https://file.aiquickdraw.com/mathbot/file/17289565002769qzfewoz.jpg

Let's break this problem down step-by-step.

The base function is:

\[
f(x) = \sqrt{x}
\]

The given function to transform is:

\[
h(x) = -\sqrt{x + 9} + 9
\]

We need to identify the **correct transformations** applied to \( f(x) = \sqrt{x} \) to result in \( h(x) \).

---

### Step 1: Horizontal Transformation
- **Inside the square root**, the expression is \( \sqrt{x + 9} \).
- This indicates a **horizontal shift to the left** by **9 units** (since \( x + 9 \) shifts left, not right).

---

### Step 2: Reflection about the x-axis
- The **negative sign** in front of the square root, \( -\sqrt{x + 9} \), means the graph is **reflected about the x-axis** (flips the graph vertically).

---

### Step 3: Vertical Shift
- The **+9** added outside the square root shifts the graph **up by 9 units**. However, this vertical shift is not needed to answer the multiple-choice question, which focuses on horizontal shifts and reflections.

---

### Answer Choice
From the above analysis:
- **Horizontal shift to the left by 9 units**
- **Reflection about the x-axis**

---

The correct answer is:

\[
\boxed{C.} \text{The graph of } \sqrt{x} \text{ should be horizontally shifted to the left by 9 units and reflected about the x-axis.}
\]

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Do you need further details on this solution, or have any additional questions?

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### Related Questions
1. What happens when a negative sign is placed inside the square root, like \( \sqrt{-x} \)?
2. How does a vertical stretch or compression affect a square root graph?
3. What is the effect of shifting \( \sqrt{x} \) vertically instead of horizontally?
4. Can you describe how both x-axis and y-axis reflections change the graph's shape?
5. How would you plot the function \( h(x) = -\sqrt{x + 9} + 9 \) step-by-step?

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**Tip:** A transformation inside the square root always affects the graph **horizontally** and behaves oppositely to the sign seen inside the function (e.g., \( +9 \) shifts left).

What transformations are needed to graph the function h(x) = -√(x + 9)? Choose the correct answer below.

Learn how to apply transformations to the square root function f(x) = √x to graph h(x) = -√(x + 9). Understand horizontal shifts, reflections about the x-axis, and their effects on the graph. This problem covers key concepts suitable for students in Grades 9-11.

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