Question

Question

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

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.

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.

Answer 1

The correct response is:

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

To elaborate, when transforming the function from \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), the \( (x - 3) \) indicates a horizontal shift to the right by 3 units, while the shape of the graph remains the same since there is no change in the coefficient in front of \( x^2 \) that would affect the width. The "+1" adjusts the vertical position but does not affect the horizontal shift.