The problem involves translating the graph of the function $f(x) = \frac{1}{2}x^2$ to obtain the graph of $g(x)$. From the image, we can determine the transformations based on the relative positions of the two parabolas.

Step-by-Step Solution:

1. Graph of $f(x) = \frac{1}{2}x^2$:

   • The original graph has its vertex at $(0, 0)$, and it is symmetric about the $y$-axis.

2. New Graph $g(x)$:

   • The vertex of $g(x)$ is at $(-3, 2)$, which means the graph has been translated 3 units to the left and 2 units up.

3. Translation Rule:

   • A horizontal translation $h$ and vertical translation $k$ modify the function $f(x)$ as: $g(x) = f(x + h) + k$ Here, $h = -3$ (left) and $k = 2$ (up).

4. Substitute $f(x)$:

   • Start with $f(x) = \frac{1}{2}x^2$.
   • Substitute $x + 3$ for $x$, and add 2: $g(x) = \frac{1}{2}(x + 3)^2 + 2$

Final Answer:

$g(x) = \frac{1}{2}(x + 3)^2 + 2$

Let me know if you want a detailed breakdown of any step or if you have further questions.

Related Questions:

1. How do you identify horizontal and vertical translations from a graph?
2. What is the effect of translating a graph horizontally by $h$ units?
3. How does a vertical shift $k$ affect the equation of a function?
4. What would the equation of $g(x)$ be if the vertex were at $(4, -1)$?
5. Can you explain how symmetry affects the translation of parabolas?

Tip:

Always track the vertex when analyzing translations, as it simplifies identifying both horizontal and vertical shifts.