The given equation is:  
\[
y = 2\sqrt{x + 2 - 7}
\]

### Step 1: Simplify the expression under the square root.
\[
x + 2 - 7 = x - 5
\]
Thus, the equation becomes:
\[
y = 2\sqrt{x - 5}
\]

### Step 2: Domain of the function  
Since the square root function \( \sqrt{u} \) is only defined for non-negative inputs (i.e., \( u \geq 0 \)), the expression \( x - 5 \) must satisfy:
\[
x - 5 \geq 0 \quad \Rightarrow \quad x \geq 5
\]
So, the domain is:
\[
\boxed{x \geq 5}
\]

### Step 3: Range of the function  
The square root function always outputs non-negative values. Therefore,  
- The minimum value of \( \sqrt{x - 5} \) occurs when \( x = 5 \), which gives \( \sqrt{5 - 5} = 0 \).  
- As \( x \to \infty \), \( \sqrt{x - 5} \to \infty \).  
Thus, the range of \( y = 2\sqrt{x - 5} \) is:
\[
\boxed{y \geq 0}
\]

### Step 4: Graph behavior and transformations  
- **Horizontal shift**: The graph of \( y = 2\sqrt{x - 5} \) is shifted **5 units to the right** compared to the parent function \( y = 2\sqrt{x} \).  
- **Vertical scaling**: The factor of 2 stretches the graph vertically by a factor of 2.  

### Step 5: Key points for plotting  
- At \( x = 5 \), \( y = 2\sqrt{5 - 5} = 0 \).  
- At \( x = 6 \), \( y = 2\sqrt{6 - 5} = 2 \).  
- At \( x = 9 \), \( y = 2\sqrt{9 - 5} = 4 \).  

The graph starts at \( (5, 0) \) and increases as \( x \) increases.  

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Do you need a more detailed graph analysis or further assistance?

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### 5 Related Questions:
1. How do you find the inverse of \( y = 2\sqrt{x - 5} \)?
2. What happens if we square both sides of the equation \( y = 2\sqrt{x - 5} \)?
3. How does changing the coefficient from 2 to another value affect the graph?
4. Can we rewrite this function in terms of a logarithmic expression?
5. What are the properties of the square root function that determine the domain and range?

### Tip:  
Always check the domain of square root functions carefully, since negative inputs will make the function undefined.

Learn how to solve and analyze the function y = 2√(x - 5). This solution explains the domain, range, and key graph transformations, including vertical stretching and horizontal shifts. Suitable for students in Grades 9-11.

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