The graph in the image represents a transformation of the function $x^2$. Based on the vertex and transformations visible, it appears that the function has been shifted to the right by 1 unit and up by 3 units from the standard position of $x^2$, which has its vertex at the origin (0,0).

The correct equation for the graphed function would be: $g(x) = (x-1)^2 + 3$

This equation reflects that the quadratic parabola has been translated right by 1 unit (hence, $x-1$) and lifted vertically by 3 units (hence, $+3$).

Would you like to know more about this transformation or any other related mathematical concepts? Here are a few follow-up questions you might find interesting:

1. What happens if you further translate the function $g(x) = (x-1)^2 + 3$ to the left by 4 units?
2. How would the graph change if the equation was modified to $g(x) = -(x-1)^2 + 3$?
3. What does the coefficient of $x^2$ affect in the graph of a quadratic equation?
4. Can you explain how to find the axis of symmetry for the graph of $g(x)$?
5. What are the intercepts of the graph $g(x) = (x-1)^2 + 3$?

Tip: When transforming a function, always remember that moving the graph horizontally involves changes inside the function's brackets, while vertical movements adjust the constant term outside the brackets.