Graphs in the xb Plane

LetŐs look at the following equation in the xb plane.

x^{2}+bx+1=0

When using Graphing Calculator, this equation will
appear as

x^{2}+yx+1=0

LetŐs look at the following graph when c=1.

What happens as the value of c changes?

LetŐs see.

We see that as the value of c is increased the graph
shifts upward in a positive direction and downward in a negative.

What happens when c is negative?

We see that a negative value of c alters the graph
completely.

Now, letŐs put all of our different values of c on the
same graph.

We also see that when c=0 the graph becomes an
asymptote in the xb plane. This line crosses throw
the graphs where c=1 and c=-1.

Now lets look at particular values of b (or y in our
case). The number of times the horizontal line y intersects the curve
corresponds to the number of roots the value has. LetŐs look at some different
values of y.

From this graph we see that when y>2 and y<-2 the
value will have 2 roots, y=2 will have one root, and -2<y<2 to value will
have no real roots.

LetŐs graph some more values of c.

We see here that we have a family of hyperbolas with
an asymptote when c=0.