Find a function of the form y = A sin(kx) + C or y = A cos(kx) + C whose graph matches the function shown below: 3- 2 19-18 -17 -16 -15 -14 -13 -11 -10 -8 -7 -6 -5 -3 -2 -1 6 7 8 -2 -3+ Leave your answer in exact form; if necessary, type pi for T. y =
Find a function of the form y = A sin(kx) + C or y = A cos(kx) + C whose graph matches the function shown below: 3- 2 19-18 -17 -16 -15 -14 -13 -11 -10 -8 -7 -6 -5 -3 -2 -1 6 7 8 -2 -3+ Leave your answer in exact form; if necessary, type pi for T. y =
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
![**Understanding Sinusoidal Functions**
**Problem Statement:**
Find a function of the form \( y = A \sin(kx) + C \) or \( y = A \cos(kx) + C \) whose graph matches the function shown below:
[Graph description]
- The x-axis ranges from -19 to 9.
- The y-axis ranges from -3 to 3.
- The graph is a sine curve that appears to move sinusoidally between -3 and 3, starting from a peak at \(x = -15\) and completing cycles.
**Instructions:**
Leave your answer in exact form; if necessary, type \(\pi\) for \(\pi\).
\[ y = \boxed{} \]
---
**Graph Explanation:**
The graph provided shows a sinusoidal function with the following characteristics:
- Amplitude: The graph has a maximum value of 3 and a minimum value of -3. Therefore, the amplitude \(A\) is 3.
- Period: Observing the graph, it completes one cycle from \(x = -15\) to \(x = -5\), a span of 10 units. The period \(T\) is 10.
- The formula for the period of a sine or cosine function is \(\frac{2\pi}{k}\). Solving for \(k\):
\[
T = 10 \implies \frac{2\pi}{k} = 10 \implies k = \frac{2\pi}{10} = \frac{\pi}{5}
\]
- Midline: The graph oscillates symmetrically around the x-axis, indicating the midline value \(C\) is 0.
- Phase shift: Because the function has a peak at \(x = -15\), it suggests that a cosine function might be more straightforward to use with a phase shift.
Thus, the cosine function with the above-stated parameters is:
\[ y = 3 \cos\left(\frac{\pi}{5}x + \phi\right) \]
Given the peak is at \(x = -15\), the phase shift \(\phi\) must satisfy:
\[ -\frac{\pi}{5}(-15) + \phi = 0 \implies 3\pi + \phi = 0 \implies \phi = -3\pi \]
Therefore](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F955ff4f4-16d4-4965-9c48-d9ab4c3976eb%2F6f634d08-825f-4b2a-8223-6c001e6547f1%2Fabu20lr_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Sinusoidal Functions**
**Problem Statement:**
Find a function of the form \( y = A \sin(kx) + C \) or \( y = A \cos(kx) + C \) whose graph matches the function shown below:
[Graph description]
- The x-axis ranges from -19 to 9.
- The y-axis ranges from -3 to 3.
- The graph is a sine curve that appears to move sinusoidally between -3 and 3, starting from a peak at \(x = -15\) and completing cycles.
**Instructions:**
Leave your answer in exact form; if necessary, type \(\pi\) for \(\pi\).
\[ y = \boxed{} \]
---
**Graph Explanation:**
The graph provided shows a sinusoidal function with the following characteristics:
- Amplitude: The graph has a maximum value of 3 and a minimum value of -3. Therefore, the amplitude \(A\) is 3.
- Period: Observing the graph, it completes one cycle from \(x = -15\) to \(x = -5\), a span of 10 units. The period \(T\) is 10.
- The formula for the period of a sine or cosine function is \(\frac{2\pi}{k}\). Solving for \(k\):
\[
T = 10 \implies \frac{2\pi}{k} = 10 \implies k = \frac{2\pi}{10} = \frac{\pi}{5}
\]
- Midline: The graph oscillates symmetrically around the x-axis, indicating the midline value \(C\) is 0.
- Phase shift: Because the function has a peak at \(x = -15\), it suggests that a cosine function might be more straightforward to use with a phase shift.
Thus, the cosine function with the above-stated parameters is:
\[ y = 3 \cos\left(\frac{\pi}{5}x + \phi\right) \]
Given the peak is at \(x = -15\), the phase shift \(\phi\) must satisfy:
\[ -\frac{\pi}{5}(-15) + \phi = 0 \implies 3\pi + \phi = 0 \implies \phi = -3\pi \]
Therefore
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning