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Answer
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.
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
The function \( y = (0.5) 2^{-x} \) can be visualized as an exponential decay function. Since \( 2^{-x} \) is equivalent to \( \frac{1}{2^x} \), the graph will start at \( 0.5 \) when \( x = 0 \) and will decrease as \( x \) increases. This illustrates how quickly the function approaches zero but never quite touches the x-axis, creating a smooth, downward-sloping curve. When sketching this graph, remember that as \( x \) approaches negative infinity, \( y \) rapidly increases towards infinity, creating a sharp rise on the left side of the graph. The curve will float gently above the x-axis, reflecting the properties of exponential functions where they always remain positive.
