Which of the following best describes the transformation that occurs when changing an equation from f(x)=x^2+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 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.

Answer 1

To understand the transformation from \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we need to analyze the changes made to the equation.

1. Identifying the transformations:

   • The expression \( (x - 3) \) indicates a horizontal shift. Specifically, it means the function is shifted to the right by 3 units (the general rule is that \( (x - h) \) shifts the graph \( h \) units to the right).
   • The \( +1 \) at the end does not affect the horizontal transformation but indicates that the function is shifted vertically upwards by 1 unit.

2. Shape of the graph:

   • The shape of the graph remains the same because the quadratic function has not been altered in terms of its coefficient in front of the squared term; it's still essentially a parabola.

With this understanding, the correct description of the transformation is:

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