(b) State the period for the graph. T = (c) State the vertical translation for the graph. k = (d) State the phase shift for the graph. h =
(b) State the period for the graph. T = (c) State the vertical translation for the graph. k = (d) State the phase shift for the graph. h =
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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(b) State the period for the graph.
T =
(c) State the vertical translation for the graph.
k =
(d) State the phase shift for the graph.
h =
![### Transcription for Educational Use
#### Graphing a Trigonometric Function
(a) **Graph one complete cycle for the following function:**
\[
y = -4 - 3 \sec\left(3\pi x + \frac{\pi}{4}\right)
\]
**Explanation of Graphs:**
You will see two separate displays:
1. **Left Graph:**
- The graph is labeled with axes marked as \(x\) on the horizontal axis and \(y\) on the vertical axis.
- The function shows multiple curves with values extending between \(y = -15\) and \(y = 15\).
- The graph illustrates several upward and downward branches typical of a secant function (sec), with vertical asymptotes where the secant function approaches infinity. This behavior occurs periodically between \(x \approx 0.1\) and \(x = 0.7\).
2. **Right Graph:**
- Similarly structured with \(x\) and \(y\) axes over the same ranges as the left graph.
- The plot is consistent with the left graph, demonstrating periodic vertical asymptotes and sections of branches forming upward and downward curves.
These depict the basic behavior of a transformed secant function over one period.
(b) **State the period for the graph.**
\(T = \) [Input Field for the period]
**Explanation:**
To determine the period \(T\) of the function \(y = -4 - 3 \sec\left(3\pi x + \frac{\pi}{4}\right)\), note that the period of \(\sec(kx)\) is \(\frac{2\pi}{k}\). Here, \(k = 3\pi\), so calculate \(T\) using \(\frac{2\pi}{3\pi} = \frac{2}{3}\).
Fill in the value of \(T\) as calculated for future verification.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe069bb76-d12a-4659-9fff-a2d2132df1d6%2F48e3ba43-3278-413b-936b-7090447c6a54%2Fzzi9zjh_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Use
#### Graphing a Trigonometric Function
(a) **Graph one complete cycle for the following function:**
\[
y = -4 - 3 \sec\left(3\pi x + \frac{\pi}{4}\right)
\]
**Explanation of Graphs:**
You will see two separate displays:
1. **Left Graph:**
- The graph is labeled with axes marked as \(x\) on the horizontal axis and \(y\) on the vertical axis.
- The function shows multiple curves with values extending between \(y = -15\) and \(y = 15\).
- The graph illustrates several upward and downward branches typical of a secant function (sec), with vertical asymptotes where the secant function approaches infinity. This behavior occurs periodically between \(x \approx 0.1\) and \(x = 0.7\).
2. **Right Graph:**
- Similarly structured with \(x\) and \(y\) axes over the same ranges as the left graph.
- The plot is consistent with the left graph, demonstrating periodic vertical asymptotes and sections of branches forming upward and downward curves.
These depict the basic behavior of a transformed secant function over one period.
(b) **State the period for the graph.**
\(T = \) [Input Field for the period]
**Explanation:**
To determine the period \(T\) of the function \(y = -4 - 3 \sec\left(3\pi x + \frac{\pi}{4}\right)\), note that the period of \(\sec(kx)\) is \(\frac{2\pi}{k}\). Here, \(k = 3\pi\), so calculate \(T\) using \(\frac{2\pi}{3\pi} = \frac{2}{3}\).
Fill in the value of \(T\) as calculated for future verification.
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