Quadratic Parameter Plane
- Judah L Schwartz
In the right hand panel is a quadratic function in the form f(x) = x² + px + q The two parameters p and q determine the parabola. The p,q plane is shown in the left hand panel with the point p,q plotted. Drag the point around the p,q plane by sliding the large BLUE tick marks on the axes. What happens in the right hand x,y plane? Why do the point and the parabola change color? Where are they RED? GREEN? Challenge - What is the shape of the red/green boundary in the p,q plane? In the p,q plane, the boundary can be thought of a a function q(p). What is this function? How is it related to the discriminant of the quadratic? Challenge – The locations of the real or complex conjugate roots of the quadratic appear in the right hand panel as large gold dots. Trace the complex roots in the x,y plane. Can you formulate a conjecture about the path they take as you move the point in the p,q plane along a horizontal line? along a vertical line? Can you prove or disprove your conjectures?