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

To analyze the transformation from \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we first recognize that the general form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

1. Original Function: \( f(x) = x^2 + 1 \)

   • This function has a vertex at \( (0, 1) \).

2. Transformed Function: \( f(x) = (x - 3)^2 + 1 \)

   • This function has a vertex at \( (3, 1) \).

The transformation involves moving the vertex from \( (0, 1) \) to \( (3, 1) \), which is a shift to the right by 3 units. Additionally, the shape of the graph remains the same since the coefficient of the squared term (which is 1) is unchanged and is the same as in the original function.

Thus, the correct description is:

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