Graph the function fx=-2 sin2x+3.

You must graph at least one full period. Follow the steps below to graph.
Draw a dotted line for the midline on the graph
Mark a point for the beginning and end of one period of the function
Mark a point for the maximum point
Mark a point for the minimum point
Draw the graph connecting these points. Double check that this is the graph of the function.

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 $.