To determine the correct formula for the transformation of the function $f(x) = x^2$ based on the graph, let's analyze the transformations applied to the original function $f(x) = x^2$.

Observations from the Graph:

1. The vertex of the original function $f(x) = x^2$ is at $(0, 0)$.
2. In the transformed graph, the vertex has shifted to $(1, -2)$, which means there has been a horizontal shift to the right by 1 unit and a vertical shift down by 2 units.

Transformation Analysis:

For a quadratic function $f(x) = (x - h)^2 + k$, the vertex is located at $(h, k)$. Thus:

• A horizontal shift to the right by 1 unit implies $h = 1$, so the formula should have $(x - 1)^2$.
• A vertical shift down by 2 units implies $k = -2$, so we should subtract 2 from the function.

Conclusion:

The transformed function that best matches the graph is: $f(x) = (x - 1)^2 - 2$

Answer:

The correct choice is B: $f(x) = (x - 1)^2 - 2$.

Would you like more details on how transformations affect the graph? Here are some related questions to explore further:

1. What are the effects of different types of transformations on the graph of a function?
2. How do horizontal and vertical shifts impact the coordinates of the vertex in a quadratic function?
3. What are the steps to identify transformations in other types of functions, such as cubic or absolute value functions?
4. How can you rewrite a transformed function to find its vertex easily?
5. What are some examples of combining multiple transformations in a single function?

Tip: When analyzing transformations, focus on shifts (horizontal/vertical), stretches/compressions, and reflections separately to make it easier to identify the changes applied.