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.