Graph y = 3 – cOS T X over a one-period interval. State its period, the five key points. %3D
Graph y = 3 – cOS T X over a one-period interval. State its period, the five key points. %3D
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°.
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![### Problem 5: Graphing a Trigonometric Function
**Question:**
Graph \( y = 3 - \cos(\pi x) \) over a one-period interval. State its period and the five key points.
**Explanation:**
To graph the function \( y = 3 - \cos(\pi x) \) over a one-period interval, we need to understand the characteristics of the cosine function, including its period, amplitude, phase shift, and vertical shift.
#### Key Characteristics:
1. **Period:** The period of the function \( y = 3 - \cos(\pi x) \) is determined by the coefficient of \( x \) inside the cosine function. For \( \cos(\pi x) \), the period \( T \) is given by:
\[
T = \frac{2\pi}{\pi} = 2
\]
Therefore, the function completes one full cycle over the interval \([0, 2]\).
2. **Amplitude:** The amplitude of the function is 1, as it is the coefficient of the cosine function, but here it is not explicitly visible because the coefficient is implicitly 1. The amplitude indicates the maximum deviation from the midline of the graph.
3. **Vertical Shift:** The function \( y = 3 - \cos(\pi x) \) has a vertical shift of 3 units upwards, meaning the midline of the cosine graph is shifted from \( y=0 \) to \( y=3 \).
4. **Phase Shift:** There is no horizontal shift in this function since there is no horizontal translation term inside the cosine function.
#### Five Key Points:
To plot the graph accurately, we can find the y-values at five key x-values within one period \( [0, 2] \):
- \( x = 0 \)
- \( x = \frac{1}{2} \)
- \( x = 1 \)
- \( x = \frac{3}{2} \)
- \( x = 2 \)
#### Calculations:
1. \( x = 0 \): \( y = 3 - \cos(0) = 3 - 1 = 2 \)
2. \( x = \frac{1}{2} \): \( y = 3 - \cos\left(\pi \cdot \frac{1}{2}\right) = 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f9e96de-8215-4532-a7d0-6f1fdf87f241%2F33fc7dd2-b155-4ddc-88b5-d43de9b1d086%2F9b6gcgi_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 5: Graphing a Trigonometric Function
**Question:**
Graph \( y = 3 - \cos(\pi x) \) over a one-period interval. State its period and the five key points.
**Explanation:**
To graph the function \( y = 3 - \cos(\pi x) \) over a one-period interval, we need to understand the characteristics of the cosine function, including its period, amplitude, phase shift, and vertical shift.
#### Key Characteristics:
1. **Period:** The period of the function \( y = 3 - \cos(\pi x) \) is determined by the coefficient of \( x \) inside the cosine function. For \( \cos(\pi x) \), the period \( T \) is given by:
\[
T = \frac{2\pi}{\pi} = 2
\]
Therefore, the function completes one full cycle over the interval \([0, 2]\).
2. **Amplitude:** The amplitude of the function is 1, as it is the coefficient of the cosine function, but here it is not explicitly visible because the coefficient is implicitly 1. The amplitude indicates the maximum deviation from the midline of the graph.
3. **Vertical Shift:** The function \( y = 3 - \cos(\pi x) \) has a vertical shift of 3 units upwards, meaning the midline of the cosine graph is shifted from \( y=0 \) to \( y=3 \).
4. **Phase Shift:** There is no horizontal shift in this function since there is no horizontal translation term inside the cosine function.
#### Five Key Points:
To plot the graph accurately, we can find the y-values at five key x-values within one period \( [0, 2] \):
- \( x = 0 \)
- \( x = \frac{1}{2} \)
- \( x = 1 \)
- \( x = \frac{3}{2} \)
- \( x = 2 \)
#### Calculations:
1. \( x = 0 \): \( y = 3 - \cos(0) = 3 - 1 = 2 \)
2. \( x = \frac{1}{2} \): \( y = 3 - \cos\left(\pi \cdot \frac{1}{2}\right) = 3
Transcribed Image Text:**Instruction for Mathematical Expressions**
When solving mathematical expressions, it is important to rationalize the denominators whenever applicable. DO NOT provide the value generated by a calculator under any circumstances!
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