Graphs in the xb
Plane

By Taylor Adams

To look at graphs in the xb plane, we want to look at the equation

In order to look at the xb plane on a Cartesian plane, we will set b=y, so we are
going to look at the equation on
the x-y plane.

In this equation, c=1. Let’s see what happens when the constant
term, c, is other values. The graph below
has c=1, 3, 5, and 7, respectively.

As the c value increases, the graph is
moving farther away from the x-axis.

Let’s look at the equation again. If we graph a particular value of b, or y, on
top of this graph, it will create a horizontal line parallel to the
x-axis. If this equation for a value of
b crosses the original curve, the intersection will be the roots of the original
equation with that value of b.

For example, if we graph b=4 along with on
the xb plane, the intersection of this line and curve
will tell us the roots of the equation

Therefore, the roots of this equation is

If we look at the graphs of b=4 and , the intersections occur at these
x-values.

If we look at the graph of on
the xb plane, different
values of b will produce a different number of roots.

When b>2, there will be two real
number solutions because the horizontal line will cross the curve twice.

When b=2, there will be one real root
because the horizontal line crosses the curve once.

When -2<b<2, there will be no real
roots because the horizontal lines will not intersect the graph of .

When b=-2, there will be one real root
because the horizontal line will intersect the equation once.

When b<-2, there will be two real
roots because the horizontal line will intersect the equation twice.

We have been looking at the equation .
This is when the constant, c, is equal to one. Let’s look at the equation when c=-1, i.e. let’s
look at the equation .

No matter what equation for a value of b
we graph, there will always be two real roots for the equation because the horizontal line will always
intersect the curve twice.

Let’s graph other values of c for the
equations and

when c=0, 1, 2, 3, 4, 5.

When c=0, it produces the equation .
Therefore, x=0 and x=-b. These
are the asymptotes for our graphs above.

The horizontal line representing a b
value will cross the curves for the equations and either 0, 1, or 2 times. Therefore, depending on the b value in the
equation, the equations will have 0, 1, or 2 real roots.