Question

Question

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)
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 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 right and would be half as wide.
The graphed function would shift 3 units to the right and would be half as wide.
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Answer 1

To analyze the transformation from the function \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \):

1. Identify the transformation: The term \( (x - 3) \) indicates a horizontal shift. Specifically, when you replace \( x \) with \( x - 3 \), it shifts the graph to the right by 3 units.

2. Remaining terms: The \( +1 \) at the end indicates that the graph is also shifted upward by 1 unit, but in both equations, this upward shift is the same. Thus, it does not affect the comparison of the shifts.

3. Shape of the graph: Both functions are parabolas (quadratic functions) with the same shape, as their leading coefficients are the same (which is 1 in both cases).

Given this analysis, the correct transformation description from the given choices is:

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