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.