Complex Zeros of Quadratic Functions
This Demonstration shows the graphs of two symmetric quadratic functions (with respect to the axis) of the form and , where and are the horizontal and vertical translations of the corresponding parabolas and , with vertices at the origin. Their complex zeros are identical and marked by red dots located in the complex plane , where the and axes (labeled in red on the graph) coincide with the Cartesian plane coordinate and axes; that is, the axis is also the real axis and the axis is also the imaginary axis. While any real zeros lie on the axis (or real axis), imaginary zeros come in pairs (complex conjugates) and lie on the vertical line that runs through the vertices (and foci) of the parabolas. Further, as complex conjugates, the zeros are symmetric with respect to the axis (real axis). To see the effects on the graph when , click on the checkbox "force to equal (vertices = zeros)" and move the slider.
To find the complex zeros, set in each equation, and , and solve for : and , which implies (in either case) , , , . When , the zeros are real and lie on the axis; when , there are two imaginary zeros (complex conjugates) that lie on the vertical line . For and fixed vertices , observe that: (1) as , the parabolas get narrower while the imaginary zeros approach the real axis along the line ; and (2) as , the parabolas flatten out while the imaginary zeros approach ± along the line . When and the imaginary part of either imaginary zero is the same as , that is when , both imaginary zeros coincide with the vertices. To solve , square both sides and get , which has solution .