To find the equation of the function $h(x)$ that transforms the graph of $f(x) = \sqrt{x}$, we need to apply two transformations:

1. Shift down by 6 units: This transformation subtracts 6 from the original function. So the new function becomes: $f(x) - 6 = \sqrt{x} - 6$

2. Reflect over the x-axis: Reflecting a function over the x-axis involves multiplying the entire function by $-1$. Applying this to the function from step 1: $h(x) = -(\sqrt{x} - 6)$ This simplifies to: $h(x) = -\sqrt{x} + 6$

Thus, the equation of the function $h(x)$ is: $h(x) = -\sqrt{x} + 6$

Would you like more details or have any questions?

Here are some related questions:

1. How do you reflect a graph over the y-axis?
2. What happens to a graph when you shift it left or right?
3. What are vertical and horizontal stretches and compressions?
4. How do you shift a graph up or down?
5. How do multiple transformations (e.g., shifts and reflections) interact with each other?

Tip: When applying multiple transformations to a function, always perform vertical shifts last.