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

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

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 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 but otherwise have the same shape.

GPT-4o mini · Answer

To analyze the transformation from $ f(x) = x^2 + 1 $ to $ f(x) = (x - 3)^2 + 1 $, we can focus on the changes in the equation. 1. The term $ (x - 3) $ indicates a horizontal shift. Specifically, the function shifts to the right if we subtract a positive number from $ x $. In this case, since we have $ -3 $, the function shifts 3 units to the right. 2. The vertical component $ +1 $ remains the same in both functions, indicating that the vertical position does not change with the y-coordinate of the vertex. The shape of the parabola does not change, as the coefficient of the squared term remains the same (which is implicit in both equations as 1). Putting this all together, the correct description of the transformation is: **The graphed function would shift 3 units to the right but otherwise have the same shape.**