The problem is asking for a formula for $g(x)$, which is a transformation of the function $f(x) = \sin(x)$.

Analyzing the Graph

1. Period: The graph completes a cycle from $0$ to $2\pi$, indicating the period is $2\pi$, which matches the standard sine function. So, there is no horizontal stretch/compression.
2. Amplitude: The maximum and minimum values of the graph are $2$ and $-2$, respectively. This suggests the amplitude has been scaled by a factor of 2.
3. Vertical Shift: The graph is centered around the x-axis, so there is no vertical shift.
4. Phase Shift: The graph starts at $(0, 0)$, similar to $\sin(x)$, so there is no horizontal shift.

Conclusion

Given these observations, $g(x)$ can be described by scaling the amplitude of $f(x) = \sin(x)$ by 2. Therefore, the formula for $g(x)$ is:

$g(x) = 2 \sin(x)$

Would you like more details on the transformations, or do you have any questions?

Here are 5 related questions for further exploration:

1. What would the formula for $g(x)$ look like if the amplitude were 3 instead of 2?
2. How would the formula change if there were a vertical shift of 1 unit up?
3. If the function had a phase shift of $\frac{\pi}{4}$, what would be the new formula for $g(x)$?
4. How would you express $g(x)$ if it were compressed horizontally by a factor of 2?
5. What effect does a negative amplitude have on the graph of a sine function?

Tip: For sine transformations, always analyze period, amplitude, vertical shift, and phase shift in that order to systematically identify the formula.