Properties of Quartics

f(2) = -12 negative

A switch from negative to positive indicates an inflection point.

f(4) = 36 positive

Find Y

Point Q f(1) = 23

Point R f(3) = 34 - 8(3)3 + 18(3)2 - 12(3) + 24 = 15

Inflection points are:

Q (1, 23) R (3, 15)

Line QR

Equation of a Line Passing through Two Points:

y - y1 = (y2 - y2)/(x2 - x1) * (x - x1)

Plug in Known Values

y - 15 = (23-15)/(1-3) * (x-3)

y = -4x + 27

Find P and S

Set f(x) Equal to g(x) and Then to Zero

-4x + 27 = x4 - 8x3 + 18x2 - 12x +24

x4 - 8x3 + 18x2 - 8x - 3 = 0

Divide by Root 1

x3 - 7x2 + 11x +3 = 0

Divide by Root 3

x2 - 4x -1 = 0

Apply to Quadratic Equation

X1,2 = 2 ± V5

Find Y

Point P f(2 - V5) = (-.236)4 - 8(-.236)3 + 18(-.236)2 - 12(-.236)+ 24 = 27.9

Point S f(2 + V5) = (.236)4 - 8(.236)3 + 18(.236)2 - 12(.236) + 24 = 22.1

P = (-.236, 27.9) S = (.236, 22.1)

Find the Ratio PQ:QR:RS

Use the Distance Formula

PQ:

d = V((1- (-.236)) 2 + (23 - 27.9) 2)

d = 1.24

QR:

d = V((3- 1) 2 + (15 - 23) 2)

d = 2

RS:

d = V((.236 - 3)2 + (22.1 - 15) 2)

d = 1.24

PQ:QR:RS = 1.24:2:1.24

3. Simplify this ratio so that PQ = 1 and comment upon your results.

Divide by PQ

(1.24/1.24):(2/1.24):(1.24/1.24)

1:1.618:1

When PQ is set to equal 1, QR becomes 1.618. This number is known as "Phi" (?) or the "Divine Proportion", as it is a number found in ratios in most things in nature, such as in spirals in Nautilus Shells or in the construction of the Human Body.

4. Choose another quartic function with a "W" shape and investigate the same ratios.

Choose a Function

f(x) = x4 + 3x3 -5x

Find Q and R

Find First Derivative:

f(x) = x4 + 3x3 -5x

f'(x) = 4x3 +9x2 -5

Find Second Derivative

f"(x) = 12x2 + 18x

Set to Zero

0 = 12x2 + 18x

Apply Quadratic Equation

x = -1.5, 0

Concavity Checks

f(-2) = 12 positive

A switch from positive to negative indicates an inflection point.

f(-1) = -6 negative

A switch from negative to positive indicates an inflection point.

f(4) = 30 positive

Find Y

Point Q f(-1.5) = (-1.5)4 + 3(-1.5)3 - 5(-1.5) = 2.4

Point R f(0) = 0

Inflection points are:

Q (-1.5, 2.4) R (0, 0)

Line QR

Equation of a Line Passing through Two Points:

y - y1 = (y2 - y2)/(x2 - x1) * (x - x1)

Plug in Known Values

y - 0 = (2.4-0)/(-1.5-0) * (x-0)

y = -1.6x

Find P and S

Set f(x) Equal to g(x) and Then to Zero

-1.6x = x4 + 3x3 -5x

x4 + 3x3 -3.4x = 0

Reduce to a Quadratic and Apply to Quadratic Equation

X1,2 = -2.7, 1

Find Y

Point P f(-2.7) = (-2.7) 4 + 3(-2.7) 3 - 5(-2.7) = 7.6

Point S f(1) = (1) 4 + 3(1) 3 - 5(1)= -1

P = (-2.7, 7.6) S = (1, -1)

Find the Ratio PQ:QR:RS

Use the Distance Formula

PQ:

d = V ((-1.5- (-2.7)) 2 + (2.4 - 7.6) 2)

d = 5.34

QR:

d = V ((0- (-1.5)) 2 + (0 - 2.4) 2)

d = 8.97

RS:

d = V ((1 - 0)2 + (.1 - 0) 2)

d = 5.34

PQ:QR:RS = 5.34:8.97:5.34

Divide by PQ

(5.34/5.34):(8.97/5.34):(5.34/5.34)

1:1.618:1

5. Form a conjecture and formally prove it using a general quartic function.

Starting With the General Equation:

f(x) = ax4 + bx3 + cx2 +dx + e

Take the First Derivative

f'(x) = 4ax3 + 3bx2 + 2cx + d

Take the Second Derivative

f"(X) = 12ax2 +6bx +2c = 0

Set the Two Derivatives Equal To Each Other (Horizontal Inflection Points Occur When Either the First or Second Derivative Equals Zero)

4ax3 + 3bx2 + 2cx + d = 12ax2 +6bx +2c

Set It Equal to Zero

4ax3 +12ax2 + 3bx2 + 6bx + 2cx + 2c + d = 0

This equation will ensure that a quartic will have at least one horizontal inflection point, a requirement for the Divine Proportion.

As show by original equation f(x) = x4 - 8x3 + 18x2 - 12x +24 and chosen equation f(x) = x4 + 3x3 -5x , quartic equations will always be proportionate in terms of ?. This is proven by how a "W" shape of a quartic will only appear if the proportions are exactly 1:1.681:1. If the ratio is not exactly this proportion, then the "w" shape will no longer be present. However, for the Divine Proportion to be present in a quatric equation, two distinct inflection points must be present, and it does not have to result in a "w" shape quatric function.

6. Extend this investigation to other quartic functions that are not strictly of a "W" shape.

For quartic functions not strictly of a "W" shape then the proportion of 1:1.681:1 does not exist. For example, take the quartic equation f(x) = x4 + 4x.

As can be seen by the graph there is no inflection point from which to derive a ratio. For other types of quartic equations, like those with one inflection point, the proportion could not be found as there is no other point for which to compare to.