Determine the​ amplitude, period, and phase shift of the function. Graph the function.

y equals negative 2 sine left parenthesis 3 x minus 2 pi right parenthesisy=−2sin(3x−2π)

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Part 1

The amplitude is 

enter your response here.

​(Simplify your​ answer.)

Part 2

The period is 

enter your response here.

​(Type an exact​ answer, using 

piπ 

as needed.  Use integers or fractions for any numbers in the​ expression.)

Part 3

The phase shift is 

enter your response here.

​(Type an exact​ answer, using 

piπ 

as needed.  Use integers or fractions for any numbers in the​ expression.)

Part 4

Use the graphing tool to graph the function.

Let's break down the given function \( y = -2 \sin(3x - 2\pi) \) step by step to determine the amplitude, period, and phase shift.

### Part 1: Amplitude
The amplitude of a sine function \( y = A \sin(Bx + C) \) is given by \( |A| \).

Here, \( A = -2 \), so the amplitude is:
\[
\text{Amplitude} = |A| = |-2| = 2
\]

### Part 2: Period
The period of a sine function \( y = A \sin(Bx + C) \) is calculated as:
\[
\text{Period} = \frac{2\pi}{|B|}
\]
In this case, \( B = 3 \), so the period is:
\[
\text{Period} = \frac{2\pi}{3}
\]

### Part 3: Phase Shift
The phase shift is given by \( \frac{-C}{B} \), where \( C \) is the constant inside the parentheses with \( x \).

Here, the equation inside the sine function is \( 3x - 2\pi \). We can factor out the 3 from this expression:
\[
y = -2 \sin(3(x - \frac{2\pi}{3}))
\]
Thus, the phase shift is:
\[
\text{Phase Shift} = \frac{2\pi}{3} \quad \text{to the right}
\]

### Part 4: Graphing the Function
To graph the function \( y = -2 \sin(3x - 2\pi) \):
- The sine wave has an amplitude of 2.
- The period is \( \frac{2\pi}{3} \), meaning one complete cycle occurs every \( \frac{2\pi}{3} \) units along the x-axis.
- The phase shift moves the graph \( \frac{2\pi}{3} \) units to the right.
- Since the amplitude is negative, the graph is flipped vertically.

You can use a graphing tool to sketch the curve by plotting key points (max, min, and intercepts) and applying the transformations described above.

### Recap:
1. Amplitude = 2
2. Period = \( \frac{2\pi}{3} \)
3. Phase Shift = \( \frac{2\pi}{3} \) to the right

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Would you like more details or clarifications on any part?

Here are 5 related questions:
1. How would the graph change if the amplitude were positive instead of negative?
2. What is the effect of increasing the value of \( B \) on the period of the sine function?
3. How does a horizontal stretch or compression affect the graph of a sine function?
4. What happens if the phase shift moves the graph to the left instead of the right?
5. How does the vertical translation of a sine function affect its maximum and minimum points?

**Tip:** Always pay attention to the sign of the amplitude when graphing, as it determines whether the sine wave starts going up or down.

Determine the amplitude, period, and phase shift of the function y = -2sin(3x - 2π). Graph the function.

Learn how to find the amplitude, period, and phase shift of the trigonometric function y = -2sin(3x - 2π). This step-by-step solution covers key trigonometric concepts, including sine function transformations. Suitable for high school students in Grades 10-12.

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