https://file.aiquickdraw.com/mathbot/file/17280650588061a9aod5r.jpg

The function given is \( f(x) = \sqrt{x + 4} - 2 \). To understand how the graph behaves, let's break it down step by step.

### Step-by-Step Analysis:

1. **Starting with the base function**:
   - The basic form is \( \sqrt{x} \), which represents a square root function. Its graph starts at \( x = 0 \) and increases gradually, always non-negative.

2. **Horizontal Shift**:
   - The function inside the square root is \( x + 4 \), so the graph of \( \sqrt{x} \) shifts to the left by 4 units. This means the graph starts at \( x = -4 \) because \( \sqrt{x+4} \) is defined when \( x + 4 \geq 0 \), or \( x \geq -4 \).

3. **Vertical Shift**:
   - The \( -2 \) outside the square root represents a vertical shift downward by 2 units. This shifts the entire graph down by 2 units.

### Key Points:
- **Domain**: The function is defined for \( x \geq -4 \), as \( \sqrt{x+4} \) requires \( x + 4 \geq 0 \).
- **Range**: Since the smallest value of \( \sqrt{x+4} \) is 0 (at \( x = -4 \)), the smallest value of \( f(x) \) will be \( 0 - 2 = -2 \). So, the range is \( f(x) \geq -2 \).
- **Graph Shape**: The graph will resemble the basic square root curve, starting at \( (-4, -2) \) and increasing as \( x \) increases.

To summarize, the graph will start at \( (-4, -2) \), rise slowly, and never dip below \( -2 \).

Would you like further details, or do you have any questions about this? 

Here are some follow-up questions:
1. What is the behavior of the graph as \( x \) approaches infinity?
2. How does the domain of this function compare to that of the basic square root function \( \sqrt{x} \)?
3. How does a vertical shift affect the range of a function?
4. Can you calculate the point where the function intersects the x-axis (if it exists)?
5. How would the graph change if the function was \( \sqrt{x + 4} + 2 \) instead?

**Tip**: Always analyze transformations (shifts and stretches) to understand how a function's graph moves from its basic form.

Which graph represents the function f(x) = √(x + 4) - 2?

Learn how to graph the function f(x) = √(x + 4) - 2, a square root function involving horizontal and vertical shifts. Understand key concepts like the domain starting at x = -4 and a vertical shift of -2 units. Suitable for high school students.

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