Graphs in the xb plane


by: Stephen Bismarck


Consider the equation

Now graph this relation in the xb plane.

Well to graph this on the xb plane we need to substitute y in for b, because graphing software will graph they xb plane but can graph the xy plane. So every place that you see y just think of it as b in the equation:

This is the graph of As you can see from the graph there is at least one solution for

This can be shown easily by substituting ­2 and 2 in for y to get

All you have to do is solve each equation and find that you get a double root x = -1 forand a double root x = 1 for


So there is no solution for when y is between ­2 and 2 and there are two solutions for when y is greater then 2 and less than ­2.

Now lets investigate changing the 1 in the equation Lets first try the values ­3, -2, -1, 0, 1, 2, 3 and see what happens.


It looks like all of these graphs are bounded by and the y axis, so lets take the x and y axis away from this picture and see what happens.

As you can see the y-axis was just hiding the other piece of the graph

As you can see by this 3-D graph, the graphis bounded between the green function which are planes representing in 3-D