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WOLFRAM|DEMONSTRATIONS PROJECT

Parameters for Plotting a Quartic

​
function:
Z(x)
R(x)
R(​
2
x
​)
parameters:
a
1
2
ϵ
2
y
N
z'
-10
y
N
z
5
show derivatives:
x
2
2
x
x range:
x
min
-4
x
max
4.5
y range
y
min
-60
y
max
40
The general quartic
F(x)≡a
4
x
+b
3
x
+c
2
x
+dx+e
can be brought into the reduced form
Z(x)≡a
4
x
-6a
2
ϵ
2
x
+
y
N
z'
x+
y
N
z
by means of the translation
x↦x-
b
4a
. If
x
N
f
=-
b
4a
, then
y
N
z
=F
x
N
f
and
y
N
z'
=F'
x
N
f
.
The
x
coordinates of the two points of inflection of
Z(x)
are
±ϵ
, where
2
ϵ
=
3
2
b
-8ac
48
2
a
.
When
2
ϵ
≥0
, there are two real points of inflection and hence three real turning points. When
2
ϵ
<0
, both points of inflection are complex and hence there is only one real turning point.
Since
ϵ
,
y
N
z
, and
y
N
z'
are directly related to the geometry of the quartic, this Demonstration offers a more intuitive insight regarding how the shape of the curve is related to the coefficients of the reduced form
Z(x)
.
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