Which of the following describe the correct transformations of $$ f(x)=2^{x-1}+3 $$ from $$ f(x)=2^{x} $$ ? Select all that apply. The graph of $$ f(x) $$ is shifted to the right 1 unit. The graph of $$ f(x) $$ is reflected over the $$ x $$-axis. The graph of $$ f(x) $$ is shifted up 3 units. The graph of $$ f(x) $$ is shifted down 3 units. The graph of $$ f(x) $$ is shifted to the left 1 unit.

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To determine the correct transformations of $$ f(x) = 2^{x-1} + 3 $$ from the base function $$ f(x) = 2^x $$, let's analyze each component of the function:

1. **Horizontal Shift**:
   - The term $$ x - 1 $$ inside the exponent indicates a horizontal shift.
   - Specifically, $$ x - 1 $$ shifts the graph **to the right by 1 unit**.

2. **Vertical Shift**:
   - The constant $$ +3 $$ outside the exponent shifts the graph vertically.
   - This means the graph is shifted **up by 3 units**.

3. **Reflection and Other Shifts**:
   - There is no reflection over the $$ x $$-axis since the coefficient of the exponential term is positive.
   - There is no shift downward or any leftward shift.

**Conclusion**:
The transformations applied to $$ f(x) = 2^x $$ to obtain $$ f(x) = 2^{x-1} + 3 $$ are:
- **Shift to the right by 1 unit**.
- **Shift upward by 3 units**.

**Therefore, the correct options are:**

- The graph of $$ f(x) $$ is shifted to the right 1 unit.
- The graph of $$ f(x) $$ is shifted up 3 units.

Answer: 
• The graph of $$ f(x) $$ is shifted to the right 1 unit.
• The graph of $$ f(x) $$ is shifted up 3 units.
The graph of $$ f(x) $$ is shifted to the right 1 unit and up 3 units.

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