# 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)