Which of the following best describes the transformation that occurs when changing an equation from f(x)=x2+1

Question

to f(x)=(x−3)2+1
?(1 point)
Responses

The graphed function would shift 3 units to the right but otherwise have the same shape.
The graphed function would shift 3 units to the right but otherwise have the same shape.

The graphed function would shift 3 units to the left and would be half as wide.
The graphed function would shift 3 units to the left and would be half as wide.

The graphed function would shift 3 units to the right and would be half as wide.
The graphed function would shift 3 units to the right and would be half as wide.

The graphed function would shift 3 units to the left but otherwise have the same shape.

Answer 1

To analyze the transformation from the function \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can consider how the components of the function change.

1. The original function \( f(x) = x^2 + 1 \) is a parabola that opens upwards with its vertex at the point (0, 1).
2. In the transformed function \( f(x) = (x - 3)^2 + 1 \), the term \( (x - 3) \) indicates a horizontal shift of the graph.
3. Specifically, the function \( (x - 3)^2 \) shifts the graph 3 units to the right compared to the original function, while the \( +1 \) maintains the same vertical position (the y-value doesn't change).

There are no changes to the width of the parabola since the coefficient of \( x^2 \) remains 1 in both cases.

Therefore, the correct response is:

The graphed function would shift 3 units to the right but otherwise have the same shape.