In the image provided, we are asked to find the equation for the function $g(x)$, which is obtained from the function $f(x) = x^2$ by a sequence of transformations.

By observing the graph, it seems that the function $g(x)$ (blue graph) is a transformation of $f(x)$ (black graph). The transformations include:

1. Reflection across the x-axis: This changes the sign of the function, making $g(x) = -f(x)$, or $g(x) = -x^2$.
2. Horizontal shift: The vertex of $g(x)$ is shifted to the right. The vertex of $f(x)$ is at the origin $(0, 0)$, while the vertex of $g(x)$ seems to be at $x = 5$, so this involves a shift 5 units to the right, giving the transformation $g(x) = -(x - 5)^2$.
3. Vertical shift: Finally, the graph of $g(x)$ is shifted downward by 5 units, leading to the transformation $g(x) = -(x - 5)^2 - 5$.

Thus, the equation for $g(x)$ is: $g(x) = -(x - 5)^2 - 5$

Would you like more details on these transformations or further explanation? Here are 5 related questions:

1. How would the equation change if the graph were shifted upward instead of downward?
2. What is the effect of reflecting a function across the y-axis instead of the x-axis?
3. Can you identify the transformation steps if the vertex of the transformed function is at $(-3, -2)$?
4. How do we perform vertical stretching or shrinking on a quadratic function?
5. How can we combine multiple transformations into a single equation?

Tip: When identifying transformations, always look at changes to the vertex and orientation of the graph as a starting point.