Maths Investigation on Properties of Quartics - Assignment Example

Properties of Quartics] IB Higher Level Investigation Portfolio I Properties of Quartics Introduction Some quartic functions produce graphs with a “w” shape. These graphs have two points of inflection, Q and R. A straight line is drawn through Q and R intersects the function at two further points, P and S. This assignment investigates the ratio PQ: QR: RS, first using specific examples to formulate a conjecture, then proving the conjecture formally. 1. Graph the function f(x) = x4 – 2x3 – 12x2 + 15x +12 The plot for the quartic function: f(x) = x4 – 2x3 – 12x2 + 15x +12 is as follows. Figure 1: Plot of the given Quartic Equation 2. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line QR intersects the quartic function again, and calculate the ratio PQ:QR:RS. Calculations for point of inflection As per definition, point of inflection is a point at which the second derivative of an equation is equal to zero (). After differentiating the above given equation, we get the first and second derivatives as follows. First derivative: 1) f() =  2) f΄() =  3) f΄ () =  Second derivative (): 1) f΄() =  2) f΄΄() = 12 3) f΄΄() =  Let f΄΄() =  This will allow us to find the x values. Solving this equation by factorizing it, we get the following values of x: ( - 2)( + 1) = 0  = 2 or  = -1 The value of the function for these values of “x” will be as follows. 1) f(2) = -2-12+15(2)+12 2) f(2) = 16-16-48+42 = -6 1) f(-1) = -2-12+15 (-1)+12 2) f(-1) = 1+2-12-3 = -12 Finding the g(x) intersection line equation: -6 = -3m     In order to find the points of intersection between f() and g() we solve f() = g().  = 2  =  It is a 4th power equation This equation has 4 real roots: 1 Coordinates for P = (-2.854; -15.708) y = 2 -10 f(-2.854) = 2(-2.854)-10 f(-2.854) = -15.708 Coordinates for Q = (-1; -12) Coordinates for R = (2; -6) Coordinates for S = (3.854; -2.292) y = 2 -10 f(3.854) = 2(3.854)-10 f(3.854) = -2.292 PQ =  RS = PQ is equal to RS If we simplify this ratio with respect to PQ and RS, then we get the ratio as follows. PQ: QR: RS = 4.146: 6.708: 4.146 3. Simplify this ratio so that PQ = 1 and comment upon your results. PQ: QR: RS = 4.146: 6.708: 4.146 = 1: 1.618: 1 (to make PQ = 1, dividing by 4.146) Therefore, we can deduct that PQ = RS and ratios of QR: PQ and QR: RS are equal and is approximately. This number (1.618) is commonly referred to as the Golden Ratio. It is a number that is abundant in nature, and is often present in numerous proportions. 4. Choose another quartic function with a “W” shape and investigate the same ratios. Taking quartic function as f(x) =  –  + -8 – 2 : it was solved by integration as below In order to have a “w” shape graph, first of all, we need to take an ordinary quadratic equation and integrate two times to get “” value in the 4th power. Step 1 Find a simple quadratic function:  Step 2 Let   (-3)(-1) = 0   = 3 or  = 1 Step 3 Using Integrals we, first of all, find a cubic function. Step 4 Yet again, using integrals we find the quartic function from the cubic function. However we can remove value and take only equation. Therefore, the equation will be (putting = ) Step 5 The value of ‘c’ and ‘d’ in the equation I made up myself (by guess work using different values for finding W shape quartic function), as it carries the “lifting” function and came up with -8 and -2 respectively. Thus, my final quartic function looks in the following way: Determining the inflection points As we know the second derivative of the above quartic function is equal to , putting this equal to zero, we can find the inflection points.  Therefore, (x-3)*(x-1) = 0  = 3 or  = 1 The value of the function for these values of “x” will be as follows. 1) f(3) = -8+18-8(3)-2 2) f(3) = 81-216+162-26 = 1 1) f(1) = -8+18-8(1)-2 2) f(1) = 1-8+18-10 = 1 Finding the g(x) intersection line equation: 0 = 2m     (Coincidence, it was not stipulated) The plot of quartic function and intersection line is as follows Figure 2: Plot for second quartic Therefore, the points of inflection are Q (1, 1) and R (3, 1) In order to find the points of intersection between f(x) and g(x); f(x) = g(x):  = 1  1 = 0 It is a 4th power equation This equation has 4 real roots: 2 Each of these points will have y =1 as the intersection line is y =1 Coordinates for P = (- 0.236; 1) Coordinates for Q = (1; 1) Coordinates for R = (3; 1) Coordinates for S = (4.236; 1) PQ = =1.236 ( is taken because values of y is constant i.e. y =1) QR = =2 RS = =1.236 Again, PQ is equal to RS If we simplify this ratio with respect to PQ and RS, then we get the ratio as follows: PQ: QR: RS = 1.236: 2: 1.236 =  1: 1.618: 1 5. Formulate a conjecture and formally prove it using a general quartic function Conjecture: Let f(x) be a quartic polynomial and assume that the graph of y = f(x) has two distinct points, Q and R of inflection. Then the straight line passing through Q and R meets the graph of y = f(x) in two other distinct points P and S; furthermore PQ = RS and, is the Golden Ratio. 6. Formally prove it using a general quartic function It is possible to obtain quartic equation by picking two inflection points and forming a quadratic equation with it and double integrating it. Below steps can be performed for obtaining a general equation of quartic: For simplicity let us assume that the point Q be at the origin, and the point R is defined by two variables, a and f(a). Therefore, point Q will be (0, 0) and point R will be (a, f(a)) As we know, the zeroes of the second derivative of a quartic are inflection points and the roots of the second derivative are the x values of the inflection points. As assumed earlier, the roots are 0 and a. By multiplying (x- 0) and (x-a) by each other we can get a quadratic equation which is the second derivative of general quartic function. Integrating above function twice, we will get the original general quartic function. (C and D are constants) As passes through origin i.e. point Q, therefore, , and the value of constant D will be zero and the equation will be reduced as shown below: Also the quartic function passes through point R (a, f(a)). Putting value of , the value of can be calculated as shown below. There x, y coordinate of point R is (, ) Let us assume that the equation of a line that goes through points Q and R is given by . Since the line passes through origin (i.e. through point Q), therefore the value of b will be zero. As the line passes through Q and R, therefore slope, ‘m’ can be calculated as follows: Now, equation of intersection line will be: For calculating all the intersection points, setting this line equation equal to quartic function equation. As assumed earlier, the two roots of equation are 0 and a. Therefore, dividing above quartic equation by and for finding other roots. Dividing by using synthetic division, the quartic equation reduces to: Now again dividing the above-reduced equation by using synthetic division, the equation further reduces to: Now, by using the quadratic equation formula for roots, other roots of the above function can be found as follows: Therefore, the ratio of PQ:QR:RS is :: :1: Or If calculated, it simplifies to -0.618033988 : 1 : -0.618033988 and 1.61803398 : 1 : 1.61803398 Hence, we can a draw a conclusion that the ratio between the points of inflection passing through the quartic as a straight line will always have the distance between them in the ratio of PQ:QR:RS =1: 1.61803398:1 (or ) also well-known as a golden ratio. 7. Extend this investigation to other quartic functions that are not strictly of a “W” shape. As we know, all quartics of a W shape share two distinct, non-horizontal inflection points. However, for other quartic functions these properties do not apply. There are four different types of quartics possible that are quartics with no inflection points; quartics with one-inflection points; quartics with two inflection points, of which one or two are horizontal; and quartics with two distinct, real inflection points that are not horizontal (i.e. W shape quartics). For W shape quartics, we have already proved the ratio will be golden ratio. The example of quartics with no inflection points is f(x) = x4 and for which it is not possible to determine the ratio because no line would be formed between the inflection points as there is no inflection points. For quartics with one inflection point it will not possible to find ratio as for calculating ratio we need to have two inflection points Q and R. however, in this case there will be one. For quartics with two inflection points, of which one or two are horizontal the ratio will be again Golden ratio as the general conjecture from question 6 applies to all quartics with two distinct inflection points. Read More

1. Graph the function f(x) = x4 – 2x3 – 12x2 + 15x +12

The plot for the quartic function: f(x) = x4 – 2x3 – 12x2 + 15x +12 is as follows.

• Plot of the given Quartic Equation

 

f(x) =

g(x) = 2  -10

S

R

Q

P

 

 

2. Find the coordinates of the points of inflection Q and R. Determine the points P and S, where the line QR intersects the quartic function again, and calculate the ratio PQ:QR: RS.

Calculations for point of inflection

As per definition, a point of inflection is a point at which the second derivative of an equation is equal to zero ( ). After differentiating the above-given equation, we get the first and second derivatives as follows.

First derivative :

• f( ) =
• f΄( ) =
• f΄ ( ) =

Second derivative ( ):

• f΄( ) =
• f΄΄( ) = 12
• f΄΄( ) =

 

Let f΄΄( ) =

This will allow us to find the x values.

Solving this equation by factorizing it, we get the following values of x:

(  - 2)(  + 1) = 0

 = 2 or  = -1

The value of the function for these values of “x” will be as follows.

• f(2) = -2 -12 +15 (2)+12
• f(2) = 16-16-48+42 = -6

 

• f(-1) = -2 -12 +15 (-1)+12
• f(-1) = 1+2-12-3 = -12

Finding the g(x) intersection line equation:

-6 = -3m

In order to find the points of intersection between f( ) and g( ) we solve f( ) = g( ).

 = 2

 =

It is a 4th power equation

This equation has 4 real roots:

Coordinates for P = (-2.854; -15.708)

y = 2  -10

f(-2.854) = 2 (-2.854)-10

f(-2.854) = -15.708

Coordinates for Q = (-1; -12)

Coordinates for R = (2; -6)

Coordinates for S = (3.854; -2.292)

y = 2  -10

f(3.854) = 2 (3.854)-10

f(3.854) = -2.292

PQ =

RS =

PQ is equal to RS

If we simplify this ratio with respect to PQ and RS, then we get the ratio as follows.

PQ: QR: RS = 4.146: 6.708: 4.146

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

PQ: QR: RS = 4.146: 6.708: 4.146 = 1: 1.618: 1 (to make PQ = 1, dividing by 4.146)

Therefore, we can deduct that PQ = RS and ratios of QR: PQ and QR: RS are equal and is approximate. This number (1.618) is commonly referred to as the Golden Ratio. It is a number that is abundant in nature and is often present in numerous proportions.

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

Taking quartic function as f(x) =  –  +  -8  – 2 : it was solved by integration as below

In order to have a “w” shape graph, first of all, we need to take an ordinary quadratic equation and integrate it two times to get the “ ” value in the 4th power.

Step 1

                    Find a simple quadratic function:  

Step 2

Let

 ( -3) ( -1) = 0

   = 3 or  = 1

Step 3

Using Integrals we, first of all, find a cubic function.

Step 4

Yet again, using integrals we find the quartic function from the cubic function. However, we can remove the value and take only the equation.

Therefore, the equation will be (putting = )

Solved by software “Equation Wizard”

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