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 \( f(x) = \sqrt{x} \) 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 \((4, 0)\).  *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 \((4, 2)\).  *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 \( f(x) = \sqrt{x} \):**  - **Transformed Function \( g(x) = \sqrt{x - 4} + 2 \):**  ### 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. --- **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 \( g(x) \).