Which of the following best describes the transformation that occurs when changing an equation from f(x)=x2+1

Question

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 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 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 but otherwise have the same shape.

Answer 1

To understand the transformation from the function \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can analyze the changes in the equation.

1. The original function \( f(x) = x^2 + 1 \) is a standard quadratic function that opens upwards and has a vertex at the point (0, 1).
2. The new function \( f(x) = (x - 3)^2 + 1 \) shows that the \( x \) variable has been replaced with \( x - 3 \). This indicates a horizontal shift of the graph.
3. When we replace \( x \) with \( x - 3 \), the graph shifts to the right by 3 units. The \( +1 \) at the end indicates that the vertical position of the graph is not changed.

Additionally, neither equation involves a vertical stretch or compression since the coefficient of \( (x-3)^2 \) is still 1.

Based on this information, the correct response is:

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