Graph the function. 10) y = sec (2x - )+ 2 A) B) 3n x 2 C) D) Зл х 2.
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°.
Related questions
Question
![### Graphing Trigonometric Functions
#### Problem
Graph the function:
\[ y = \sec \left(2x - \frac{\pi}{4}\right) + 2 \]
The task is to identify the correct graph of the given secant function from the four options labeled A, B, C, and D.
#### Options
**A)**
- The graph features upward-facing vertical parabolas.
- Vertical asymptotes are located at \( x = -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4} \).
- The graph appears to be vertically shifted upwards by 2 units.
**B)**
- The graph features downward-facing vertical parabolas.
- Vertical asymptotes are located at \( x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4} \).
- The graph appears shifted to the right and also upwards by 2 units.
**C)**
- The graph features upward-facing vertical parabolas.
- Vertical asymptotes are located at multiples of \( \frac{\pi}{4} \).
- No obvious vertical shift is present.
**D)**
- The graph features both upward and downward vertical parabolas.
- Vertical asymptotes’ locations suggest a different phase shift.
- There’s a significant downward shift observed.
#### Analysis
Secant functions, \( y = \sec k(x - d) + c \), have properties such as vertical repeats (period) and asymptotes influenced by phase shifts (\( d \)) and vertical shifts (\( c \)).
#### Conclusion
To graph \( y = \sec \left(2x - \frac{\pi}{4}\right) + 2 \):
1. **Phase Shift:** The horizontal shift is \( \frac{\pi}{4} \) units to the right.
2. **Vertical Shift:** The graph is shifted 2 units upwards.
3. **Period:** Since the coefficient of \( x \) is 2, the period is \( \pi \).
Based on these observations, identify which graph reflects the correct transformation of the secant function according to these shifts and properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8492a80-fb1a-4bc0-8e68-f5fde27cb209%2F04fb168d-f51a-402b-ab75-6ac703a54718%2Fe3ydc4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing Trigonometric Functions
#### Problem
Graph the function:
\[ y = \sec \left(2x - \frac{\pi}{4}\right) + 2 \]
The task is to identify the correct graph of the given secant function from the four options labeled A, B, C, and D.
#### Options
**A)**
- The graph features upward-facing vertical parabolas.
- Vertical asymptotes are located at \( x = -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4} \).
- The graph appears to be vertically shifted upwards by 2 units.
**B)**
- The graph features downward-facing vertical parabolas.
- Vertical asymptotes are located at \( x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4} \).
- The graph appears shifted to the right and also upwards by 2 units.
**C)**
- The graph features upward-facing vertical parabolas.
- Vertical asymptotes are located at multiples of \( \frac{\pi}{4} \).
- No obvious vertical shift is present.
**D)**
- The graph features both upward and downward vertical parabolas.
- Vertical asymptotes’ locations suggest a different phase shift.
- There’s a significant downward shift observed.
#### Analysis
Secant functions, \( y = \sec k(x - d) + c \), have properties such as vertical repeats (period) and asymptotes influenced by phase shifts (\( d \)) and vertical shifts (\( c \)).
#### Conclusion
To graph \( y = \sec \left(2x - \frac{\pi}{4}\right) + 2 \):
1. **Phase Shift:** The horizontal shift is \( \frac{\pi}{4} \) units to the right.
2. **Vertical Shift:** The graph is shifted 2 units upwards.
3. **Period:** Since the coefficient of \( x \) is 2, the period is \( \pi \).
Based on these observations, identify which graph reflects the correct transformation of the secant function according to these shifts and properties.
Expert Solution
Step 1
graph the function
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