## Some Different Ways to Examine

### James W. Wilson and Ken Hayakawa University of Georgia

It has now become a rather standard exercise, with availble technology, to construct graphs to consider the equation

and to overlay several graphs of

for different values of a, b, or c as the other two are held constant.
From these graphs discussion of the patterns for the roots of

can be followed. For example, if we set

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following
picture is obtained.

b=-3(blue), b=-2(red), b=-1(violet), b=0(green), b=1(sky), b=2(yellow), b=3(black)

We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point
on the y-axis (the point (0,1)). For b < -2 the parabola will intersect the x-axis in two points with
positive x values (i.e. the original equation will have two real roots, both positive). For b = -2, the parabola is
tangent to the x-axis and so the original equation has one real and positive root at the point of tangency. For -2 < b
< 2, the parabola does not intersect the x-axis -- the original equation has no real roots. Similarly for b = 2 the
parabola is tangent to the x-axis (one real negative root) and for b > 2, the parabola intersets the x-axis twice to
show two negative real roots for each b.

Now consider the locus of the vertices of the set of parabolas graphed from

Expand this eqation.

The locus of the verticesis given by and.Eliminate parameter b.

We get the equation of (red), this is loci of the vertices.

### Graphs in the xb-plain

Consider again the equation

Change the values of coefficient "b" and graph. That is to say, consider the coefficient "b" as the variable 'b".

The graph is hyperbola,following.

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.

We put constant "c".

The graph is following.

sky curve c<0 , red line c=0 , blue curve c>0

Consider the case when c<0 .Add the line y(=b)=5. The graph is

For any value of b , the hyperbola(sky curve c<0)will intersect the line y(=b)=5 in two points.One is in negative region and another one is in positive region (i.e. the equation will have one real root and one positive root)

### Graphs in the xc-plain

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 on xc-plain ,there are parabola(*)(blue)

and lines (c=n). n>2.25(red), n=2.25(sky) , 0<n<2.25(vioret) ,n=0(green) ,n<0(black)

Consider the and lines,

referece:

1) n>2.25(red) :

The willhave no real roots for c > 2.25.

2) n=2.25(sky) :

Thewill have only 1 real root -- at c = 2.25

3) 0<n<2.25(vioret) :for c < 2.25 the equation will have two roots.

Thewill have two positive roots----approximately 0.28 and 2.71

4) n=0(green) :

Thewill have also two roots ; one negative and one 0 when c = 0

5) n<0(black) :

Thewill have also two roots ;one negative and one positive when c < 0.

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