Answer
The graph of \( f(x) = \sqrt{x} \) is shifted 4 units to the right and 2 units up to become \( g(x) = \sqrt{x - 4} + 2 \).
Solution
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 \( g(x) \).
Reviewed and approved by the UpStudy tutoring team
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