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 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 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 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

The transformation from the equation \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \) involves shifting the graph.

In the function \( f(x) = (x - 3)^2 + 1 \), the expression \( (x - 3) \) indicates a horizontal shift. Specifically, subtracting 3 from \( x \) shifts the graph to the right by 3 units.

Additionally, both functions maintain the same shape since they are both quadratic functions (parabolas) with the same vertical stretch.

Therefore, the correct description of the transformation is:

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