Locus of the Vertices of Parabolas

The relationship between y = ** a**x

By Pei-Chun Shih

From
the previous write-up, we found that the locus of the vertices of the set of
parabolas graphed from y = x^{2} + bx + 1 is the locus of the
parabola y = -x^{2} + 1:

[The red parabola, y = - x^{2} + 1, connects
the vertices of the rest parabolas]

It
looks like there exists relationship between the equation y = ** a**x

y =
2x^{2} + bx + 1 & y = -2x^{2} + 1 (let ** a** = 2,

The
equation y = -2x^{2} + 1 passes through the following points: (-1, -1),
(1, -1), (-0.75, -0.125), (0.75, -0.125), (-0.5, 0.5), (0.5, 0.5), (-0.25,
0.875), (0.25, 0.875), and (0, 1) which are the vertices of the parabolas of
equation y = 2x^{2} + bx + 1, with b = 0, ±1, ±2, ±3, and ±4.

y =
-3x^{2} + bx + 2 & y = 3x^{2} + 2 (let ** a** = -3,

Again,
the equation y = 3x^{2} + 2 passes through the points which are the
vertices of the parabolas of equation y = -3x^{2} + bx + 2, with b = 0,
±1, ±2, ±3, and ±4.

Therefore,
we can generalize the equation of the locus, for any equation of the form y = ** a**x

y =
-** a**x

Well,
generalizing by graphics along is not complete enough for a rigorous proof in
mathematics. Therefore, here I am going to use algebra to prove the
relationship between the equations of y = ** a**x

As I
have discussed in my previous write-up 2 that the standard form of the equation
of a parabola with a vertical axis of symmetry can be written as (x - h)^{2}
= 4p (y - k), where (h, k) is the vertex, (h, k + p) is the focus, x = h is the
axis of symmetry, and y = k – p is the directrix. Since we have known
that the equation, y = ** a**x

y =
ax^{2} + bx + c

ax^{2}
+ bx = y – c

x^{2}
+x = (y – c)

x^{2}
+ 2* x *+ ()^{2} = (y – c) + ()^{2}

(x+)^{2} = (y – c+)

(x+)^{2} = 4*[ y – (c -)]

Therefore,
h = -, k = c -, and the vertex of y = ** a**x

Since
we want to prove that every vertex from the equation y = ** a**x

(-, c -) into the equation y = -** a**x

y =
-** a**x

Let
x = - and y = c -.

c - = -** a***(-)

c - = -+ *c*

Thus,
0 = 0. This confirms that every point of the form (-, c -) is on the equation y = -** a**x

By
the algebraic statement above, we have proved that the locus of the vertices of
the set of parabolas graphed from

y = ** a**x