Which of the following best describes the transformation that occurs when changing an equation from: f (x) = x^2 + 1, to: f (x) = (x - 3)^2 + 1.

Question

A. The graphed function would shift 3 units to the right but otherwise have the same shape.
B. The graphed function would shift 3 units to the right and would be half as wide.
C. The graphed function would shift 3 units to the left and would be half as wide.
D. The graphed function would shift 3 units to the left but otherwise have the same shape.

Answer 1

To analyze the transformation from the equation \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can break down the components:

1. The original function \( f(x) = x^2 + 1 \) is a standard parabola that opens upwards and is shifted up by 1 unit.

2. The transformed function \( f(x) = (x - 3)^2 + 1 \) can be understood by recognizing that the term \( (x - 3) \) indicates a horizontal shift.

In this case, the transformation \( (x - 3) \) means that the graph of the function shifts 3 units to the right.

The shape of the parabola remains the same because neither the coefficient of \( x^2 \) nor the vertical shift (the "+1") has changed. The width of the parabola is also unchanged, as no vertical stretch or compression is applied.

Thus, the best description of the transformation is:

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