Assignment 3: Graphing in the xb Plane


Jonathan Sabo

Graphs in the xb plane.

Consider again the equation

Now graph this relation in the xb plane. We get the following graph.
In order to graph this equation in the software we must plug in y instead of b.

If we take any particular value of b, say b = 5, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.

For each value of b we select, we get a horizontal line. It is clear on a single graph that we get two negative real roots of the original equation when b > 2, one negative real root when b = 2, no real roots for -2 < b < 2, One positive real root when b = -2, and two positive real roots when b < -2.

Consider the case when c = - 1 rather than + 1.

Observe, in this graph we see that there are 2 real roots when c is less than 1.  Lets observe several other cases where c is less than 1.

In each of these cases we can see that there are always 2 real roots.

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