In the set of the parabolas which are graphed from

 y = x2 + bx + 1, why the locus of the vertices is the locus of the parabola y =  - x2 + 1?

By

Ronnachai  Panapoi

 

 

 

 

          Firstly, let’s talk about the definition of the parabola.

 

          A parabola is defined as the locus of the points to which the distance from a given points (the focus) and a given line (the directrix) are equal.

 

          Now, what we have to know are the focus and the directrix of the parabola y = - x2 + 1.

 

          Since this parabola with the axis which is parallel to y-axis, it must be in the form y – h = 4c(x – h)2 such that

(h, k) is its vertex and c  is the distance from the vertex to the focus.

         

          Also, we obtain that the directrix of this parabola is the line: y = k – c and the focus at (h, k + c).

 

          We, then, rearrange the form of the parabola y = -x2 + 1 into. Therefore, this parabola has the

line:  as the directrix and the  as the focus.

 

          Looking at the equation , we obtain that it is the parabola with the vertex .

 

          We want to show that this vertex is the locus of the parabola y = -x2 + 1.

          If it is true, the distance from this vertex to the line:   and to the focus  is the same according to the definition of a parabola.

 

          Consider, the distance () between vertex and the directrix.

 

          Consider, the distance () between vertex and the focus.

 

                                                                          =

                                                                          =

                                                                          =

                                                                   

                             Thus, .

                   This suggests that the locus of the vertices of the set of parabolas graphed from  y = x2 + bx + 1 is the locus of the parabola  y = - x2 + 1.

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