Explain how the graph of the function $$ h(x) = -\sqrt{x} $$ is affected by the transformation compared to the original $$ f(x) = \sqrt{x} $$.

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The graph of $$ h(x) = -\sqrt{x} $$ is a transformation of the original function $$ f(x) = \sqrt{x} $$ through **reflection over the x-axis**. Here's a detailed explanation of how this transformation affects the graph:

### Original Function: $$ f(x) = \sqrt{x} $$
- **Shape**: This is the standard square root function, which starts at the origin $$(0, 0)$$ and increases gradually to the right.
- **Domain**: $$ x \geq 0 $$ (since you can't take the square root of a negative number in the set of real numbers).
- **Range**: $$ y \geq 0 $$.

### Transformed Function: $$ h(x) = -\sqrt{x} $$
- **Reflection Over the x-axis**: Every y-coordinate of the original function is multiplied by $$-1$$. This means that for every point $$(x, y)$$ on $$ f(x) $$, the corresponding point on $$ h(x) $$ is $$(x, -y)$$.
- **Shape**: The graph now starts at the origin $$(0, 0)$$ and decreases gradually to the right, mirroring the original shape below the x-axis.
- **Domain**: Remains the same, $$ x \geq 0 $$.
- **Range**: Now $$ y \leq 0 $$, since all output values are the negatives of the original function.

### Visual Comparison

- **$$ f(x) = \sqrt{x} $$**:
  
  ![Graph of f(x) = sqrt(x)](https://i.imgur.com/8zA9tUJ.png)

- **$$ h(x) = -\sqrt{x} $$**:
  
  ![Graph of h(x) = -sqrt(x)](https://i.imgur.com/bV6Lg5F.png)

In summary, the transformation from $$ f(x) = \sqrt{x} $$ to $$ h(x) = -\sqrt{x} $$ involves reflecting the graph of the square root function across the x-axis, which changes all positive y-values to their negative counterparts while keeping the same domain.
The graph of $$ h(x) = -\sqrt{x} $$ is the reflection of $$ f(x) = \sqrt{x} $$ over the x-axis. This means all the y-values of $$ f(x) $$ are multiplied by -1, making the graph start at the origin and decrease as x increases.

Explain how the graph of the function \( h(x) = -\sqrt{x} \) is affected by the transformation compared to the original \( f(x) = \sqrt{x} \).

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