For the given function y=−3 sin 2 x+
Find:
π
4
Amplitude :
Period:
Phase shift:
The interval:

For the function \( y = -3 \sin(2x + \frac{\pi}{4}) \), let's break it down step by step to find the amplitude, period, phase shift, and the interval of one cycle.

### 1. **Amplitude**:
The amplitude of a sine function \( y = a \sin(bx + c) \) is the absolute value of the coefficient of the sine function, which is \( |a| \).

Here, \( a = -3 \), so:
\[
\text{Amplitude} = | -3 | = 3
\]

### 2. **Period**:
The period of a sine function is determined by the coefficient \( b \) in \( y = a \sin(bx + c) \). The period formula is:
\[
\text{Period} = \frac{2\pi}{|b|}
\]
In this case, \( b = 2 \), so:
\[
\text{Period} = \frac{2\pi}{2} = \pi
\]

### 3. **Phase Shift**:
The phase shift of the sine function is given by \( -\frac{c}{b} \), where \( c \) is the constant added to the argument \( bx + c \).

Here, \( c = \frac{\pi}{4} \) and \( b = 2 \), so the phase shift is:
\[
\text{Phase shift} = -\frac{\frac{\pi}{4}}{2} = -\frac{\pi}{8}
\]
This means the graph is shifted \( \frac{\pi}{8} \) units to the left.

### 4. **Interval of One Cycle**:
The interval for one cycle of the sine function is from the starting point (after applying the phase shift) to one full period later. The period is \( \pi \), and the phase shift moves the graph to the left by \( \frac{\pi}{8} \).

- Start of the interval: \( -\frac{\pi}{8} \)
- End of the interval: \( -\frac{\pi}{8} + \pi = \frac{7\pi}{8} \)

Thus, the interval is \( \left[ -\frac{\pi}{8}, \frac{7\pi}{8} \right] \).

---

### Summary:
- **Amplitude**: 3
- **Period**: \( \pi \)
- **Phase Shift**: \( -\frac{\pi}{8} \) (shifted left)
- **Interval**: \( \left[ -\frac{\pi}{8}, \frac{7\pi}{8} \right] \)

Do you have any questions or would you like more details on any part of the solution?

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Here are 5 related questions:
1. How do you find the amplitude of cosine functions?
2. What happens to the graph of a sine function if the amplitude is negative?
3. How does a vertical shift affect the sine or cosine graph?
4. What is the effect of increasing the coefficient \( b \) in the sine function on its period?
5. Can you calculate the phase shift of \( y = \sin(3x - \frac{\pi}{6}) \)?

**Tip**: The negative sign in front of the amplitude of sine or cosine reflects the graph vertically across the x-axis!

For the given function y=−3 sin(2x + π/4), find the amplitude, period, phase shift, and the interval of one cycle.

This solution explains how to find the amplitude, period, phase shift, and interval of one cycle for the trigonometric function y = -3sin(2x + π/4). Suitable for high school students in Grades 10-12 learning trigonometry.

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