Assignment 3: Quadratics in the xb plane

By Kendyl Wade

For this exploration, we will check out graphs in the xb plane for varying values of c in the quadratic equation. The parameter a will remain at 1.

Here we have a graph of the function x^2 + yx + c, using integers -4 to 4 for c.

Here is an animation for values of c from -20 to 20:

Notice that the for c = 0, it is a straight line at x = -b and a straight line at x = 0. Observe what happens when we graph b = 4 and b = 2. The intersection of these horizontal lines and x^2 + bx + c represent the real roots of the original quadratic function.

Here we see that for b = 4, there are three cases:

Case 1: There are two real roots for the original equation, when 4 > c.

Case 2: There is one real root for the original equation, when c = 4.

Case 3: There are no real roots for the original equation, when c > 4.

To get a more general understanding of roots, we can look at the quadratic formula.

Since we let a = 1 for this exploration, we will apply this to the quadratic formula as well.

In mathematics, we cannot take the square root of a negative number so if 4c > b^2, there are no real roots. If 4c = b^2, the square root of 0 is 0. So we would not have two answers because adding zero and subtracting zero would result in the same answer. If 4c < b^2, we would be taking the square root of a positive number; so we would have two roots.