Ib Hl Coursework Properties Of Quartics

Properties of Quartics Abstract: The aim of this paper is to investigate on a property of quartic polynomials. Considering a “W” shaped function with inflection points Q and R, a line was drawn through the points to meet the function again at P and S. The ratio of PQ:QR:RS is found to correspond to the Golden Ratio hence a conjecture is produced and formally proven in order to demonstrate this. The function

hence points of inflection are Q(1,23) and

f (x) = x 4 - 8x 3 + 18x 2 - 12x + 24 is a “W” shape quartic with two points of inflection. Its first derivative f ′(x) = 4 x 3 - 24 x 2 + 36x - 12 is a cubic function and its second derivative f ′′(x) = 12x 2 - 48x + 36 a quadratic.

R(3,15) The equation of the straight line joining Q and R is: y - y1 = m(x - x1 ) y - 23 = - 4(x - 1)         y = - 4 x + 27 Let points P and S be the intersection of the

Fig 1

line QR and the quartic function. They are found by equating, factorizing using long division and solving for x.

−4 x + 27 = x 4 - 8x 3 +18x 2 - 12x + 24 0 = x 4 - 8x 3 + 18x 2 - 8x - 3 0 = (x 2 - 4 x + 3)(x 2 - 4 x +1) 4 ± 20 2 x = 2 + 5   and  x = 2 x= x

5 

Note that the other quadratic equation gives the x-values of Q and R. The coordinates of Fig 1 shows the function f (x) and its first and second derivatives. Note that the points where f ′(x) = 0 are the function’s inflection points.

points P and S are therefore P((2 -

5), f (2 -

5)) and

S((2 + 5), f (2 + 5)).

The quartic’s points of inflection are found by equating the second derivative to zero.

The investigation must then consider the ratio

f ′′(x) = 12x 2 - 48x + 36        0 =12(x - 1)(x - 3)

of PQ:QR:RS. This can be done using

x = 1 and x = 3 ,

Pythagoras Theorem, however, because the coordinates of P and S are irrational, the process is long and tedious. The features of similar triangles enable us to concentrate on x

Substituting for x, f (1) = 23 and f (3) = 15

coordinates only. Since all four points lie on

the same line, the triangles they form are

occur when the second derivative equals zero:

similar and the ratio of their distances is the

′′ = 24 x 2 - 24 x g (x)

same as the ratio of their components.

0 = 24 x(x - 1)          x = 0        x = 1

Fig 2

g(0) = 0   g(1) = - 2

Hence coordinates are Q(0,0) and R(1,-2). Since the line QR goes through the origin, its equation is y = - 2x which intersects with the quartic when: 2x 4 - 4 x 3 = - 2x 2x 4 - 4 x 3 + 2x = 0 Long division using the factor (x - 1) simplifies the equation into two quadratics, which can then be solved using the formula. 2x 4 - 4 x 3 + 2x = 0 (2x 2 - 2x)(x 2 - x - 1) = 0

In Fig 2,points P, Q, R, and S lie on the line y = - 4 x + 27 . The dotted lines illustrate the similar triangles they form. The ratio PQ:QR:RS is found using the previously calculated x-values. Let xP = 2 -

5 , xS = 2 + 5 , xq = 1

and x R = 3

x=

The x-coordinates of points P and S are therefore

2

and

1+ 5 respectively. 2

since the point of inflection Q lies on the x P - xQ : xQ - x R : x R - x S

5) - 1 = 5 - 1

x R - x S = 3 - (2 + 5) = 5 - 1

1:

5

Calculating the ratio is simpler in this case

5-1 : 2

xQ - x R = 1- 3 = 2

through by

1-

origin. The ratio PQ:QR:RS corresponds to

x P - xQ = (2 -

PQ:QR:RS is

4± 4 1± 5   and  x = 4 2

5 - 1: 2 : 5 - 1and dividing 5 - 1 simplifies to

1+ 5 :1 2

Considering the simpler quartic function g(x) = 2x 4 - 4 x 3 , the points of inflection also

1

:

and dividing through by 1:

1+ 5 :1 2

5-1 2 5-1 simplifies to 2

Fig 3 Let us consider the general case of a quartic polynomial function f (x) . It will have two distinct points of inflection, Q and R, if its second derivative has two real roots. This means that the quadratic function f ′′(x) must be in the form of y = bx(x - a) where a and b are real numbers. Fig 4 Fig 3 shows the graph of g(x) and y = - 2x . Points Q and R are inflection points and P and S intersection points.

y=

1 b(x 4 - ax 3 + cx) 12

In both cases, we assist to the emergence of the golden ratio defined as the positive solution of x 2 - x - 1 = 0 or

1+ 5 . 2

We therefore produce a conjecture based of the previous investigations. Conjecture: If f (x) is a quartic polynomial with two distinct points of inflection, Q and R, then the straight line joining Q and R meets the graph of y = f (x) in two other points P and S. In its simplest form, the ratio of PQ:QR:RS is equal to 1:

1+ 5 :1 so that 2

QR 1+ 5 = or the Golden Ratio. PQ 2

Fig 4 shows the relationship between the function and its second derivative. The quartic function is obtained by integrating f ′′(x) twice. 1

òòbx(x - a) dx = 12 b(x

4

- 2ax 3 + cx + d)

For simplicity, we assume that the point Q lies on the origin (0,0). The constant d must therefore equal zero so that f (0) = 0 meaning that the quartic is expressed as

f (x) =

1 b(x 4 - 2ax 3 + cx) . 12

Now if Q is the origin, the other root of the second derivative is a hence the second point of inflection R has coordinates R(a, f (a)) . Substituting for f (a) , the coordinates are R(a,

Using factor (x - a) first: x 3 - ax 2 - a 2 x x - a)    x - 2ax 3 + a 3 x 4

- (x 4 - ax 3 )

1 b(- a 4 + ac)) . 12

        - ax 3 + a 3 x     - (- ax 3 + a 2 x 2 )

The gradient of the straight line joining points

                 - a 2 x 2 + a 3 x

1 bx(- a 4 + ac) 1 Q and R is 12 = bx(- a 3 + c) so a 12

its equation is y =

1 bx(- a 3 + c) because the 12

y-intercept is zero. The line intersects the quartic at points P and S and the x-coordinates of these points are the solutions of the

             - (- a 2 x 2 + a 3 x)                                   0                    and then factor (x - 0) x 2 - ax - a 2 x - 0)     x 3 - ax 2 - a 2 x - (x 3 )        

equation

       - ax 2 - a 2 x

1 1 bx(- a 3 + c) = bx(x 3 - 2ax 2 + c) 12 12

       - (- ax 2 )                  - a 2 x

x(- a 3 ) = x(x 3 - 2ax 2 )

             - (- a 2 x)

x 4 - 2ax 3 + a 3 x = 0

                       0

The easiest way to obtain the solutions of the

Simplifies the equation into the quadratic

equation in terms of a is to use the quadratic

y = x 2 - ax - a 2 whose roots are found using the

formula. This means simplifying the quartic into a quadratic by long division using the

formula x =

already known real factors (x - a) and (x - 0) . xS =

a ± a 2 - 4(- a 2 ) hence 2

a + 5a 2 a - 5a 2 x = 0 , xP = , Q and x R = a . 2 2

The first type has no points of inflection since its second derivative has no real roots. The quadratic f ′′(x) does not cross the x-axis Once again, properties of similar triangles

because its determinant b 2 - 4ac is negative. A

enable us to use x-coordinates only to find the

quartic with no points of inflection will have a

ratio PQ:QR:RS.

“U” shape as shown on fig.5

x P - xQ : xQ - x R : x R - x S

Fig 5

5a 2 - a 5a 2 - a :    a    : 2 2

The ratio can be greatly simplified: 5a 2 - a 5a 2 - a :a: 2 2

(dividing through by 1:

2a 5a 2 - a

5a 2 - a ) 2

:1 x4

(simplifying the fraction)

a   =   ´ a 5a 2 - a 2a

2 1+ 5  =  2 5-1

Hence the ratio equals 1: Golden Ratio

1+ 5 :1 or the 2

x2

Fig 5 shows the graph of y = 12 + 2 and its second derivative y = x 2 + 1. Note that the quadratic has no real roots hence the quartic is “U” shaped.

The second type of quartic does not have any points of inflection because its second derivative has only one real solution. Indeed,

It is important to mention the limitations of this conjecture that cannot be extended to quartic functions that are not strictly of a “W” shape. The ratio of lengths can only be found if the function has two distinct points of inflection meaning its second derivative must have two real roots. There are therefore two types of quartics that will not illustrate the golden ratio.

since its determinant equals zero, it touches the x-axis at one point and does not change sign. Quartics of such type have a flattened U shape. Fig6

As a result, the conjecture only4 holds if the Fig6 shows the graph of y = x and its determinant of the second is greater y =12x 2derivative second derivative . than zero. Integrating the general function twice: ax 4 bx 3 cx 2 òò(ax + bx + c)dx = 12 + 6 + 2 + dx + e 2

Hence the conjecture only affects quartic function with b 2 - 4ac > 0 Using the proof of the conjecture and the analysis of its limitations, it is possible to produce a formal theorem on this specific property of quartic polynomials.

Theorem: Let f (x) be a quartic polynomial with two distinct points of inflection Q and R, the straight line joining these points will meet the function again at P and S. If points P,Q,R,S are ordered by increasing coordinates, then PQ = RS and QR 1+ 5 = , the Golden Ratio RS 2

x-