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 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 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.

Answer 1

To analyze the transformation from the original equation \( f(x) = x^2 + 1 \) to the new equation \( f(x) = (x - 3)^2 + 1 \):

1. The term \((x - 3)\) indicates a horizontal shift. Specifically, subtracting 3 from \(x\) shifts the graph 3 units to the right.
2. The \(+1\) at the end does not affect the horizontal position of the graph; it maintains the same vertical position.
3. The shape of the graph remains a parabola (it is still a quadratic function), and there are no changes to the width because the coefficient in front of the squared term is still \(1\).

Thus, the description that fits this transformation is:

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