To find the formula for $g(x)$, let's analyze the transformation applied to the basic cosine function $f(x) = \cos(x)$.

1. Amplitude: The graph oscillates between $3$ and $-3$, so the amplitude is $3$. This indicates a vertical stretch by a factor of $3$, making the formula start as $g(x) = 3 \cos(\dots)$.

2. Vertical Shift: Since the graph centers around $y = 0$, there is no vertical shift applied.

3. Period: The cosine function normally has a period of $2\pi$. Observing the graph, $g(x)$ completes one full cycle from $0$ to $2\pi$, indicating that there is no horizontal scaling or stretching of the period. Thus, the period remains $2\pi$.

4. Reflection: The graph appears to start from $(0, -3)$, which is the minimum point. The cosine function normally starts from its maximum at $x = 0$, so this graph represents a reflection over the x-axis. This reflection adds a negative sign to the function, making it $g(x) = -3 \cos(x)$.

Putting it all together, the formula for $g(x)$ is: $g(x) = -3 \cos(x)$

Would you like a further breakdown of any of these steps, or have any other questions?

Here are some related questions to deepen your understanding:

1. What would the formula for $g(x)$ be if the amplitude were doubled to 6?
2. How would a phase shift of $\frac{\pi}{2}$ to the right affect the function $g(x)$?
3. If the period of $g(x)$ was halved, what would the new function be?
4. How would a vertical shift by 2 units affect the formula for $g(x)$?
5. How does the negative sign in front of the amplitude affect the shape of the cosine graph?

Tip: For trigonometric transformations, start by identifying amplitude changes, period adjustments, reflections, and shifts to systematically build the function.