The function \( h(x) = \sqrt{x} + 4 \) is a transformation of the parent square root function \( f(x) = \sqrt{x} \).

Here’s a breakdown of how it is derived from the parent function \( f(x) = \sqrt{x} \):

1. **Parent function**: \( f(x) = \sqrt{x} \) is the basic square root function, which starts at \( (0, 0) \) and only takes non-negative values of \( x \) (i.e., its domain is \( [0, \infty) \)).
   
2. **Vertical shift**: Adding 4 to the square root function, \( \sqrt{x} + 4 \), shifts the graph vertically upward by 4 units. This means that instead of starting at \( (0, 0) \), the graph now starts at \( (0, 4) \).

### Key Characteristics of \( h(x) = \sqrt{x} + 4 \):
- **Domain**: Since the square root function is defined only for non-negative \( x \), the domain of \( h(x) \) is \( [0, \infty) \).
- **Range**: The minimum value of \( \sqrt{x} \) is 0 (when \( x = 0 \)), and since the graph is shifted up by 4 units, the range is \( [4, \infty) \).
- **Graph behavior**: The graph starts at \( (0, 4) \) and increases slowly as \( x \) increases.

### Summary of transformations:
- **Vertical shift**: The graph of \( \sqrt{x} \) is shifted up by 4 units.

Let me know if you'd like a visual representation or more details!

### Expansion Questions:
1. What is the domain and range of the parent function \( f(x) = \sqrt{x} \)?
2. How would the graph change if the function were \( h(x) = \sqrt{x} - 4 \)?
3. What would happen to the graph if there were a horizontal shift involved, e.g., \( h(x) = \sqrt{x-2} + 4 \)?
4. How does the graph of \( h(x) = \sqrt{x} + 4 \) differ from \( h(x) = 2\sqrt{x} + 4 \)?
5. Can you find the inverse of the function \( h(x) = \sqrt{x} + 4 \)?

### Tip:
Always check the domain and range of transformed functions by considering how the parent function behaves under transformations!

Explore the function h(x) = √x + 4, a transformation of the square root function. Learn how vertical shifts affect the graph, understand the domain and range, and see a step-by-step breakdown of how the parent function changes. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation