https://file.aiquickdraw.com/mathbot/file/1729368688874rxdq4sk5.jpg

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.

Write the expression for g(x) given that f(x) = (1/2)x^2 is translated horizontally and vertically.

Learn how to find the translated function g(x) from the original quadratic function f(x) = (1/2)x^2. Understand horizontal and vertical shifts using transformations, and write the new function g(x). Perfect for Grades 9-11 students.

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