https://file.aiquickdraw.com/mathbot/file/17301695419849ibsgo2j.jpg

To analyze the function \( y = -3 \sin\left(\frac{\pi}{4} x - \frac{\pi}{4}\right) + 2 \), let's break down each part as follows:

1. **Midline**: The midline of a sinusoidal function \( y = A \sin(Bx - C) + D \) is given by \( y = D \).
   - Here, \( D = 2 \).
   - So, the midline is \( y = 2 \).

2. **Amplitude**: The amplitude is the absolute value of \( A \) in \( y = A \sin(Bx - C) + D \).
   - Here, \( A = -3 \), so the amplitude is \( |A| = 3 \).

3. **Period**: The period of a sine function \( y = A \sin(Bx) \) is given by \( \frac{2\pi}{|B|} \).
   - In this function, \( B = \frac{\pi}{4} \).
   - Therefore, the period is \( \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \).

4. **Phase Shift**: The phase shift is found by solving \( \frac{C}{B} \) where \( C \) and \( B \) are the coefficients in \( y = A \sin(Bx - C) + D \).
   - Here, \( C = \frac{\pi}{4} \) and \( B = \frac{\pi}{4} \).
   - The phase shift is \( \frac{\frac{\pi}{4}}{\frac{\pi}{4}} = 1 \).
   - Since the expression inside the sine function is \( \left(\frac{\pi}{4} x - \frac{\pi}{4}\right) \), the phase shift is **positive 1 unit** to the right.

To summarize:

- **Midline**: \( y = 2 \)
- **Amplitude**: 3
- **Period**: \( 8 \)
- **Phase Shift**: \( 1 \) (to the right)

Would you like further details, or do you have any questions?

---

### Related Questions:

1. How would these values change if the amplitude were doubled?
2. What would happen to the period if \( B \) was doubled?
3. How does a negative amplitude affect the graph of the function?
4. How can we express this function in terms of a cosine function?
5. What are the x-intercepts of this function?

### Tip:
Remember, a positive phase shift means a shift to the right, while a negative phase shift means a shift to the left.

Let y = -3 sin((π/4)x - π/4) + 2. Determine the midline, amplitude, period, and phase shift.

Learn how to find the midline, amplitude, period, and phase shift of the sinusoidal function y = -3 sin((π/4)x - π/4) + 2. This problem covers key trigonometric concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation