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

To understand how the graph of the radical function \( f(x) = \sqrt{x} \) transforms into \( g(x) = \sqrt{x - 4} + 2 \), let's break down the changes step by step. ### 1. **Horizontal Shift (Right)** - **Inside the Square Root:** The expression inside the square root in \( g(x) \) is \( x - 4 \). - **Effect:** This indicates a **horizontal shift** of the graph **4 units to the right**. - **Interpretation:** Every point \( (a, b) \) on the original graph \( f(x) \) moves to \( (a + 4, b) \) on the transformed graph \( g(x) \). ### 2. **Vertical Shift (Up)** - **Outside the Square Root:** The "+ 2" outside the square root in \( g(x) \) indicates a **vertical shift**. - **Effect:** This results in the graph being shifted **2 units upward**. - **Interpretation:** After the horizontal shift, every point \( (a + 4, b) \) moves to \( (a + 4, b + 2) \). ### 3. **Combined Transformation** - **Starting Point:** The original graph \( f(x) = \sqrt{x} \) starts at the origin \( (0, 0) \). - **After Shift:** Applying both transformations: - **Horizontal Shift:** The starting point moves from \( (0, 0) \) to \( (4, 0) \). - **Vertical Shift:** Then it shifts up to \( (4, 2) \). ### 4. **Visual Summary** - **Shape Preservation:** The shape of the graph remains the same (a square root curve), maintaining its characteristic increase. - **New Starting Point:** The lowest point of \( g(x) \) is now at \( (4, 2) \) instead of the origin. ### **Graphical Representation** - **Original Function \( f(x) = \sqrt{x} \):** - Starts at \( (0, 0) \). - Extends to the right, increasing gradually. - **Transformed Function \( g(x) = \sqrt{x - 4} + 2 \):** - Starts at \( (4, 2) \). - Extends to the right, mirroring the original shape but positioned higher and to the right. ### **Conclusion** The transformation from \( f(x) = \sqrt{x} \) to \( g(x) = \sqrt{x - 4} + 2 \) involves: 1. **Shifting the graph 4 units to the right**. 2. **Shifting the graph 2 units upward**. These shifts reposition the entire graph without altering its fundamental shape.