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

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.