# Complex Zeros of Quadratic Functions

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.

x

y=a+v

2

(x-h)

y=--v

2

a(x-h)

h

v

y=a

2

x

y=-a

2

x

Im

Re

2

x

y

x

y

x

x=h

x

a=1/v

a

1/v

v