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.