The function $g(x)$ is a translation of $f(x) = x$, which implies that $g(x)$ involves shifting, stretching, compressing, or reflecting $f(x)$. Let's analyze the given information.

1. Vertex: The vertex is at $(0, 6)$. This means $h = 0$ and $k = 6$ in the form $a(x - h) + k$.

2. Parent function: Since $f(x) = x$, the transformation depends only on scaling ($a$) and vertical or horizontal shifts.

3. Rule for $g(x)$: Given $h = 0$ and $k = 6$, and assuming no further transformations (e.g., stretches or compressions), the rule for $g(x)$ becomes:

$g(x) = x + 6$

Thus, the function rule for $g(x)$ is:

$g(x) = x + 6$

If you have additional details about $g(x)$ (e.g., stretches or reflections), please share them for further refinement.

Would you like a detailed explanation of how these transformations work?

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Expanding Questions

1. How does shifting affect the graph of $f(x) = x$?
2. What role does the $h$ value play in the vertex form of a linear equation?
3. How would a reflection over the x-axis affect $g(x)$?
4. What are the differences between vertical and horizontal shifts?
5. Can we describe $g(x)$ if $a \neq 1$?

Tip:

Always observe the vertex and slope changes in a transformation to determine the precise effect on the function.