Rayen Antillanca.
Assignment 3
The
general form of quadratic equation is: .
We are interested in exploring how the parameter b affects the roots of this
equation.
First
Investigation
I
am going to consider the equation .
To explore the effect of parameter b on the roots, I will solve the equation for b, so that I can plot the relationship
between the roots and b on the xb plane. Solving for
b gives us whose graph is shown in figure 1.
This graph shows the relation between
x and b 

The purple curve is
a hyperbola with vertical asymptote and with a diagonal asymptote given by 
Figure 1 
If we fix the value of b, for
instance b=4, there will be two values for x represented by the intersection
of b = 4 and the red curve. These values of x coincide with the roots of the equation . 

Summary 
Different values of b are represented
by the horizontal purple line above. Varying b implicates to move
horizontally the purple line, and the number of intersections of this line
with the red curve represent the number and the location of the roots for .
In other words, if b>2 we get 2 real negatives roots if b=2 we get one
negative root; when 2<b<2 we get no real roots, when b=2 we get one
positive root n when b<2 we get two real roots 
Second
Investigation
Now,
let’s see what happens when c=1
The next figure shows the graph of for c=1 (purple curve) and c=1 (blue curve). 

As you can see,
with c=1 the new function is a hyperbola whose branches are in the first and
third quadrant with asymptotes and .
When ,
regardless the value of b, we always get two intersections between the
horizontal line and the blue curve. That is, the equation has always two real roots (one positive and
the other negative). 
The next figure shows what happens
with the graph of as c < 0 varies. 



Summary 
We can observe that as long as c<0, the equation has two real roots, regardless the value of
b. When c<0, this gives a family of hyperbolas with one branch on the
first and fourth quadrants, and the other one on the second and third quadrants.
These hyperbolas have a vertical asymptote and at diagonal asymptote given by . 
Note: All graphs of this webpage were made with
Graphing Calculator 4.0 