Assignment Three
**The Joy of Parabolas**
Collaborative Effort by
Lucia Zapata and Claudette Tucker

We will explore the quadratic formula
of

Let's examine this equation when b
= -3, -2, -1, 0, 1, 2, 3.
What can you infer from the graph above?
The graph always passes through the
same point, (0, 2) on the y-axis.
When b < -2, the parabola will have
two positive real roots.
When b = - 2 and b = 2, the parabola's
tangent is the x-axis.
For -2 < b > 2, the parabola
does not intersect the x-axis, which implies that the original
equation has no real roots.

How many real roots are there a b =
-2 and b = 2? Are the roots positive or negative?
What happens when b > 2?

Do you think that the vertices of these
parabolas share something in common?
Yes, they have something in common.
We will prove this by showing that the locus is the parabola .
Before we examine their relationship,
we should define locus. Click
here for definition
Let's use this information to explore
the relationship of the parabolas.

Let's see if we can pick some points
to verify the locus.
We will the use the points (0, 1), (-1, 0),
and (1,0) to form a system of equations using quadratic formula,

This implies that c = 1.

This implies that a - b + c = 0.

This implies that a + b + c = 0.

We will use c = 1 and the sum of the
second and third equations to solve our system of equations.
a - b + c + a + b +c = 0 This implies
that 2a + 2c = 0. Since we know that c = 1, we can solve for a.
a = -1. Let's use the facts to determine the value of b. -1 +
b +1 = 0 This implies that b = 0. Thus, is the equation that passes
through the points (0, 1), (-1, 0), and (1,0).

Let's examine the graph of and .
Therefore the locus of the vertices
of the set of parabolas form a parabola as well at .
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