Find amplitude period and phase shift of y=-2sin(3x- pi/4)+2. Give exact value

To analyze the function \( y = -2\sin(3x - \frac{\pi}{4}) + 2 \), we need to identify its amplitude, period, and phase shift. This function is a transformation of the standard sine function \( y = \sin(x) \).

### 1. Amplitude

The amplitude of a sine function \( y = A\sin(Bx - C) + D \) is given by the absolute value of the coefficient \( A \). 

For the function \( y = -2\sin(3x - \frac{\pi}{4}) + 2 \), the amplitude \( A \) is:
\[
|A| = |-2| = 2
\]
So, the **amplitude** is \( 2 \).

### 2. Period

The period of a sine function \( y = A\sin(Bx - C) + D \) is determined by the coefficient \( B \). The period \( T \) can be calculated using the formula:
\[
T = \frac{2\pi}{|B|}
\]
For \( y = -2\sin(3x - \frac{\pi}{4}) + 2 \), the coefficient \( B \) is \( 3 \). Thus, the period \( T \) is:
\[
T = \frac{2\pi}{|3|} = \frac{2\pi}{3}
\]
So, the **period** is \( \frac{2\pi}{3} \).

### 3. Phase Shift

The phase shift of a sine function \( y = A\sin(Bx - C) + D \) is given by the value of \( C \) divided by \( B \). The phase shift can be calculated using:
\[
\text{Phase Shift} = \frac{C}{B}
\]
In the function \( y = -2\sin(3x - \frac{\pi}{4}) + 2 \), the value of \( C \) is \( \frac{\pi}{4} \) and \( B \) is \( 3 \). Thus, the phase shift is:
\[
\text{Phase Shift} = \frac{\frac{\pi}{4}}{3} = \frac{\pi}{12}
\]
Since the expression inside the sine function is \( 3x - \frac{\pi}{4} \), the phase shift is to the **right** by \( \frac{\pi}{12} \).

### Summary

- **Amplitude**: \( 2 \)
- **Period**: \( \frac{2\pi}{3} \)
- **Phase Shift**: \( \frac{\pi}{12} \) (to the right)

Would you like to see more details or have any other questions?

Here are some related questions:
1. What is the effect of the vertical shift in the function?
2. How does the negative sign in front of the sine affect the graph?
3. Can you explain how to graph this function step-by-step?
4. What are the maximum and minimum values of this function?
5. How would you find the x-intercepts of this sine function?

**Tip**: The period of a trigonometric function is the length of one complete cycle on the graph. Adjusting the value of \( B \) changes the number of cycles within a given interval, altering the period.

Discover how to find the amplitude, period, and phase shift of the sine function y = -2sin(3x - π/4) + 2. This detailed analysis covers trigonometric transformations and key formulas for amplitude, period, and phase shift calculations. Suitable for advanced high school students and anyone interested in trigonometry.

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