Quadratics and its zeros

by Hwa Young Lee

In assignment 2, we investigated the role of the parameters a, b, and c in the quadratic function

.

Here, we are going to further our investigation of ‘b’ in

.

We found out that the value of b has something to do with the vertex, thus determines the location of the vertex of the graph. Thus, the value of 'b' might have something to do with the zers of the graph. We are to investigate this by considering quadratic functions and equations.

Also, we are going to find the locus of the vertices of the different parabolas as b varies. Let's start with this first. Recall the graphs of when b varied from -3 to 3:

   

 

  Also, we rewrote the quadratic to get the idea of the vertex:

.

We found that the vertex, which is changes as the value of b changes. To find the relationship between our x-value and y-value of the vertex, we let

and .

Then .

Thus, the locus of the vertices of the parabolas formed as b is varied, forms a parabola itself, with an equation . (See parametric equations in assignment 10)

 

Now let’s consider two functions and .

The graphs of the two functions would be a parabola and the x-axis, respectively. Also, the x-values of the intersection of these two graphs would be the roots of the quadratic equation . Now we consider this equation  and graph this relation in the xb-plane.

For each value of b, we can graph a horizontal line on the xb-plane. Since the curve of the equation shows us the relationship between the value of b and the roots (values of x) of the equation, the x-coordinates of the intersection points of that horizontal line for a value of b and the curve of the equation are the roots of the quadratic equation for that value of b. Let’s see how many intersections we get as the value of b varies.

We can see that  has:

two distinct positive real roots when ,

one (double) positive real root when ,

no real roots for ,

one (double) negative real root when ,

two distinct negative real roots when .

In assignment 2, we found out that the value of c also affects the vertex, so let’s consider the case when our c is equal to -1

.

We can see that has two distinct real roots despite the value of b.

Let's vary the value of c, and see what the graph of looks like on the xb-plane.

When c>0,

Similar to the case in , depending on the range of b (where we draw the horizontal line), there are either two distinct real roots or a double real root or no real roots. We can check this by using the discriminant of the equation.

The quadratic has two distinct real roots when ;

The quadratic has one double real root when ;

The quadratic has no real roots when.

When c=0,

 

 

In this case, it seems like there is only one line but actually, there are two lines: When x=0 (the y-axis) and x=-b (the pink line above). Therefore, the equation has two real roots zero or -b when b is not equal to zero, one double real root zero when b=0.

When c<0,

In this case, regardless to the value of b, the equation has two distinct real roots.

We can see from the discriminant that when c<0, and thus the quadratic equation has two distinct real roots.