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We are given two functions:

- \( f(x) = x^2 \)
- \( g(x) = (x - 2)^2 - 4 \)

The transformation from \( f(x) \) to \( g(x) \) involves two shifts:

1. **Horizontal shift**: The expression \( (x - 2) \) indicates a **shift to the right** by 2 units.
2. **Vertical shift**: The constant term \( -4 \) outside the squared term indicates a **shift downward** by 4 units.

Thus, the graph of \( g(x) \) is a translation of the graph of \( f(x) = x^2 \) **2 units to the right** and **4 units downward**.

Now, for the missing terms in the statement:

"The graph of \( g \) is a translation **2 units to the right** and **4 units down** of the graph of \( f \)."

Do you need further explanation or clarification?

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Here are 5 questions to deepen your understanding:

1. What is the general form of a quadratic transformation?
2. How would the graph change if the transformation were \( g(x) = (x + 2)^2 - 4 \)?
3. Can you identify other transformations that affect the shape of the graph, like stretching or reflection?
4. How do translations impact the vertex of a parabola?
5. What are the key features (vertex, axis of symmetry) of the graph of \( g(x) \)?

**Tip**: When identifying transformations, always look for horizontal and vertical shifts, stretches, and reflections by examining the equation's structure.

Describe the transformation of f(x) = x^2 represented by g(x) = (x - 2)^2 - 4. Then identify the graph of each function.

Understand how the graph of f(x) = x^2 is transformed into g(x) = (x - 2)^2 - 4. This problem explores horizontal and vertical translations of quadratic functions, with a focus on algebraic transformations.

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