Answer:
Connect the points marked, and you'll have the graph of the function \( [tex]f(x) = -2 \sin(2x) + 3 \).[/tex]
Step-by-step explanation:
To graph the function \( f(x) = -2 \sin(2x) + 3 \), we'll follow these steps:
1. **Determine the midline**: The midline is the average value of the function. For a sine function, the midline is the horizontal line that the function oscillates around. Since the amplitude is 2 and the vertical shift is 3, the midline is at \( y = 3 \).
2. **Determine the period**: The period of a sine function is \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function. In this case, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
3. **Identify key points**: We'll mark points for the beginning and end of one period, the maximum point, and the minimum point.
4. **Draw the graph**: We'll connect these points to graph one full period of the function.
Let's start:
1. **Midline**: \( y = 3 \) (draw a dotted line)
2. **Period**: \( \pi \)
3. **Key Points**:
- Beginning of one period: \( x = 0 \)
- End of one period: \( x = \pi \)
[tex]- Maximum point: \( \left( \frac{\pi}{4}, 5 \right) \) (when \( \sin(2x) = 1 \))[/tex]
[tex]- Minimum point: \( \left( \frac{3\pi}{4}, 1 \right) \) (when \( \sin(2x) = -1 \))[/tex]
4. **Graph**:
```
|
6 +-------------------------------+---------+
| | |
| | * | Minimum Point
| | |
5 +-------------------------------+---------+
| | |
| | |
| | Maximum | Point
4 +-------------------------------+---------+
| | |
| | |
| | |
3 +-------------------------------+---------+-- Midline
| | |
| | |
| | |
2 +-------------------------------+---------+
| | |
| | |
| | |
1 +-------------------------------+---------+
| | |
| | |
+-------------------------------+---------+------- x
0 π/2 π 3π/2
```
Connect the points marked, and you'll have the graph of the function \( f(x) = -2 \sin(2x) + 3 \).
