To see an animated version of
from a = -3...3, click on the graph above.
Consider again the equation
If we graph this relation
in the xb plane we will get the following graph.
To see how the equation
of the above graph was derived click on the graph.
If we take any particular
value of b, say b = 3, 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.
for c = -3,
-2, -1, 0, 1, 2, and 3.Initially, there are two things that stand out about these graphs. First, the point of intersection through the y-axis of all of the parabolas corresponds with the c value in each equation. Secondly, all of the parabolas have the same axis of symmetry, x = -1/2, which would make sense because all of the a and b values in each equation are the same. Recall that to find the axis of symmetry we use the equation,
To see how this equation was derived, click on the equation.
In the following example
the equation
is considered. If the equation is graphed in the xc plane, it is easy to see that the curve will be a parabola. For each value of c considered, its graph will be a line crossing the parabola in 0, 1, or 2 points -- the intersections being at the roots of the orignal equation at that value of c. In the graph, the graph of c = 1 is shown. The equation
will have two negative
roots -- approximately -0.2 and -4.8.
There is one value of c where the equation will have only 1 real root, at c = 6.25. For c > 6.25 the equation will have no real roots and for c < 6.25 the equation will have two roots, both negative for 0 < c < 6.25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.
Send e-mail to Dr. Jim Wilson or Kevin Mylod
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