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 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.
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

The correct response is:

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

Here's why:

• The original function is \( f(x) = x^2 + 1 \).
• The transformed function is \( f(x) = (x - 3)^2 + 1 \).

The term \( (x - 3) \) indicates a horizontal shift to the right by 3 units. The \( +1 \) outside the squared term does not affect the width of the parabola; it simply shifts the graph vertically. Therefore, the shape of the parabola remains the same, confirming that the transformation is a rightward shift of 3 units without changing the width.