he maximum height of the graph of sin 2x is ___________. It is at x =_________.

Understanding the Problem

1. We need to determine the maximum height of the graph of the function $\sin(2x)$ and the values of $x$ where this maximum height occurs.

Relevant Concepts

2. The sine function, $\sin(x)$, has a maximum value of 1. Therefore, the maximum height of $\sin(2x)$ will also be 1.

Analysis and Detail

3. The function $\sin(2x)$ reaches its maximum height of 1 at specific points determined by the equation: $2x = \frac{\pi}{2} + 2k\pi$ where $k$ is any integer (since the sine function is periodic).

   To find the corresponding values of $x$, we solve for $x$: $x = \frac{\pi}{4} + k\pi$ where $k \in \mathbb{Z}$.

Verify and Summarize

4. Thus, the maximum height of the graph of $\sin(2x)$ is 1, and this occurs at $x = \frac{\pi}{4} + k\pi$ for any integer $k$.

Final Answer

The maximum height of the graph of $\sin(2x)$ is 1. It is at $x = \frac{\pi}{4} + k\pi$, where $k$ is any integer.