https://file.aiquickdraw.com/mathbot/file/1726358530618dvr0rbrh.jpg

To solve this, let's analyze the transformation and graphs.

The transformation from \( f(x) = x^2 \) to \( g(x) = (x + 2)^2 \) represents a **horizontal translation**.

### Step-by-step analysis:

1. The given function \( f(x) = x^2 \) is a basic parabola with its vertex at \( (0, 0) \).
2. The function \( g(x) = (x + 2)^2 \) shifts the graph of \( f(x) \) **2 units to the left**, since adding 2 inside the parentheses translates the graph left by 2 units.

So, the answer to the sentence would be:
- **The graph of g is a translation 2 units to the left of the graph of f.**

### Identifying the graphs:

- In the top graph (option 1), the red graph is shifted to the right, which is incorrect because \( g(x) \) should be shifted left.
- In the bottom graph (option 2), the red graph (labeled \( g \)) is shifted **2 units left** compared to the blue graph (labeled \( f \)), which is correct.

Thus, **the correct graph is the second option (bottom graph)**, where the red parabola represents \( g(x) \) and the blue one represents \( f(x) \).

Would you like a further breakdown or clarification on any part of this?

Here are some follow-up questions to deepen your understanding:
1. What happens if the equation changes to \( g(x) = (x - 2)^2 \)?
2. How would the graph change if we modify \( g(x) = x^2 + 2 \)?
3. Can you describe the effect of a vertical stretch or compression on \( f(x) \)?
4. How would the graph change for \( g(x) = -(x + 2)^2 \)?
5. What if \( g(x) = (x + 2)^2 + 3 \)? What combined transformations occur?

**Tip:** Remember that translations inside parentheses affect the horizontal shift, while additions or subtractions outside parentheses affect vertical movement.

Describe the transformation of f(x) = x^2 represented by g(x) = (x + 2)^2. Identify the graph of each function.

Explore the transformation of the quadratic function f(x) = x^2 into g(x) = (x + 2)^2. Learn how the graph shifts 2 units to the left and identify the correct graph in this algebraic transformation problem, suitable for students in Grades 9-10.

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