Which of the following describe the correct transformations of $$ g(x)=-2(0.5)^{x-3} $$ from $$ f(x)=0.5^{2} $$ ? Select all that apply. The graph of $$ g(x) $$ is shifted to the right 3 units. The graph of $$ g(x) $$ is compressed vertically The graph of $$ g(x) $$ is shifted to the left 3 units. The graph of $$ g(x) $$ is reflected over the $$ x $$-axis. The graph of $$ g(x) $$ is stretched vertically.

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To determine the correct transformations of $$ g(x) = -2(0.5)^{x-3} $$ from the base function $$ f(x) = 0.5^x $$, let's analyze each component of the transformation step by step.

### Base Function:
$$ f(x) = 0.5^x $$

### Transformed Function:
$$ g(x) = -2(0.5)^{x-3} $$

### Breakdown of Transformations:

1. **Horizontal Shift**:
   - The term $$ (x - 3) $$ inside the exponent indicates a **shift to the right by 3 units**.
   
2. **Vertical Stretch/Compression**:
   - The coefficient $$ -2 $$ outside the function affects the vertical stretch and reflection.
   - The absolute value $$ | -2 | = 2 $$ means the graph is **stretched vertically by a factor of 2**.
   
3. **Reflection**:
   - The negative sign $$ -2 $$ indicates a **reflection over the $$ x $$-axis**.

### Summary of Transformations:
- **Shifted to the right by 3 units**.
- **Stretched vertically by a factor of 2**.
- **Reflected over the $$ x $$-axis**.

### Incorrect Transformations:
- **Compressed vertically**: This is incorrect because the graph is actually stretched.
- **Shifted to the left by 3 units**: The graph is shifted to the right, not the left.

### Conclusion:
The correct transformations applied to $$ f(x) = 0.5^x $$ to obtain $$ g(x) = -2(0.5)^{x-3} $$ are:

1. **The graph of $$ g(x) $$ is shifted to the right 3 units.**
2. **The graph of $$ g(x) $$ is reflected over the $$ x $$-axis.**
3. **The graph of $$ g(x) $$ is stretched vertically.**

**Therefore, the correct options are:**
- The graph of $$ g(x) $$ is shifted to the right 3 units.
- The graph of $$ g(x) $$ is reflected over the $$ x $$-axis.
- The graph of $$ g(x) $$ is stretched vertically.

**Answer:**

All except “compressed vertically” and “shifted to the left 3 units”. Specifically:
• Shift right 3 units;
• Reflect over the x‑axis;
• Stretch vertically.
The graph of $$ g(x) $$ is shifted to the right 3 units, reflected over the $$ x $$-axis, and stretched vertically.

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