The graph of a sinusoidal function intersects its midline at (0, 1) and then has a maximum point at 7x Write the formula of the function, where is entered in radians. f(x) =
The graph of a sinusoidal function intersects its midline at (0, 1) and then has a maximum point at 7x Write the formula of the function, where is entered in radians. f(x) =
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![### Understanding the Sinusoidal Function
The problem presented involves identifying the formula of a sinusoidal function that satisfies given conditions. Let’s break down the information step by step.
#### Problem Statement:
The graph of a sinusoidal function intersects its midline at \( (0, 1) \) and then has a maximum point at \( \left(\frac{7\pi}{4}, 5\right) \).
**Task:** Write the formula of the function, where \( x \) is entered in radians.
This suggests that the function passes through the coordinates \( (0, 1) \) and \( \left(\frac{7\pi}{4}, 5\right) \).
### Explanation:
A sinusoidal function generally takes the form:
\[ f(x) = A \sin(Bx - C) + D \text{ or } f(x) = A \cos(Bx - C) + D \]
Where:
- \( A \) is the amplitude
- \( B \) affects the period of the function
- \( C \) is the horizontal shift (phase shift)
- \( D \) is the vertical shift (midline)
#### Step-by-Step Solution:
1. **Identify the Vertical Shift (Midline \( D \)):**
Since the function intersects its midline at \( y = 1 \), the vertical shift \( D \) is 1.
\[ D = 1 \]
2. **Identify the Maximum Value (Amplitude \( A \)):**
Given that the maximum point is \( \left(\frac{7\pi}{4}, 5\right) \), we can determine the amplitude \( A \). The difference between the maximum point and the midline is:
\[ A = 5 - 1 = 4 \]
3. **Determine the Period and Frequency ( \( B \)):**
The period \( P \) of a sinusoidal function is calculated from the frequency \( B \) using the formula:
\[ P = \frac{2\pi}{B} \]
To find \( B \), examine the period between points where the function reaches its maximum or another corresponding point on the sinusoidal curve. By analyzing the points provided (\( 0 \) at midline and maximum at \( \frac{7\pi}{4} \)), the information suggests a pattern established every \( 2\pi/B \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fbacb6c-bf79-4add-ae44-d1379ae5bc9f%2F86f734c1-3b9c-4faa-8a27-8813c8447357%2Fexndp8s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Sinusoidal Function
The problem presented involves identifying the formula of a sinusoidal function that satisfies given conditions. Let’s break down the information step by step.
#### Problem Statement:
The graph of a sinusoidal function intersects its midline at \( (0, 1) \) and then has a maximum point at \( \left(\frac{7\pi}{4}, 5\right) \).
**Task:** Write the formula of the function, where \( x \) is entered in radians.
This suggests that the function passes through the coordinates \( (0, 1) \) and \( \left(\frac{7\pi}{4}, 5\right) \).
### Explanation:
A sinusoidal function generally takes the form:
\[ f(x) = A \sin(Bx - C) + D \text{ or } f(x) = A \cos(Bx - C) + D \]
Where:
- \( A \) is the amplitude
- \( B \) affects the period of the function
- \( C \) is the horizontal shift (phase shift)
- \( D \) is the vertical shift (midline)
#### Step-by-Step Solution:
1. **Identify the Vertical Shift (Midline \( D \)):**
Since the function intersects its midline at \( y = 1 \), the vertical shift \( D \) is 1.
\[ D = 1 \]
2. **Identify the Maximum Value (Amplitude \( A \)):**
Given that the maximum point is \( \left(\frac{7\pi}{4}, 5\right) \), we can determine the amplitude \( A \). The difference between the maximum point and the midline is:
\[ A = 5 - 1 = 4 \]
3. **Determine the Period and Frequency ( \( B \)):**
The period \( P \) of a sinusoidal function is calculated from the frequency \( B \) using the formula:
\[ P = \frac{2\pi}{B} \]
To find \( B \), examine the period between points where the function reaches its maximum or another corresponding point on the sinusoidal curve. By analyzing the points provided (\( 0 \) at midline and maximum at \( \frac{7\pi}{4} \)), the information suggests a pattern established every \( 2\pi/B \
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