What actually happens when we decide to look at the xb-plane instead of the xy-plane?

In the xy-plane our quadratic equation

gives us two vertical lines that represent the roots at those points. When we change our y-axis to a b-axis, these roots take on the form of a hyperbola. The xb-plane represents values (x,b) for which this equation hold true. For each value of b, there are different x values that satisfy the given equation. With these specific values of a=1 and c=1, the be values that actually satisfy the equation contain all real numbers except (-2,2). This means that from -2 to 2, there are no roots to the given quadratic equation. Therefore, the art of changing our axis allowed us to make a graphical representation of the roots of our quadratic equation.