F(x)=x^2+1 to f(x)=(x-3)^2+1 which of the following best describes the transformation that occurs

Question

The graph function was shift 3 units to the left and would be half as wide the graph function would shift 3 units to the right but otherwise have the same shape the graph function would shift 3 units to the left left but otherwise had the same shape the graph function would shift three units to the right and would be half as wide

Answer 1

To analyze the transformation of the function \( F(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can break down the changes step by step.

1. Original Function: \( F(x) = x^2 + 1 \)

   • This is a standard parabola that opens upwards with its vertex at the point (0, 1).

2. Transformed Function: \( f(x) = (x - 3)^2 + 1 \)

   • Here, we see that the expression \( (x - 3) \) indicates a horizontal shift. Specifically, the function has been shifted 3 units to the right.

3. Shape of the Function:

   • The transformation does not affect the width of the parabola. Since the coefficient in front of the squared term remains 1, the width remains the same.

Given this analysis, the correct description of the transformation is:

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

This option accurately captures both the horizontal shift and the fact that the shape of the graph remains unchanged.