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The problem asks to find the equation of the function after applying the following transformations to \( y = \sqrt{x} \):

1. Shift down 9 units.
2. Reflect about the x-axis.
3. Reflect about the y-axis.

Let's go step by step:

### 1. Start with the original function:
\[
y = \sqrt{x}
\]

### 2. Apply the first transformation: Shift down 9 units
To shift the graph down by 9 units, subtract 9 from the function:
\[
y = \sqrt{x} - 9
\]

### 3. Apply the second transformation: Reflect about the x-axis
Reflecting about the x-axis means multiplying the entire function by \(-1\):
\[
y = -(\sqrt{x} - 9) = -\sqrt{x} + 9
\]

### 4. Apply the third transformation: Reflect about the y-axis
Reflecting about the y-axis involves replacing \(x\) with \(-x\):
\[
y = -\sqrt{-x} + 9
\]

### Final Equation:
The final transformed equation is:
\[
y = -\sqrt{-x} + 9
\]

Would you like further clarification or have additional questions?

### 5 Related Questions:
1. What is the effect of reflecting a function over the x-axis?
2. How does a vertical shift impact the graph of a square root function?
3. What does reflecting over the y-axis do to the graph of any function?
4. How do you identify transformations on a graph given the equation?
5. Can you explain how horizontal shifts differ from vertical shifts in transformations?

### Tip:
Always apply transformations step by step, especially with reflections, as they can change the orientation of the graph drastically.

Find the function that is finally graphed after the following transformations are applied to the graph of y = √x: 1) Shift down 9 units, 2) Reflect about the x-axis, 3) Reflect about the y-axis.

Learn how to apply transformations to the square root function y = √x, including a vertical shift, reflection over the x-axis, and reflection over the y-axis. This problem is suitable for high school students and covers key transformation concepts in algebra.

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