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)

.