Complex Zeros of Quadratic Functions

coefficient of leading term, a

1

horizontal translation, h

1

vertical translation, v

1

show equations & zeros

force a to equal 1/v (vertices = zeros)

reset

This Demonstration shows the graphs of two symmetric quadratic functions (with respect to the

x

axis) of the form

y=a

2

(x-h)

+v

and

y=-

2

a(x-h)

-v

, where

h

and

v

are the horizontal and vertical translations of the corresponding parabolas

y=a

2

x

and

y=-a

2

x

, with vertices at the origin. Their complex zeros are identical and marked by red dots located in the complex plane



, where the

Im

and

Re

axes (labeled in red on the graph) coincide with the Cartesian plane

2



coordinate

x

and

y

axes; that is, the

x

axis is also the real axis and the

y

axis is also the imaginary axis. While any real zeros lie on the

x

axis (or real axis), imaginary zeros come in pairs (complex conjugates) and lie on the vertical line

x=h

that runs through the vertices (and foci) of the parabolas. Further, as complex conjugates, the zeros are symmetric with respect to the

x

axis (real axis). To see the effects on the graph when

a=1/v

, click on the checkbox "force

a

to equal

1/v

(vertices = zeros)" and move the

v

slider.