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**Exploring Loci of Parabolas**

**By **

**Princess Browne**

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We will
look at the graph of y = ax^{2} + bx + c

(b is varied
and a and c are held constant)

First, we will graph the equation where a, b, and c are
equal to 1.

Like any parabola the graph
is concave upward when a is positive. Therefore the graph will be concave
downward when a is negative. Because c = 1 the graph is not symmetrical around
the y-axis. The parabola passes through the points (0.1) because c = 1. When x
= 0, the graph would be symmetrical around the y-axis. Next, we will graph the
equation where b is varied and a and c remain 1.

Yellow

Purple

Red

Blue

Green

Teal

The following graphs intersect
the y-axis at (0,1) because c = 1. The vertex is always left of the graph when
b is greater than 0. When b = 0 the vertex is on the x-axis. The real root of
the equation occurs where the parabola intersects the x-axis. The graph of y =
x^{2 }+ 2x + 1 have a real root because it intersect the x-axis. The
graph of y = x^{2 }+ 1x + 1 does not have any real root because it does
not intersect the x-axis. When b is positive the real roots occur on the
negative side of the y-axis.

Green

Purple

Red

Teal

Yellow

When b is negative the graphs intersect the y-axis at
(o,1). The vertex is to the right of the graph and the vertex is on the y-axis
when b = 0. The graph have no real root when b = -1.

Purple y = x^{2 }+ 4x
+ 2

Red y
= 2x^{2 }+ 4x + 2

Blue y =
3x^{2 }+ 4x + 2

Green y = - x^{2 }+
4x + 2

Teal y =
-2x^{2 }+ 4x + 2

Yellow y = -3x^{2 }+ 4x + 2

From the picture above, we can
conclude that each parabola passes through the point (0,2). When a is positive
the parabola is concave up and the parabola is concave down when a is negative.
The x-coordinate of the vertex is x = -b/2a.