Sletch the Graph Of y= 3sec(ax) 2X
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
![**Title:** Sketch the Graph of \( y = 3\sec(2x) \)
**Objective:**
Learn how to sketch the graph of the function \( y = 3\sec(2x) \).
**Step 1: Understanding the Function**
The function \( y = 3\sec(2x) \) is a transformation of the basic secant function \( y = \sec(x) \).
- **Amplitude:** The factor of 3 indicates a vertical stretch.
- **Period:** The period of the secant function is \( \pi \). The 2 inside the secant function affects the period. The new period is calculated as:
\[
\text{Period} = \frac{\pi}{2}
\]
**Step 2: Key Points and Asymptotes**
- The function has vertical asymptotes where the cosine function is equal to zero because the secant function is undefined wherever its reciprocal, cosine, equals zero.
**Key Points:** The key points for \( \sec(x) \) would be adjusted for \( \frac{\pi}{2} \).
**Step 3: Sketching the Graph**
1. **Determine Asymptotes:** For every half period, the secant function will have an asymptote. Calculate these points within one period and replicate them across the x-axis.
2. **Plot Critical Points:** The points where the cosine is at maximum or minimum.
3. **Draw the Secant Curves:** Based on the critical points and asymptotes, sketch the upward and downward branches of the secant curves.
**Conclusion:**
Sketching \( y = 3\sec(2x) \) involves understanding the effect of transformations on the basic \( \sec(x) \) graph. Key transformations are the vertical stretch by a factor of 3 and the horizontal compression by adjusting the period to \( \frac{\pi}{2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65dbacaf-b2d6-4ef7-85e7-1004f2132644%2Fd8ae9dde-384b-4e38-b3c0-5c5e1e998325%2Fx2c3ul7.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Sketch the Graph of \( y = 3\sec(2x) \)
**Objective:**
Learn how to sketch the graph of the function \( y = 3\sec(2x) \).
**Step 1: Understanding the Function**
The function \( y = 3\sec(2x) \) is a transformation of the basic secant function \( y = \sec(x) \).
- **Amplitude:** The factor of 3 indicates a vertical stretch.
- **Period:** The period of the secant function is \( \pi \). The 2 inside the secant function affects the period. The new period is calculated as:
\[
\text{Period} = \frac{\pi}{2}
\]
**Step 2: Key Points and Asymptotes**
- The function has vertical asymptotes where the cosine function is equal to zero because the secant function is undefined wherever its reciprocal, cosine, equals zero.
**Key Points:** The key points for \( \sec(x) \) would be adjusted for \( \frac{\pi}{2} \).
**Step 3: Sketching the Graph**
1. **Determine Asymptotes:** For every half period, the secant function will have an asymptote. Calculate these points within one period and replicate them across the x-axis.
2. **Plot Critical Points:** The points where the cosine is at maximum or minimum.
3. **Draw the Secant Curves:** Based on the critical points and asymptotes, sketch the upward and downward branches of the secant curves.
**Conclusion:**
Sketching \( y = 3\sec(2x) \) involves understanding the effect of transformations on the basic \( \sec(x) \) graph. Key transformations are the vertical stretch by a factor of 3 and the horizontal compression by adjusting the period to \( \frac{\pi}{2} \).
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