https://file.aiquickdraw.com/mathbot/file/172706617664031dm4ba3.jpg

The problem asks for the transformation rule between two functions, \( f(x) = x^2 \) (red graph) and the blue graph representing \( g(x) \).

### Analyzing the Graphs:
The graph of \( f(x) = x^2 \) is a standard parabola centered at the origin. Looking at the blue graph \( g(x) \), it appears to be a translation of the red parabola.

From the image:
- The blue parabola seems shifted horizontally to the right.
  
### Translation Rule:
A horizontal translation to the right by a units results in the function \( g(x) = f(x - a) \).

### Steps:
1. Identify the horizontal shift by looking at how far the vertex of the blue graph (parabola) moves from the origin (which is the vertex of the red graph).
2. It seems that the vertex of the blue graph is at \( x = 2 \).

Thus, \( g(x) \) is a horizontal translation of \( f(x) = x^2 \) by 2 units to the right. The function for \( g(x) \) would be:

\[
g(x) = (x - 2)^2
\]

This represents the translation of the parabola \( f(x) = x^2 \) two units to the right.

Would you like any further details or explanation?

### Related Questions:
1. What happens if the graph shifts to the left instead of the right?
2. How would you describe a vertical shift instead of a horizontal one?
3. What is the general form of a function that has both horizontal and vertical shifts?
4. How does stretching or compressing a parabola affect its equation?
5. Can you determine the function if the graph was reflected as well?

**Tip:** When identifying transformations, always check if there's any vertical or horizontal shift by analyzing the movement of the vertex.

The graph of f(x) = x^2 is sketched in red, and the graph of g(x) is sketched in blue. Use the translation rule and f(x) = x^2 to identify the function g(x).

Analyze the transformation of the quadratic function f(x) = x^2 to identify g(x) by applying the horizontal translation rule. This problem covers function transformations and is ideal for students in Grades 9-11.

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