Let's break down the function $y = -2\sin(3x - \frac{\pi}{4}) + 2$ again to provide the exact values for amplitude, period, and phase shift:

1. Amplitude

The amplitude of the function is given by the absolute value of the coefficient of the sine function. In this case:

$A = -2$

The amplitude is:

$|A| = |-2| = 2$

2. Period

The period of a sine function $y = A\sin(Bx - C) + D$ is calculated using the formula:

$T = \frac{2\pi}{|B|}$

Here, $B = 3$. Thus, the period is:

$T = \frac{2\pi}{3}$

3. Phase Shift

The phase shift of the function is determined by the expression inside the sine function $(Bx - C)$. The phase shift is given by:

$\text{Phase Shift} = \frac{C}{B}$

In the function $y = -2\sin(3x - \frac{\pi}{4}) + 2$, $C = \frac{\pi}{4}$ and $B = 3$. Therefore, the phase shift is:

$\text{Phase Shift} = \frac{\frac{\pi}{4}}{3} = \frac{\pi}{12}$

Summary of Exact Values

• Amplitude: $2$
• Period: $\frac{2\pi}{3}$
• Phase Shift: $\frac{\pi}{12}$ to the right

These values are provided in exact terms, using fractions and $\pi$ as needed.

Would you like further explanation or have more questions?

Here are some related questions you might find interesting:

1. What is the vertical shift of this function, and how does it affect the graph?
2. How would you determine the range of the function $y = -2\sin(3x - \frac{\pi}{4}) + 2$?
3. What are the x-intercepts and y-intercepts of this function?
4. How would you find the points where the function reaches its maximum and minimum values?
5. How would the graph of the function change if the sine function were replaced with a cosine function?

Tip: The phase shift tells you how the graph of the sine function is shifted horizontally. If the phase shift is positive, the graph shifts to the right; if negative, it shifts to the left.