Which of the following best describes the transformation that occurs when changing an equation from f(x)=x2+1 to f(x)=(x−3)2+1?(1 point)

Question

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 and would be half as wide.

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 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 left but otherwise have the same shape.
The graphed function would shift 3 units to the left but otherwise have the same shape.
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Answer 1

To analyze the transformation from \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can focus on the change in the \( x \) variable.

1. The original function, \( f(x) = x^2 + 1 \), is a standard parabola that opens upwards and is vertically shifted 1 unit up.
2. The modified function, \( f(x) = (x - 3)^2 + 1 \), indicates a horizontal shift of the graph. The expression \( (x - 3) \) means that every point on the graph of the original function will move 3 units to the right.

The "+1" in the equation remains the same, indicating that the vertical shift does not change.

Thus, the best description of the transformation is:

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