Math 210
Jerry L. Kazdan
Quadratic Polynomials
Polynomials in One Variable.
After studying linear functions
y
=
ax
+
b
, the next step is to study
quadratic poly
nomials
,
y
=
ax
2
+
bx
+
c
, whose graphs are parabolas. Initially one studies the simpler
special case
y
=
ax
2
+
c
(1)
If
a >
0 these parabolas have a minimum at
x
= 0 and open upward, while if
a <
0 they
have a maximum at
x
= 0 and open downward.
One can reduce the more general quadratic polynomial
y
=
ax
2
+
bx
+
c
(2)
to the special case (1) by a change of variable,
x
=
v
+
r
translating
x
by
r
, where
r
is to
be found. Substituting this into (2) we find
y
=
a
(
v
+
r
)
2
+
b
(
v
+
r
) +
c
=
av
2
+ (
b
+ 2
ar
)
v
+
ar
2
+
br
+
c.
We now pick
r
to remove the linear term in
v
, that is,
b
+ 2
ar
= 0 so
r
=
−
b/
(2
a
). Then
y
=
av
2
+
k
=
a
(
x
−
r
)
2
+
k,
(3)
where
k
=
c
−
b
2
/
(4
a
). Thus,
x
=
r
is the
axis of symmetry
of this parabola.
This procedure is equivalent to “completing the square”, a
procedure that should be familiar from algebra.
Another way
to find
r
is to observe that the only critical point of (2) is at
x
=
−
b/
(2
a
).
Thus translating by
b/
(2
a
) places this critical
point on the vertical axis.
0
5
10
15
20
25
4 2
2
4
6
Shifted Parabola
Example
.
On the right are the graphs of
y
=
x
2
+ 1 and
y
= (
x
−
2)
2
+ 1 =
x
2
−
2
x
+ 2. They clearly shows the graph
on the right is merely a translation of the graph on the left.
Polynomials in Several Variables.
Maxima, minima, and saddle points.
There are more interesting possibilities for quadratic polynomials in two variables.
w
= 2
x
2
+
y
2
w
=
−
(2
x
2
+
y
2
)
w
=
−
2
x
2
+
y
2
From the graphs it is clear that the first has a
minimum
at the origin, the second a
maxi
mum
, while the third has a
saddle point
there.
1
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It is less obvious how to treat polynomials such as
w
= 3
x
2
−
2
xy
+
y
2
(4)
or
w
= 3
x
2
−
2
xy
+
y
2
+
z
2
(5)
with
xy
terms and possibly more variables.
These two examples can be handled if one
recognizes that
x
2
−
2
xy
+
y
2
= (
x
−
y
)
2
so, if one makes the change of variables
r
=
x, s
=
x
−
y
and
t
=
z
then in the new coordinates these polynomials become
w
= 2
x
2
+ (
x
−
y
)
2
= 2
r
2
+
s
2
and
w
= 2
r
2
+
s
2
+
t
2
,
which clearly have minima at the origin in
rs
and
rst
space, respectively.
The primary task of this section is to give useful criteria for a quadratic polynomial in
several variables to have a maximum, minimum, or saddle point. This will then be used in
the next section to generalize the calculus of one variable second derivative test for a local
maximum to functions of several variables such as our quadratic polynomials.
The first step is to be a bit more systematic. Rewrite (4) as
w
= 3
x
2
−
xy
−
yx
+
y
2
and
observe that using the inner (=dot) product it can be written in the more compact form
w
=
X
·
A
X
,
(6)
where
X
=
parenleftBigg
x
y
parenrightBigg
and
A
is the
symmetric
matrix
A
=
parenleftBigg
3
−
1
−
1
1
parenrightBigg
.
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 Spring '12
 STAFF
 Polynomials, Matrices, Diagonal matrix, Orthogonal matrix, Symmetric matrix, positive definite

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