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Graphs in the xb Plane

Let’s look at the following equation in the xb plane.

x²+bx+1=0

When using Graphing Calculator, this equation will appear as

x²+yx+1=0

Let’s look at the following graph when c=1.

What happens as the value of c changes?

Let’s 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, let’s 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. Let’s 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.

Let’s graph some more values of c.

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

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