write-up #3 Assignment 3

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.

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.

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.

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)

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.