Describe how the graph of the radical function $$ f(x) = \sqrt{x} $$ changes when transformed to $$ g(x) = \sqrt{x - 4} + 2 $$.

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To understand how the graph of the radical function $$ f(x) = \sqrt{x} $$ changes when transformed to $$ g(x) = \sqrt{x - 4} + 2 $$, let's break down the transformations step by step.

### 1. **Start with the Basic Function:**
   $$
   f(x) = \sqrt{x}
   $$
   - **Graph Characteristics:**
     - **Domain:** $$ x \geq 0 $$
     - **Range:** $$ y \geq 0 $$
     - **Key Point:** Starts at the origin $$(0, 0)$$ and increases to the right.

### 2. **Horizontal Shift (Inside the Square Root):**
   $$
   h(x) = \sqrt{x - 4}
   $$
   - **Transformation:** The expression $$ x - 4 $$ inside the square root indicates a **horizontal shift**.
   - **Direction & Magnitude:** Shifts the graph **4 units to the right**.
   - **New Domain:** $$ x - 4 \geq 0 \Rightarrow x \geq 4 $$
   - **New Key Point:** The starting point moves from $$(0, 0)$$ to $$(4, 0)$$.

![Horizontal Shift](https://i.imgur.com/3n7dM8H.png)
   
   *Graph of $$ h(x) = \sqrt{x - 4} $$: Shifted 4 units to the right.*

### 3. **Vertical Shift (Outside the Square Root):**
   $$
   g(x) = \sqrt{x - 4} + 2
   $$
   - **Transformation:** The $$ +2 $$ outside the square root indicates a **vertical shift**.
   - **Direction & Magnitude:** Shifts the graph **2 units upward**.
   - **New Range:** Since $$ h(x) = \sqrt{x - 4} $$ has a range of $$ y \geq 0 $$, adding 2 shifts the range to $$ y \geq 2 $$.
   - **New Key Point:** The starting point moves from $$(4, 0)$$ to $$(4, 2)$$.

![Vertical Shift](https://i.imgur.com/O4cRvhl.png)
   
   *Graph of $$ g(x) = \sqrt{x - 4} + 2 $$: Shifted 4 units to the right and 2 units upward.*

### 4. **Summary of Transformations:**
- **Horizontal Shift:** 4 units **to the right**.
- **Vertical Shift:** 2 units **upward**.

### 5. **Final Graph Characteristics of $$ g(x) $$:**
- **Domain:** $$ x \geq 4 $$
- **Range:** $$ y \geq 2 $$
- **Key Point:** Starts at $$(4, 2)$$ and increases to the right.

### 6. **Visual Comparison:**

- **Original Function $$ f(x) = \sqrt{x} $$:**

![Original Function](https://i.imgur.com/HTpCQMQ.png)

- **Transformed Function $$ g(x) = \sqrt{x - 4} + 2 $$:**

![Transformed Function](https://i.imgur.com/O4cRvhl.png)

### 7. **Real-World Interpretation:**
Imagine the square root function as the shape of a curve representing some real-world relationship (like the speed of a car increasing over time). Shifting it right by 4 units could represent a delay in the reaction time, and shifting it up by 2 units could indicate an initial boost or offset in the starting condition.

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**In Conclusion:**
When transforming $$ f(x) = \sqrt{x} $$ to $$ g(x) = \sqrt{x - 4} + 2 $$, the graph undergoes two key transformations:
1. **Shifts 4 units to the right**, altering the starting point horizontally.
2. **Shifts 2 units upward**, raising the entire graph vertically.

These transformations adjust both the position and the range of the original radical function, providing a new location and starting point for $$ g(x) $$.
The graph of $$ f(x) = \sqrt{x} $$ is shifted 4 units to the right and 2 units up to become $$ g(x) = \sqrt{x - 4} + 2 $$.

Describe how the graph of the radical function changes when transformed to .

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Solución

1. Start with the Basic Function:

2. Horizontal Shift (Inside the Square Root):

3. Vertical Shift (Outside the Square Root):

4. Summary of Transformations:

5. Final Graph Characteristics of :

6. Visual Comparison:

7. Real-World Interpretation:

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