By: Melissa Wilson

In this assignment we will investigate quadratics of the form:

We will graph this function on the xb-plane, which is where values for b are graphed where you would commonly find the y-values. This function graphed in the xb-plane is shown to the right.

Now, we take the line b=5 and plot it on the same graph as our original function, shown as the purple line.

Notice that the b=5 line passes through the other function in two places. These two intersection points correspond to the roots of the quadratic equation shown below when b, the coefficient for x, is 5.

If you solve for these roots they are approximately -4.791 and -0.209.

Next, we will look at various values of b. Notice when b>2, we can find 2 real roots, which are negative.

At b=2 we have one negative root.

When -2<b<2, there are no real roots.

For b=-2, there is one positive real root.

Lastly, for b<-2, we can find 2 positive real roots.

Previously we had the coefficients for the other two terms as 1. Now, we will vary c. Let's first look at when c=-1.

The graph is shown to the right in black. Notice how now the function has two real roots for every value of b.

Also, notice that both this function and the previous one approach the same asymptotes from opposite sides. This asymptote is shown in the next graph for when c=0.

This graph looks at varying values of c (1,-1,0,-3,3) in the xb-plane.

When c is negative (black and red graphs), the function has two real roots at every value of b.

When c=0 (purple graph), the function has one real root at each value of b.

Lastly, when c is positive (blue and green graphs), then the previous analysis holds for when the functions has various roots.