To write the function $g(x)$ as a translation of $f(x) = x^2$ in the form $a(x-h)^2 + k$, we need to identify $a$, $h$, and $k$.

Step-by-Step Solution:

1. Vertex Form: The equation of $g(x)$ is in the vertex form: $g(x) = a(x-h)^2 + k$ Here, $(h, k)$ is the vertex. From the problem, the vertex is $(0, -1)$, so $h = 0$ and $k = -1$.

2. Transformation: Since the vertex is at $(0, -1)$, the graph is translated down by 1 unit, but it remains symmetric about the $y$-axis (no horizontal stretch or compression). This means $a = 1$ (no vertical stretch or compression).

3. Equation: Substitute $h = 0$, $k = -1$, and $a = 1$ into the equation: $g(x) = 1(x-0)^2 - 1$

   Simplify: $g(x) = x^2 - 1$

Final Answer:

$g(x) = x^2 - 1$

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Would you like me to explain transformations in more detail? Here are some related questions to explore further:

1. What does changing $a$ do to the graph of $g(x)$?
2. How would the equation change if the vertex were at $(2, 3)$?
3. What happens to the graph when $h$ and $k$ are negative?
4. How can you determine the direction of the parabola from $a$?
5. What is the role of the vertex in determining translations?

Tip: Always rewrite equations step by step to ensure accuracy, especially when identifying transformations like vertex shifts or stretches.