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Let's break down the given transformation.

You are asked to describe the transformation of \( f(x) = x^2 \) represented by \( g(x) = 3(x - 1)^2 \).

1. **Step 1: Stretching** – The equation \( g(x) = 3(x - 1)^2 \) introduces a vertical stretch. The factor of 3 causes a vertical stretch by a factor of 3.
   
2. **Step 2: Translation** – The \( (x - 1)^2 \) inside the parentheses indicates a horizontal translation. Specifically, it translates the graph 1 unit **right** (since \( x - 1 \) means shifting right).

Thus, the graph of \( g \) is a **vertical stretch by a factor of 3** followed by a translation 1 unit **to the right** of the graph of \( f \).

### Identifying the graphs:
- The graph labeled as \( g \) should be stretched vertically compared to \( f \), and it should be shifted to the right by 1 unit.

Now, let's analyze the provided graphs:

- **First graph**: It shows two parabolas, where \( g(x) \) (red) appears steeper than \( f(x) \) (blue), and shifted slightly to the right.
- **Second graph**: The red graph \( g(x) \) is wider than \( f(x) \), which does not match the stretch by 3.
- **Third graph**: \( g(x) \) is steeper but shifted incorrectly.

Thus, the correct answer is the **first graph**.

Would you like further details or have questions? Here are some follow-up questions for better understanding:

1. What would happen if the coefficient 3 was replaced with 1/2?
2. How does a horizontal translation affect the vertex of a parabola?
3. What is the general effect of a vertical stretch on a function?
4. How would the transformation change if the translation was left instead of right?
5. What other transformations could be applied to a quadratic function?

**Tip:** When identifying transformations, always look at vertical stretches/compressions and horizontal shifts independently to break down the function's behavior.

Describe the transformation of f(x) = x^2 represented by g(x) = 3(x - 1)^2. The graph of g is a __ followed by a translation 1 unit __ of the graph of f. Identify the graph of each function.

Explore the transformation of the quadratic function f(x) = x^2 to g(x) = 3(x - 1)^2, including vertical stretching and horizontal translation. Learn how to identify the correct graph and analyze changes in function behavior. Suitable for students in Grades 8-10.

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