By Nami Youn
Write-up #3
Some Different Ways to Examine
by
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.
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.
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.
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.
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.