Complex Zeros of Quadratic Functions

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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.
Contributed by: Barry Cherkas (March 2011)
(Hunter College and WebGraphing.com)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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
.
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