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.