions to Analyze:**
- \( y = 2 \csc(x + \pi) - 1 \)
2. **Determine the Following:**
- **Period**: The basic period of the cosecant function, modified by any horizontal transformations.
- **Horizontal Shift**: The shift along the x-axis due to phase changes.
- **Vertical Shift**: The vertical translation from adding or subtracting constants.
- **Domain**: The set of x-values for which the function is defined.
- **Range**: The set of possible y-values.
- **Vertical Asymptotes**: Values of x where the function is undefined, typically where the related sine function is zero.
3. **Chart of Key Points:**
- Base Function: \( y = \sin x \)
- Key Points: \((0,0), \left(\frac{\pi}{2}, 1\right), (\pi, 0), \left(\frac{3\pi}{2}, -1\right), (2\pi, 0)\)
- Transformations: Fill in the table for \( y = 2\sin(x + \pi) - 1 \) and \( y = 2\csc(x + \pi) - 1 \).
4. **Graphing:**
- Draw one cycle of the transformed function, labeling axes accordingly. Focus on clarity and scale for accurate representation.
This exercise helps in understanding how trigonometric graphs transform with changes in function parameters, vital for higher mathematics involving periodic functions.
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