Find the Amplitude, Period, VIsual Shift, Phase Shift 

1) y = -2sin(2/3x - 4) - 6
2) y = 3sin(2x - 1) + 4



Let's break down both equations step by step to find the amplitude, period, vertical shift, and phase shift.

### Equation 1: \( y = -2\sin\left(\frac{2}{3}x - 4\right) - 6 \)

The general form for a sine function is:
\[
y = A\sin(Bx - C) + D
\]
Where:
- \(A\) is the **amplitude**.
- \(B\) affects the **period** (which is \( \frac{2\pi}{B} \)).
- \(C\) affects the **phase shift** (which is \( \frac{C}{B} \)).
- \(D\) is the **vertical shift**.

#### 1. **Amplitude**:
The amplitude is given by the absolute value of \(A\):
\[
\text{Amplitude} = |A| = |-2| = 2
\]

#### 2. **Period**:
The period is given by:
\[
\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{\frac{2}{3}} = 3\pi
\]

#### 3. **Phase Shift**:
The phase shift is given by:
\[
\text{Phase Shift} = \frac{C}{B} = \frac{4}{\frac{2}{3}} = 6
\]
Since the sine term is \( \sin\left(\frac{2}{3}x - 4\right) \), the phase shift is 6 units to the **right**.

#### 4. **Vertical Shift**:
The vertical shift is determined by \(D\):
\[
\text{Vertical Shift} = -6
\]
This means the graph is shifted 6 units **down**.

---

### Equation 2: \( y = 3\sin(2x - 1) + 4 \)

Again, the general form is:
\[
y = A\sin(Bx - C) + D
\]

#### 1. **Amplitude**:
\[
\text{Amplitude} = |A| = |3| = 3
\]

#### 2. **Period**:
\[
\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{2} = \pi
\]

#### 3. **Phase Shift**:
\[
\text{Phase Shift} = \frac{C}{B} = \frac{1}{2}
\]
This means the graph is shifted \( \frac{1}{2} \) units to the **right**.

#### 4. **Vertical Shift**:
\[
\text{Vertical Shift} = 4
\]
This means the graph is shifted 4 units **up**.

---

### Summary of results:

1) For \( y = -2\sin\left(\frac{2}{3}x - 4\right) - 6 \):
   - **Amplitude**: 2
   - **Period**: \( 3\pi \)
   - **Phase Shift**: 6 units to the right
   - **Vertical Shift**: 6 units down

2) For \( y = 3\sin(2x - 1) + 4 \):
   - **Amplitude**: 3
   - **Period**: \( \pi \)
   - **Phase Shift**: \( \frac{1}{2} \) units to the right
   - **Vertical Shift**: 4 units up

---

Would you like more details on any part of this, or do you have any other questions?

Here are 5 related questions for further practice:
1. How do you calculate the period of a cosine function?
2. What is the effect of a negative amplitude on the graph of a sine function?
3. How do you determine the phase shift for \( y = \sin(Bx + C) \)?
4. How would the graph of \( y = \sin(x) + 5 \) differ from \( y = \sin(x) \)?
5. Can you explain how to find the amplitude and period of a tangent function?

**Tip:** Always check the sign of the amplitude. If it's negative, the graph of the sine function will be reflected across the horizontal axis.

Find the Amplitude, Period, Vertical Shift, Phase Shift for the following functions:

1) y = -2sin(2/3x - 4) - 6
2) y = 3sin(2x - 1) + 4

Learn how to calculate the amplitude, period, phase shift, and vertical shift for two sine functions: y = -2sin(2/3x - 4) - 6 and y = 3sin(2x - 1) + 4. This guide provides a step-by-step breakdown of each concept, making it suitable for high school students in Grades 10-12.

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