**Further Explorations with Parabolas**

By: Lauren Lee

**b = -3****, b = -2, b = -1, b = 0, b = 1, b = 2, b = 3**

We
can see that all of the parabolas pass through the same point on the y-axis,
(0, 1). We will also notice for* ***b = 0**, the parabola is centered
around the y - axis. Also we noted
before that as the value of b decreases, the parabola dips further down to the
right. Similarly, as b increases, the
parabola dips further down to the left.

Now let’s explore the locus
of the vertices of these parabolas that are graphed. The vertices are:

(1.5, -1.25) for
b = -3

(1, 0) for b = -2

(0.5, 0.75) for
b = -1

(0, 1) for b = 0

(-0.5, 0.75) for
b = 1

(-1, 0) for b =
2

(-1.5, -1.25)
for b = 3

Let’s
plot the vertices and see if we can make any conjectures.

First notice that the vertices appear to form an upside-down parabola. Also from looking at the vertices, we can see that the roots of this new parabola are x = -1 and x = 1.

Let’s try to find the equation of this concave down parabola.

Using our original equation, y = ** a**x

y = (x + 1)(x – 1)

By
simplifying we get y = x^{2} - 1.

But
since parabola is concave down, the locus of the vertices when ** a **=
1 and