1. The problem states that the graph of $g(x)$ is a translation of $f(x) = x^2$. 2. The general form for a translated parabola is $$g(x) = a(x - h)^2 + k$$ where $a$ controls the vertical stretch/compression and direction, $h$ is the horizontal shift, and $k$ is the vertical shift. 3. Since the parabola opens upward and is centered on the y-axis, $a = 1$ and $h = 0$. 4. The vertex of $g(x)$ is at $(0, -7)$, so the vertical shift $k = -7$. 5. Therefore, the function rule is $$g(x) = 1(x - 0)^2 - 7 = x^2 - 7$$. 6. This matches the given graph shape and vertex position.