The given graph shows $g(x)$, which is a translation of the parent function $f(x) = x^2$. The general form of the function is:

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

Where:

• $h$ is the horizontal shift (from the vertex $(h, k)$).
• $k$ is the vertical shift.
• $a$ determines the vertical stretch or compression.

From the vertex $(0, -1)$:

• $h = 0$ (no horizontal shift),
• $k = -1$ (shifted down by 1 unit).

Assuming $a = 1$ (no vertical stretch/compression or reflection), the function rule is:

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

If you'd like to confirm the value of $a$ or delve further, let me know!

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Do you want a detailed step-by-step explanation or graph comparison?
Here are 5 related questions for deeper exploration:

1. How do you find $a$ if $g(x)$ is not simply stretched/compressed?
2. What happens to the graph of $f(x) = x^2$ when $h$ or $k$ changes?
3. How would $a$ affect the width or direction of the parabola?
4. How do you verify a transformation using points from the graph?
5. Can we graph $g(x) = x^2 - 1$ and compare it to $f(x) = x^2$?

Tip: Always check the vertex and one more point to ensure accuracy in transformations!