Let's begin our discussion of quadratic equations by considering the following equation and its graph.

First, let's notice that the curve is a parabola.
In ancient times the Greeks studied the parabola along with associated
curves called Conics or Conic Sections. The properties of a parabola
are very interesting as we shall see. In the above graph notice
that the point (0,1). We shall now overlay several parabolas which
will share that same point by choosing different values of **bx**.
We may call the collection of curves a family related to the original
quadratic equation.

The key feature to notice is that all the curves
go through the same point (0,1) even though changing **bx**
has the effect of horizontally shifting the curve parallel to
the x-axis.

Now, recall that the solutions to a quadratic
equation are called the *roots* or *zeros*. Simply,
the solutions of a quadratic can be real numbers, complex numbers,
or a combination of reals and complex. Graphically, if the curve
intersects the x-axis then that intersection point is a real solution
or root of the equation. However, if the curve does not touch
or cross the x-axis then the root would be complex. As an example
look where the following equations intersect the x-axis in the
above graph. The yellow curve has one root at approximately (1.5,
0) and (2.5, 0).

Now, to further our discussion, let's take
a look at the *locus of vertices* of the family of curves.
Think of the the locus as a collection of points and vertices
as the bottom most point in all the above curves. Note,however,
the vertex would be upper most point on the curve if the curve
opens downward. In this graph we will make the family of curves
on one color so see can see how the locus of vertices relates
to the family.

The above graph is telling us that the collection of points which represent the vertices of each of the parabolas traces the blue curve which is defined by the quadratic equation . As with the family of curves, the blue curve also shares the point (0,1)!

Notice, also, that the blue curve opens downward as opposed to the family of curves which all open upward. We, therefore, can conclude that if the first coefficient is positive the curve opens upward and if the coefficient is negative then it opens downward.

Now, returning to the roots of a quadratic
equation, we will explore the **bx** term of the quadratic
equation in more detail. We want to understand the relationship
between this term and the roots of the original equation. On face
value, one may think that the **bx** would not tell us about
a solution of an equation which also includes the **ax^2 **and
**cx** terms. Let's see what occurs when we fix the values
of the two other terms and graph the **bx** term on the *bx
plane.*

The bx plane is similar to plane except that
we allow the bx term to be expressed on its own plane. A simple
way to think of this is we are simply substituting the y-axis
for the b-axis and graphing the quadratic equation by fixing the
values of the two other terms. What we see above is a graph where
the b-axis is an *asymptote*, that is, the curve will come
very close to the b-axis but will never touch or cross the axis.
When the x values are negative the curves lies in second quadrant.
Likewise, when x values are positive the curve lies in the fourth
quadrant.

Let's consider adding an other curve such as b = 3 to the graph. In this case the b = 3 relates to our original equation where the coefficient of b was 3. Where the curve of b = 3 intersects the red curve our points that tell us what the x values are. That is, the x values in the below graph are the x values of the original quadratic equation, i.e. the roots!

As a comparison, let's reproduce the original equation to convince ourselves that the roots are indeed the same. If you draw a line down from the rightmost intersection point to the x-axis you will see the x is approximately -2.5. Likewise, the below graph has a rightmost x-intercept at -2.5! Therefore, we could select different values of b on the bx plane and match the x values to the roots of a corresponding equation on the xy plane.