Which of the following best describes the transformation that occurs when changing an equation from f(x)=x2+1 to f(x)=(x−3)2+1?(1 point)

Question

Responses

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

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

Answer 1

To analyze the transformation from the equation \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can break it down step by step.

1. The original function \( f(x) = x^2 + 1 \) is a parabola that opens upward with its vertex at the point \( (0, 1) \).
2. In the new function \( f(x) = (x - 3)^2 + 1 \), the expression \( (x - 3) \) indicates a horizontal shift. Specifically, substituting \( x - 3 \) means the graph shifts to the right by 3 units.
3. The "+1" at the end does not affect the shape or width of the parabola, but it is important to note that it maintains the same vertical position of the vertex.

Thus, the correct description of the transformation is:

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