The problem is asking you to write down the expression for the function $g(x)$, which is a translation of the function $f(x) = \frac{1}{2}x^2$.

From the graph, it appears that the function $f(x)$ has been translated both horizontally and vertically. Here's how we can deduce the translation:

1. The original function $f(x) = \frac{1}{2}x^2$ is a parabola with its vertex at $(0, 0)$.
2. In the graph of $g(x)$, the vertex of the parabola has shifted to a new point. From the image, it seems that the vertex has moved to $(-2, -4)$.

Horizontal and Vertical Translations:

• Horizontal shift: The function has shifted 2 units to the left. For a horizontal shift, the new function becomes $f(x + 2)$.
• Vertical shift: The function has also shifted 4 units down. For a vertical shift, we subtract 4 from the function, so it becomes $f(x + 2) - 4$.

Putting it together:

The function $g(x)$ can be written as: $g(x) = \frac{1}{2}(x + 2)^2 - 4$

This represents the function after both horizontal and vertical translations.

Would you like any further details or clarification on this?

Here are 5 related questions you might find interesting:

1. How do you apply translations to functions in general?
2. How would the function change if the translation was 3 units to the right and 5 units up?
3. What is the effect of a negative coefficient in front of the quadratic term?
4. How do you determine the vertex of a parabola from its equation?
5. How can you find the axis of symmetry of the parabola $g(x)$?

Tip: When translating functions, always remember that horizontal shifts affect the $x$-term inside the function, while vertical shifts affect the entire function outside of the $x$-term.