By Nami Youn

Write-up  #3

Some Different Ways to Examine

by

James W. wilson and Nami Youn
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(purple),  b = -2(Blue),  b = -1(green), b = 0(sky), b = 1(yellow), b =2(gray), b = 3(red)

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) with this equation). 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

Show that the locus is the parabola, y = -x^2+1 (black curve)

Now, I try to examine the roots of a quadratic equation through the graph.

First, the possible way is to graph in the xa plane instead of in the xy plane. This means that we will substitute y for a into a quadratic equation and the graph.
Next, instead of graphing in the xy plane, I will graph in the xb plane. This also means I will substitute y for b into a quadratic equation and the graph.
Finally, instead of graphing in the xy plane, I will graph in the xc plane. This  means I will substitute y for c into a quadratic equation and the graph.

1. Graphs in the xa plane

Consider the equation

Let's graph this equation in the xa plane. Before doing it, I need to graph the equation

Let's take any particular value of a, and notice the equation y = a on the graph.
The intersection points between the line parallel to X-axis (y=a) and yx^2+x+1=0 are the roots of the a quadratic equation.

yx^2+x+1=0(blue)
a = 2, the equation y = 2 (purple)
a = 0.25, the equation y = 0.25(red)
a = -1, the equation y = -1 (green)

For a = 2, there is real root of the equation.
For a = 0.25, the equation has one negative real root.
For a = -1, there are one positive and negative real roots.

Generalize :

when a < 0 , there is real root of the equation.
when 0 < a < 0.25 , the equation has one negative real root.
When a > 0.25 , there are one positive and negative real roots.

2. Graphs in the xbplane

Consider the equation

Let's graph this equation in the xb plane. Before doing it, I need to graph the equation.

Let's take any particular value of b, and notice the equation y = b on the graph.
The intersection points between the line parallel to X-axis (y=a) and yx^2+x+1=0 are the roots of the a quadratic equation.

x^2+yx+1=0(blue)
b = 3 and -3, the equation y = 3 , -3 (purple)
b = 2 and -2, the equation y = 2 , -2(red)
b = 1 and -1, the equation y = 1, -1(green)

For b = -1, 1, there is real root of the equation.
For b = -2, 2  the equation has one positive or one negative real root.
For b = -3, 3  there are two positive roots or two negative real roots.

Generalize :

when b <-2  , there is two positive real roots.
when b = -2 , the equation has one negative real root.
when -2 < b < 2 , the equation has no real roots.
when b = 2 , there are one negative real root.
when b > 2 , there are two negative real roots.

3. Graphs in the xcplane

In the following example, consider the equation

Let's graph this equation in the xc plane. Before doing it, I need to graph the equation.

Let's take any particular value of c, and notice the equation y = c on the graph.
The intersection points between the line parallel to X-axis (y=c) and yx^2+x+1=0 are the roots of the a quadratic equation.

x^2+5x+y = 0(blue)
c = -2, the equation y = -2 (purple)
c = 0 , the equation y = 0 (x-axis)
c = 2,   the equation y = 2(red)
c = 6.25, the equation y = 6.25(green)

For c = -2, there are one negative root and one positive root.
For c =  0, the equation 0 and one negative real root.
For c =  2, the equation has two negative real roots.
For c = 6.25,  there is no real root.

Generalize :

when c < 0 , there one negative root and one positive root.
when c = 0 , tthe equation 0  and one negative real root.
when 0 < c < 6.25 ,the equation has two negative real roots.
when c > 6.25, there is no real root.