The image shows two functions, $f(x) = x^2$ and another function $g(x)$, which is a transformed version of $f(x)$. The function $f(x) = x^2$ is a standard parabola opening upwards, while $g(x)$ appears to be a parabola that has been reflected, shifted, and possibly stretched or compressed.

To determine the equation of $g(x)$, we need to analyze the transformations that have been applied to $f(x) = x^2$.

Step 1: Reflection

The graph of $g(x)$ is opening downwards, which indicates a reflection over the x-axis. This means that the function has been multiplied by $-1$. So, the equation becomes: $g(x) = -f(x)$

Step 2: Horizontal Shift

The vertex of $g(x)$ appears to be shifted from the origin to the right. If the vertex of $g(x)$ is at $(h, k)$, then the equation will include a horizontal shift: $g(x) = -f(x-h)$ Based on the graph, the vertex of $g(x)$ seems to be at $(2, 0)$. So, the equation becomes: $g(x) = -(x-2)^2$

Step 3: Vertical Shift

Next, there is a vertical shift. The function $g(x)$ appears to have been shifted downward by 4 units. So, the final equation is: $g(x) = -(x-2)^2 - 4$

Thus, the equation for $g(x)$ is: $\boxed{g(x) = -(x-2)^2 - 4}$

Would you like more details or have any questions? Here are some related questions you might consider:

1. What would be the effect of shifting $f(x)$ to the left instead of the right?
2. How does the equation of a function change if it is reflected over the y-axis instead of the x-axis?
3. What is the general form of the equation for a parabola that has been reflected, shifted, and stretched?
4. How do vertical stretches and compressions affect the graph of $f(x) = x^2$?
5. If $g(x) = -(x-2)^2 + 3$, what transformations have been applied to $f(x) = x^2$?

Tip: When analyzing transformations, always start with reflections and shifts before considering stretches and compressions.