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.† Letís 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.
Letís 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.† Letís look at the equation when c=-1, i.e. letís 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.
Letís 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.