The problem asks for the expression of the function $g(x)$ when the graph of the function $f(x) = x^2$ is translated a whole number of units horizontally and vertically.

Analysis of the translation:

1. Horizontal translation: From the graph, we can observe that the graph of $g(x)$ has been shifted 2 units to the left. This horizontal shift is represented by $x \to x + 2$.

2. Vertical translation: The vertex of the parabola is now at $( -2, 0 )$, compared to the vertex of $f(x) = x^2$ which is at $(0, 0)$. It appears that the parabola has been reflected vertically (flipped) but not shifted vertically.

3. Reflection: The graph is reflected across the x-axis. This is represented by the negative sign in front of the expression.

Expression for $g(x)$:

Given these transformations (shift left by 2 and a reflection across the x-axis), the expression for $g(x)$ is:

$g(x) = -(x + 2)^2$

This matches the given answer in the image.

Would you like more details, or have any further questions?

Here are 5 related questions:

1. What effect does reflecting a parabola across the x-axis have on its equation?
2. How do you determine horizontal and vertical shifts from a graph?
3. How would the equation change if the graph was shifted 3 units up instead?
4. What would the equation of the function be if it was shifted 2 units right instead of left?
5. How do you find the vertex of a parabola from its equation in vertex form?

Tip: Always pay attention to the direction of shifts and reflections when transforming a graph!