Parameters for Plotting a Quartic
Parameters for Plotting a Quartic
The general quartic can be brought into the reduced form
F(x)≡a+b+c+dx+e
4
x
3
x
2
x
Z(x)≡a-6a+x+
4
x
2
ϵ
2
x
y
N
z'
y
N
z
x↦x-
b
4a
x
N
f
b
4a
y
N
z
x
N
f
y
N
z'
x
N
f
The coordinates of the two points of inflection of are , where =.
x
Z(x)
±ϵ
2
ϵ
3-8ac
2
b
48
2
a
When ≥0, there are two real points of inflection and hence three real turning points. When <0, both points of inflection are complex and hence there is only one real turning point.
2
ϵ
2
ϵ
Since , , and 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 .
ϵ
y
N
z
y
N
z'
Z(x)