To solve this problem, we need to determine the transformation that takes the function $f(x) = \sqrt{x+4} - 2$ to $g(x)$. The transformation could involve shifts, stretches, compressions, or reflections. Since the graph of $g(x)$ is a transformation of $f(x)$, we should investigate the graph of both functions to identify what changes have been made.

However, since you mentioned doing 10 problems and provided a list format (1 to 9), it seems you are asking for multiple questions or examples of similar transformations.

For this first example:

Step-by-Step Process:

1. Analyze the function $f(x)$: $f(x) = \sqrt{x+4} - 2$

   • This is the graph of $y = \sqrt{x}$, but shifted left by 4 units (due to $x + 4$) and down by 2 units (due to the $-2$).

2. Analyze the graph of $g(x)$:

   • Since the graph of $g(x)$ is a transformation of $f(x)$, we can assume different types of transformations, like horizontal or vertical shifts, stretches, or reflections.
   • To find $g(x)$, we need to figure out what changes occurred based on the graph. If the graph of $g(x)$ is a shift of $f(x)$ or any other transformation, we will modify the expression for $f(x)$ accordingly.

Without a detailed description or visual of the graph, I can provide common transformations. Here are some transformations we might encounter:

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Example Transformations:

1. Vertical Shift: If $g(x)$ is a vertical shift of $f(x)$ by 3 units up: $g(x) = \sqrt{x+4} - 2 + 3 = \sqrt{x+4} + 1$

2. Horizontal Shift: If $g(x)$ is a horizontal shift of $f(x)$ by 5 units to the right: $g(x) = \sqrt{x-1+4} - 2 = \sqrt{x+3} - 2$

3. Reflection: If $g(x)$ is a reflection of $f(x)$ across the y-axis: $g(x) = \sqrt{-x+4} - 2$

4. Vertical Stretch: If $g(x)$ is a vertical stretch by a factor of 2: $g(x) = 2(\sqrt{x+4} - 2) = 2\sqrt{x+4} - 4$

5. Horizontal Stretch: If $g(x)$ is a horizontal stretch by a factor of 2: $g(x) = \sqrt{\frac{x+4}{2}} - 2$

6. Reflection over x-axis: If $g(x)$ is a reflection over the x-axis: $g(x) = -\sqrt{x+4} + 2$

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To clarify this transformation and answer more questions, I'd need to know what specific transformation happened to $f(x)$ to get $g(x)$. Would you like to provide additional details or clarify how the graph of $g(x)$ looks in relation to $f(x)$?

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Here are 5 related questions to deepen your understanding:

1. How does the function $y = \sqrt{x}$ behave with horizontal and vertical shifts?
2. What happens to the graph of a square root function if it's reflected over the x-axis?
3. How does a horizontal stretch affect the equation of a square root function?
4. How do vertical shifts affect the equation of a square root function?
5. What is the general effect of a vertical stretch or compression on the graph of a function?

Tip:

Whenever you deal with transformations of functions, remember that horizontal shifts affect the input $x$ inside the function, while vertical shifts affect the output of the function. Keep these principles in mind to help identify the transformation.