1. The graph of $$ f(x)=\sqrt{x} $$ was transformed to create the graph of $$ g(x)=-\sqrt{x-2}+1 $$. What transformations took place to create the new graph?

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To transform the graph of $$ f(x) = \sqrt{x} $$ into $$ g(x) = -\sqrt{x - 2} + 1 $$, follow these steps:

1. **Horizontal Shift**: 
   - **$$ x - 2 $$** inside the square root shifts the graph **2 units to the right**.
   
2. **Reflection**:
   - The **negative sign** in front of the square root reflects the graph **across the x-axis**.
   
3. **Vertical Shift**:
   - The **$$ +1 $$** outside the square root shifts the graph **1 unit upward**.

**Summary of Transformations:**
- **Shift right by 2 units**
- **Reflect across the x-axis**
- **Shift upward by 1 unit**

These combined transformations take the original graph of $$ \sqrt{x} $$ and produce the new graph $$ -\sqrt{x - 2} + 1 $$.

**Graphically:**

1. Start with $$ f(x) = \sqrt{x} $$.
2. Shift right by 2 units to get $$ \sqrt{x - 2} $$.
3. Reflect across the x-axis to obtain $$ -\sqrt{x - 2} $$.
4. Shift upward by 1 unit to arrive at $$ g(x) = -\sqrt{x - 2} + 1 $$.

**Final Answer:**
They moved the graph two units right, reflected it across the x‑axis, and then shifted it up by one unit.
The graph was shifted 2 units to the right, reflected across the x-axis, and then moved up 1 unit.

1. The graph of \( f(x)=\sqrt{x} \) was transformed to create the graph of \( g(x)=-\sqrt{x-2}+1 \). What transformations took place to create the new graph?

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