Select the graph of $$ y=(0.5) 2^{-x} $$

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To identify the graph of the function $$ y = 0.5 \cdot 2^{-x} $$, let's analyze its key characteristics:

### 1. **Rewrite the Function for Clarity**

The given function can be rewritten using properties of exponents:

$$
y = 0.5 \cdot 2^{-x} = \frac{1}{2} \cdot \left(\frac{1}{2}\right)^{x} = \left(\frac{1}{2}\right)^{x + 1}
$$

This shows that the function is an exponential decay function with a base of $$ \frac{1}{2} $$ and a vertical scaling factor of $$ \frac{1}{2} $$.

### 2. **Key Features of the Graph**

- **Y-Intercept:** When $$ x = 0 $$,
  $$
  y = 0.5 \cdot 2^{0} = 0.5 \cdot 1 = 0.5
  $$
  So, the graph crosses the y-axis at $$ (0, 0.5) $$.

- **Behavior as $$ x $$ Increases:**
  As $$ x \rightarrow \infty $$,
  $$
  y = 0.5 \cdot 2^{-x} \rightarrow 0
  $$
  The graph approaches the x-axis but never touches it (horizontal asymptote at $$ y = 0 $$).

- **Behavior as $$ x $$ Decreases:**
  As $$ x \rightarrow -\infty $$,
  $$
  y = 0.5 \cdot 2^{-x} \rightarrow \infty
  $$
  The graph increases without bound.

- **Plotting Additional Points:**

| $$ x $$ | $$ y = 0.5 \cdot 2^{-x} $$ |
  |--------|--------------------------|
  | -2     | $$ 0.5 \cdot 2^{2} = 2 $$    |
  | -1     | $$ 0.5 \cdot 2^{1} = 1 $$    |
  |  0     | $$ 0.5 \cdot 2^{0} = 0.5 $$  |
  |  1     | $$ 0.5 \cdot 2^{-1} = 0.25 $$ |
  |  2     | $$ 0.5 \cdot 2^{-2} = 0.125 $$ |

### 3. **Graph Description**

- **Shape:** The graph is a downward-sloping exponential curve.
- **Asymptote:** The x-axis ($$ y = 0 $$) is a horizontal asymptote.
- **Slope:** The function decreases as $$ x $$ increases.
- **Quadrants:** The graph exists in both the first and fourth quadrants due to positive $$ y $$-values.

### 4. **Visual Representation**

While I can't display images directly, here's how you can sketch the graph based on the information:

1. **Plot the Y-Intercept** at $$ (0, 0.5) $$.
2. **Plot Additional Points** such as $$ (-2, 2) $$, $$ (-1, 1) $$, $$ (1, 0.25) $$, and $$ (2, 0.125) $$.
3. **Draw the Asymptote:** Draw a dashed horizontal line along the x-axis ($$ y = 0 $$) to indicate that the graph approaches but never touches it.
4. **Sketch the Curve:** Connect the points smoothly, showing the exponential decay as $$ x $$ increases and exponential growth as $$ x $$ decreases.

### 5. **Sample Graph**

Here’s a rough sketch outline:

```
y
|
|          *
|         /
|        /
|       /
|------*-------- x
|     /
|    *
|
```

- The point where the graph crosses the y-axis is at $$ y = 0.5 $$.
- As $$ x $$ increases, $$ y $$ approaches 0.
- As $$ x $$ decreases, $$ y $$ increases without bound.

### 6. **Conclusion**

The graph of $$ y = 0.5 \cdot 2^{-x} $$ is an exponential decay curve that starts at $$ (0, 0.5) $$, decreases towards the x-axis as $$ x $$ increases, and increases without bound as $$ x $$ decreases. If you're presented with multiple graph options, look for these characteristics to select the correct one.
The graph of $$ y = 0.5 \cdot 2^{-x} $$ is an exponential decay curve that starts at $$ (0, 0.5) $$, decreases towards the x-axis as $$ x $$ increases, and increases without bound as $$ x $$ decreases.

Select the graph of

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