Use technology to find points and then graph the function $$ y=-2 \log _{2}(x+3)+2 $$, following the instructions below.

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Certainly! Let's graph the function $$ y = -2 \log_{2}(x + 3) + 2 $$ by finding key points using technology (like a graphing calculator or software) and then plotting them. Here's a step-by-step guide:

### 1. Understand the Function Components

Before plotting, it's essential to understand how each part of the function affects its graph:

- **$$ \log_{2}(x + 3) $$**: This is the basic logarithmic function $$ \log_{2}(x) $$ shifted **3 units to the left**.
- **$$ -2 \log_{2}(x + 3) $$**: The negative sign reflects the graph across the **x-axis**, and multiplying by 2 **stretches** it vertically by a factor of 2.
- **$$ + 2 $$**: Finally, adding 2 shifts the entire graph **upward by 2 units**.

### 2. Determine the Domain

The logarithmic function is defined only when the argument is positive:

$$
x + 3 > 0 \quad \Rightarrow \quad x > -3
$$

**Domain:** $$ x > -3 $$

### 3. Identify Key Points

Let's choose several values of $$ x $$ within the domain to calculate the corresponding $$ y $$-values.

| $$ x $$ | $$ x + 3 $$ | $$ \log_{2}(x + 3) $$ | $$ -2 \log_{2}(x + 3) $$ | $$ y = -2 \log_{2}(x + 3) + 2 $$ |
|---------|------------|-----------------------|---------------------------|------------------------------------|
| -2      | 1          | 0                     | 0                         | 2                                  |
| -1      | 2          | 1                     | -2                        | 0                                  |
| 0       | 3          | $$ \log_{2}(3) \approx 1.585 $$ | $$ -2 \times 1.585 \approx -3.17 $$ | $$ -3.17 + 2 \approx -1.17 $$       |
| 1       | 4          | 2                     | -4                        | -2                                 |
| 5       | 8          | 3                     | -6                        | -4                                 |

**Key Points:**

- $$ (-2, 2) $$
- $$ (-1, 0) $$
- $$ (0, -1.17) $$
- $$ (1, -2) $$
- $$ (5, -4) $$

### 4. Plot the Key Points

Using graphing software or a graphing calculator:

1. **Plot the points** you've calculated:
   - Start by plotting $$ (-2, 2) $$, $$ (-1, 0) $$, $$ (0, -1.17) $$, $$ (1, -2) $$, and $$ (5, -4) $$.
   
2. **Draw the Vertical Asymptote:**
   - The function has a vertical asymptote at $$ x = -3 $$. Draw a dashed vertical line at $$ x = -3 $$ to represent this.

3. **Sketch the Graph:**
   - Connect the points smoothly, approaching the asymptote as $$ x $$ approaches -3 from the right.
   - The graph should extend infinitely downward to the right since logarithmic functions increase without bound, but due to the negative coefficient, it decreases.

### 5. Final Graph

Below is a conceptual representation of the graph:

```
y
|
4 |                   
3 |                   
2 |      * (-2,2)      
1 |                   
0 |          * (-1,0) 
-1|              * (0,-1.17)
-2|                 * (1,-2)
-3|                     
-4|                      * (5,-4)
   -------------------------------
          -3  -2  -1   0   1   2   3
```

- The vertical asymptote is at $$ x = -3 $$ (not shown in the simplistic sketch above).
- The graph approaches the asymptote as $$ x $$ approaches -3 from the right.
- As $$ x $$ increases, $$ y $$ decreases without bound.

### 6. Using Technology

If you're using graphing software or a calculator:

1. **Enter the Function:**
   - Input $$ y = -2 \log_{2}(x + 3) + 2 $$.
   
2. **Set the Window:**
   - Ensure your viewing window includes $$ x > -3 $$.
   - Adjust the y-range based on the key points (e.g., from -5 to 5).

3. **Graph the Function:**
   - Observe the plotted curve and verify it matches the key points and transformations discussed.

### 7. Summary of Transformations

- **Shift Left 3 Units:** Transforms $$ \log_{2}(x) $$ to $$ \log_{2}(x + 3) $$.
- **Reflect Across x-axis & Vertical Stretch by 2:** Applied through the multiplication by -2.
- **Shift Up 2 Units:** Applied through the addition of 2.

By following these steps, you can accurately plot and understand the behavior of the function $$ y = -2 \log_{2}(x + 3) + 2 $$.
To graph the function $$ y = -2 \log_{2}(x + 3) + 2 $$, follow these steps: 1. **Determine the Domain:** $$ x > -3 $$. 2. **Find Key Points:** - $$ (-2, 2) $$ - $$ (-1, 0) $$ - $$ (0, -1.17) $$ - $$ (1, -2) $$ - $$ (5, -4) $$ 3. **Plot the Points and Asymptote:** Draw a vertical asymptote at $$ x = -3 $$ and plot the key points. 4. **Sketch the Graph:** Connect the points smoothly, ensuring the graph approaches the asymptote as $$ x $$ approaches -3 from the right and decreases without bound as $$ x $$ increases. Using graphing technology, input the function and adjust the window to visualize the graph accurately.

Use technology to find points and then graph the function \( y=-2 \log _{2}(x+3)+2 \), following the instructions below.

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