If the function $$ g(x) = -\cos(x) $$ is a reflection of $$ h(x) = \cos(x) $$ over the x-axis, what can you say about their graphs?

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To analyze the relationship between the graphs of the functions $$ g(x) = -\cos(x) $$ and $$ h(x) = \cos(x) $$, we can follow these steps:

1. **Understanding the Functions**:
   - The function $$ h(x) = \cos(x) $$ is a standard cosine function, which oscillates between 1 and -1 with a period of $$ 2\pi $$.
   - The function $$ g(x) = -\cos(x) $$ is the reflection of $$ h(x) $$ over the x-axis. This means that for every point $$ (x, y) $$ on the graph of $$ h(x) $$, there is a corresponding point $$ (x, -y) $$ on the graph of $$ g(x) $$.

2. **Graph Characteristics**:
   - The graph of $$ h(x) = \cos(x) $$ starts at $$ (0, 1) $$, reaches $$ ( \frac{\pi}{2}, 0) $$, goes to $$ (\pi, -1) $$, returns to $$ ( \frac{3\pi}{2}, 0) $$, and completes the cycle at $$ (2\pi, 1) $$.
   - The graph of $$ g(x) = -\cos(x) $$ starts at $$ (0, -1) $$, reaches $$ ( \frac{\pi}{2}, 0) $$, goes to $$ (\pi, 1) $$, returns to $$ ( \frac{3\pi}{2}, 0) $$, and completes the cycle at $$ (2\pi, -1) $$.

3. **Key Observations**:
   - The maximum value of $$ h(x) $$ is 1, while the maximum value of $$ g(x) $$ is -1.
   - The minimum value of $$ h(x) $$ is -1, while the minimum value of $$ g(x) $$ is 1.
   - The x-intercepts of both functions occur at the same x-values, specifically at $$ x = \frac{\pi}{2} + n\pi $$ for $$ n \in \mathbb{Z} $$.

4. **Conclusion**:
   - The graphs of $$ g(x) $$ and $$ h(x) $$ are identical in shape but are vertically inverted relative to each other. This means that wherever $$ h(x) $$ is above the x-axis, $$ g(x) $$ is below it, and vice versa.

In summary, the graphs of $$ g(x) = -\cos(x) $$ and $$ h(x) = \cos(x) $$ are reflections of each other over the x-axis, exhibiting the same periodic behavior but with opposite vertical positions.
The graphs of $$ g(x) = -\cos(x) $$ and $$ h(x) = \cos(x) $$ are mirror images of each other over the x-axis. They have the same shape but are flipped vertically.

If the function \( g(x) = -\cos(x) \) is a reflection of \( h(x) = \cos(x) \) over the x-axis, what can you say about their graphs?

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