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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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.