Graphs in the xb Plane
LetŐs look at the following equation in the xb plane.
When using Graphing Calculator, this equation will appear as
LetŐs look at the following graph when c=1.
What happens as the value of c changes?
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, letŐs 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. LetŐs 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.
LetŐs graph some more values of c.
We see here that we have a family of hyperbolas with an asymptote when c=0.