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The problem asks to find the equation for the function \( g \), which is obtained from \( f \) through a sequence of transformations. From the graph:

- The function \( f \) appears to be a parabola with its vertex at \( (0,0) \), suggesting a form of \( f(x) = x^2 \).
- The function \( g \) is a transformation of \( f \), and its vertex is shifted to \( (6, 4) \).

To determine the transformation from \( f(x) = x^2 \) to \( g(x) \), consider the following sequence:

1. **Horizontal shift**: The vertex has shifted from \( (0,0) \) to \( (6,4) \), which suggests a horizontal shift by \( +6 \). This implies the transformation \( f(x) \) is shifted horizontally: 
   \[
   f(x - 6) = (x - 6)^2
   \]

2. **Vertical shift**: The vertex has also shifted vertically by \( +4 \), so the entire graph is moved upwards by 4 units. The transformation is:
   \[
   g(x) = (x - 6)^2 + 4
   \]

Thus, the equation for \( g(x) \) is:
\[
g(x) = (x - 6)^2 + 4
\]

This is the equation for the function \( g \) after the transformations.

Would you like a more detailed explanation, or do you have any further questions?

Here are some related questions to explore further:
1. How do horizontal and vertical shifts affect the graph of a function?
2. Can transformations change the shape of a parabola or just its position?
3. How would the equation change if there were a reflection involved?
4. What if the function \( f(x) \) had a different vertex? How would you account for that?
5. Can you describe the sequence of transformations for a more complex function?

**Tip**: Whenever you're unsure about transformations, focus first on horizontal and vertical shifts, and then consider reflections and stretches.

The functions f and g are graphed in the same rectangular coordinate system. If g is obtained from f through a sequence of transformations, find an equation for g.

Learn how to find the equation of a function obtained through transformations. This problem involves understanding horizontal and vertical shifts of a parabola, with a detailed step-by-step explanation of how g(x) = (x - 6)^2 + 4 is derived.

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