Which statement describes the effect on the parabola y equals x squared plus 1 when it is changed to y equals 2 x squared plus 1

Question

Which statement describes the effect on the parabola y equals x squared plus 1 when it is changed to y equals 2 x squared plus 1

GPT-4o mini · Answer

The parabola described by the equation $ y = x^2 + 1 $ opens upwards and has its vertex at the point (0, 1). When the equation changes from $ y = x^2 + 1 $ to $ y = 2x^2 + 1 $, the following effects can be observed: 1. **Vertical Stretch**: The coefficient of $ x^2 $ changes from 1 to 2. This means the parabola is vertically stretched by a factor of 2. As a result, for any given value of $ x $, the $ y $ value of the parabola will be double that of the original parabola. 2. **Vertex Positioning**: The vertex remains at the same horizontal position (x = 0) and the same vertical position (y = 1) since there are no changes to the constant term (1) or the linear term. In summary, the effect on the parabola is that it becomes narrower due to the vertical stretch, while its vertex remains unchanged at (0, 1).