The graph shown in the image is a downward-opening parabola, which is a transformation of the function $f(x) = x^2$.

We can determine the transformed function $g(x)$ by analyzing the graph:

1. Reflection: The graph opens downward, which means there is a vertical reflection across the x-axis. This is represented by multiplying the function by $-1$, giving $-x^2$.

2. Vertical Shift: The vertex of the graph appears to be at $(0, -4)$, which indicates the function has been shifted down by 4 units. This adds $-4$ to the function.

Thus, the equation for the function $g(x)$ is: $g(x) = -x^2 - 4$

Let me know if you need more details or explanations.

Here are 5 related questions you can explore:

1. What is the effect of horizontal shifts on the function $f(x) = x^2$?
2. How would the equation change if the graph had a vertical stretch or compression?
3. How do you determine the domain and range of a parabola like the one in the graph?
4. What would the function be if the vertex were shifted to $(2, -4)$?
5. How would you describe the transformation if the parabola opened upward instead?

Tip: Always check the vertex and orientation of a parabola to identify vertical shifts and reflections when analyzing transformations.