To write the function rule for $g(x)$, we use the vertex form of a quadratic function:

$g(x) = a(x-h)^2 + k$

Here:

• $h$ and $k$ represent the vertex of the parabola.
• $a$ determines the vertical stretch or compression and direction of the parabola.

Given that the vertex is $(0, -1)$, we substitute $h = 0$ and $k = -1$:

$g(x) = a(x-0)^2 - 1$ $g(x) = ax^2 - 1$

Next, we determine the value of $a$ by using another point on the graph. If a specific point on the graph is given (e.g., $(1, y)$), substitute it into $g(x)$ to find $a$. Let me know if additional points are provided or confirm the graph's properties, and I’ll complete the solution.

Would you like detailed steps or help confirming values for $a$?

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Questions for further exploration:

1. What happens to $g(x)$ if $a$ is a fraction?
2. How do translations affect the vertex form of a quadratic?
3. How do you verify $g(x)$ matches the graph using algebra?
4. Can you convert this function into standard form?
5. How would $g(x)$ change if the vertex were shifted to $(h, k)$?

Tip: Always check points other than the vertex to determine the stretch/compression factor $a$.