D = |SL – SR| = 0.823 – 0.323 = 0.5 = Error! Reference source not found..
→ It can be seen that when D-value is Error! Reference source not found.when the parabolas have a = 2 and there are four intersections between the parabola and the lines.
-
Consider the parabolas with a = 3
When the parabola y = 3(x – 2)2 + 3 and the lines y = x and y = 2x are graphed, the parabola does not cut the line y = x so there is only two x-values and the D-value cannot be found.
When the parabola y = 3(x – 1)2 + 1 and the lines y = x and y = 2x are graphed, four values of x are obtained.
x1 = 0.667 = Error! Reference source not found. x2 = 1.000 x3 = 1.333 = Error! Reference source not found. x4 = 2.000
SL = x2 – x1 = 1.000 – Error! Reference source not found. = Error! Reference source not found.
SR = x4 – x3 = 2.000 – Error! Reference source not found. = Error! Reference source not found.
D = |SL – SR| = Error! Reference source not found.– Error! Reference source not found. = Error! Reference source not found.
When the parabola y = 3(x – 4)2 + 2 and the lines y = x and y = 2x are graphed, four values of x are obtained.
x1 = 2.880 x2 = 3.333 x3 = 5.000 x4 = 5.786
SL = x2 – x1 = 3.333 – 2.880 = 0.453
SR = x4 – x3 = 5.786 – 5.786 = 0.786
D = |SL – SR| = 0.786 – 0.453 = 0.333 = Error! Reference source not found.
→ It can be seen that when D-value is Error! Reference source not found.when the parabolas have a = 3 and there are four intersections between the parabola and the lines.
-
Consider the parabolas with a = Error! Reference source not found.
When the parabola y = Error! Reference source not found. (x – 4)2 + 2 and the lines y = x and y = 2x are graphed, four values of x are obtained.
x1 = 2.000 x2 = 2.764 x3 = 7.236 x4 = 10.000
SL = x2 – x1 = 2.764 – 2.000 = 0.764
SR = x4 – x3 = 10.000 – 7.236 = 2.764
D = |SL – SR| = 2.764 – 0.764 = 2
→ It can be seen that when D-value is 2 when the parabolas have a = Error! Reference source not found.and there are four intersections between the parabola and the lines.
-
Consider the parabolas with a = Error! Reference source not found.
When the parabola y = Error! Reference source not found. (x – 4)2 + 2 and the lines y = x and y = 2x are graphed, four values of x are obtained.
x1 = 1.804 x2 = 2.628 x3 = 8.372 x4 = 12.196
SL = x2 – x1 = 2.628 – 1.804 = 0.824
SR = x4 – x3 = 12.196 – 8.372 = 3.824
D = |SL – SR| = 3.824 – 0.824 = 3
→ It can be seen that when D-value is 3 when the parabolas have a = Error! Reference source not found. and there are four intersections between the parabola and the lines.
From these examples, the conjecture can be obtained.
“For any four intersections of the two lines y = x, y = 2x and the parabola ax2 + bx + c (a > 0 and its vertex is in quadrant 1); the value of D which is calculated by D = |(x2 – x1) – (x4 – x3)| equals to Error! Reference source not found..”
Testing the Validity of the Statement
To test for the validity of the conjecture above, other parabolas which have a as a real number and have vertices place in any quadrants are investigated.
Firstly, the pattern in intersections of any parabolas that have real value of a and the lines y = x and y = 2x is investigated out to see whether the conjecture above still holds true.
-
a = - 4 (a Error! Reference source not found. Z)
When the parabola y = - 4x2 + 32x - 45 and the lines y = x and y = 2x are graphed, four values of x are obtained.
x1 = 1.934 x2 = 2.073 x3 = 5.427 x4 = 5.816
SL = x2 – x1 = 2.073 – 1.934 = 0.139
SR = x4 – x3 = 5.816 – 5.427 = 0.389
D = |SL – SR| = 0.389 – 0.139 = 0.25 = Error! Reference source not found.
-
a = Error! Reference source not found.a Error! Reference source not found. Q)
When the parabola y = - ½ x2 + 3x + 5 and the lines y = x and y = 2x are graphed, four values of x are obtained.
x1 = -2.317 x2 = -1.742 x3 = 5.742 x4 = 4.317
SL = x2 – x1 = -1.742 + 2.317 = 0.575
SR = x4 – x3 = 4.317 – 5.742 = -1.425
D = |SL – SR| = 0.575 + 1.425 = 2.
[…]
Formal proof
To prove the conjecture mathematically, the parabolas with a = k (k Error! Reference source not found. R) is considered.
x2 and x3 – the x-value of the intersections of the line y = x and the parabola y = kx2 + bx + c
Error! Reference source not found.kx2 + bx + c = x
Error! Reference source not found.kx2 + (b – 1)x + c = 0
Δ = (b – 1)2 - 4kc
Error! Reference source not found. x2, x3 = Error! Reference source not found.
Error! Reference source not found.x2 + x3 = Error! Reference source not found. = Error! Reference source not found.
x1 and x4 – the x-value of the intersections of the line y = 2x and the parabola y = kx2 + bx + c
Error! Reference source not found.kx2 + bx + c = x
Error! Reference source not found.kx2 + (b – 2)x + c = 0
Δ = (b – 2)2 - 4kc
Error! Reference source not found. x1, x4 = Error! Reference source not found.
Error! Reference source not found.x1 + x4 = Error! Reference source not found. = Error! Reference source not found.
D = |(x2 – x1) – (x4 – x3)| = |(x2 + x3) – (x1 + x4)| = | Error! Reference source not found. – Error! Reference source not found. Error! Reference source not found.
Therefore, the conjecture is mathematically proven.
Further investigation
[The pattern in intersections of the parabolas and two different intersecting lines]
For further investigation, the intersecting lines are changed to see whether the conjecture holds true.
Many different combinations of intersecting lines other than y = x and y = 2x are considered.
When the parabola y = 5(x – 4)2 + 1 and the lines y = x and y = 3x are graphed, four values of x are obtained.
x1 and x4 – the x-values of the intersections between the line y = 3x and the parabola on the left and right hand side of the graph respectively.
x2 and x3 – the x-values of the intersections between the line y = x and the parabola on the left and right hand side of the graph respectively.
x1 = 2.787 x2 = 3.319 x3 = 4.881 x4 = 5.813
SL = x2 – x1 = 3.319 – 2.787 = 0.532
SR = x4 – x3 = 5.813 – 4.881 = 0.932
D = |SL – SR| = 0.932 – 0.532 = 0.400 = Error! Reference source not found.
When the parabola y = 5(x – 4)2 + 1 and the lines y = 2x and y = 3x are graphed, four values of x are obtained.
x1 and x4 – the x-values of the intersections between the line y = 3x and the parabola on the left and right hand side of the graph respectively.
x2 and x3 – the x-values of the intersections between the line y = 2x and the parabola on the left and right hand side of the graph respectively.
x1 = 2.787 x2 = 3.000 x3 = 5.400 x4 = 5.813
SL = x2 – x1 = 3.000 – 2.787 = 0.213
SR = x4 – x3 = 5.813 – 5.400 = 0.413
D = |SL – SR| = 0.413 – 0.213 = 0.2 = Error! Reference source not found.
When the parabola y = 5(x – 4)2 + 1 and the lines y = x and y = 5x are graphed, four values of x are obtained.
x1 and x4 – the x-values of the intersections between the line y = 5x and the parabola on the left and right hand side of the graph respectively.
x2 and x3 – the x-values of the intersections between the line y = x and the parabola on the left and right hand side of the graph respectively.
x1 = 2.488 x2 = 3.319 x3 = 4.881 x4 = 6.513
SL = x2 – x1 = 3.319 – 2.488 = 0.832
SR = x4 – x3 = 6.513 – 4.881 = 1.632
D = |SL – SR| = 1.632 – 0.832 = 0.8 = Error! Reference source not found.
Justification
The conjecture above can be proved as below.
x2 and x3 – the x-value of the intersections of the line y = mx and the parabola y = kx2 + bx + c
Error! Reference source not found.kx2 + bx + c = mx
Error! Reference source not found.kx2 + (b – m)x + c = 0
Δ = (b – m)2 - 4kc
Error! Reference source not found. x2, x3 = Error! Reference source not found.
Error! Reference source not found.x2 + x3 = Error! Reference source not found. = Error! Reference source not found.
x1 and x4 – the x-value of the intersections of the line y = nx and the parabola y = kx2 + bx + c
Error! Reference source not found.kx2 + bx + c = nx
Error! Reference source not found.kx2 + (b – n)x + c = 0
Δ = (b – 2)2 - 4kc
Error! Reference source not found. x1, x4 = Error! Reference source not found.
Error! Reference source not found.x1 + x4 = Error! Reference source not found. = Error! Reference source not found.
D = |(x2 – x1) – (x4 – x3)| = |(x2 + x3) – (x1 + x4)| = | Error! Reference source not found. – Error! Reference source not found. Error! Reference source not found.
[The pattern in intersections of the graph of cubic polynomial and the two lines y = x and y = 2x]
In this section, the parabola is replaced by the cubic polynomial. The cubic polynomial will drawn together with the two lines y = x and y = 2x to see whether there is any pattern in intersections of these lines.
Consider the cubic y = x3 – 7x2 + 10x – 2 and the lines y = x and y = 2x. There are 6 intersections formed.
x1, x2 and x3– the x-values of the intersections between the line y = x and the cubic from left to right respectively.
x4, x5 and x6 – the x-values of the intersections between the line y = 2x and the cubic from left to right respectively.
x1 = 0.281 x2 = 1.316 x3 = 5.403 x4 = 0.354 x5 = 1.000 x6 = 5.646
SL is the difference of the two left hand roots x1 and x4.
SL = x4 – x1 = 0.354 – 0.281 = 0.073
SM is the difference of the two middle roots x2 and x5.
SM = x2 – x5 = 1.316 – 1.000 = 0.316
SR id the difference of the two right hand roots x5 and x6.
SR = x6 – x5 = 5.646 – 5.403 = 0.243
→ It can be seen that SM = SL + SR (0.243 + 0.073 = 0.316) or x1 + x2 + x3 = x4 + x5 + x6 (= 7).
Consider the cubic y = 2x3 – 4x + 5. There are only two intersections between the line y = 2x and the cubic therefore there is no relationship deduced from it.
Consider the cubic y = 2x3 – 8x2 + 5x + 4. There are 6 intersections.
x1 = - 0.481 x2 = 1.311 x3 = 3.170 x4 = - 0.520 x5 = 1.138 x6 = 3.382
SL = 0.520 – 0.481 = 0.039
SM = 1.311 – 1.138 = 0.173
SR = 3.382 – 3.170 = 0.212
→ SR = SM + SL (0.039 + 0.173 = 0.212)
From these examples, the conjecture can be deduced:
“For any 6 intersections of any cubic polynomial y = ax3 + bx2 + cx + d (aError! Reference source not found.) and the lines y = x and y = 2x, there are three differences between the x-values of six roots SL, SM, SR. The sum of the two smaller differences equals to the bigger difference, i.e. |x1 – x4| + |x2 – x5| = |x3 – x6|.”
Justification
For the general cubic equation ax3 + bx + cx + d = 0, the sum of the roots is Error! Reference source not found. [Viete’s theorem]
x1, x2 and x3 are the roots of the equation
ax3 + bx2 + cx + d = x
ax3 + bx2 + (c – 1)x + d = 0
→ x1 + x2 + x3 = Error! Reference source not found.
x4, x5 and x6 are the roots of the equation
ax3 + bx2 + cx + d = 2x
ax3 + bx2 + (c – 2)x + d = 0
→ x1 + x2 + x3 = Error! Reference source not found.
→ It can be seen that the sum of the roots of the cubic equations does not depends on the coefficient of x.
→ x1 + x2 + x3 = x4 + x5 + x6
→ |x1 – x4| + |x2 – x5| = |x3 – x6|
[The pattern in intersections of the graph of higher order polynomial and the two lines y = x and y = 2x]
In this section, the parabola is replaced by the higher order polynomial (start from 4). The cubic polynomial will drawn together with the two lines y = x and y = 2x to see whether there is any pattern in intersections of these lines.
Consider y = x4 – 6x3 + 9x2 – 2. There are 8 intersections.
x1 = -0.377 x2 = 0.726 x3 = 2.000 x4 = 3.651 x5 = -0.343 x6 = 1.000
x7 = 1.529 x8 = 3.814
x1, x2, x3, x4 are the x-values of the intersections of the quadric polynomial and the y = x
x5, x6, x7, x8 are the x-values of the intersections of the quadric polynomial and the y = 2x
D = (x1 + x2 + x3 + x4) - (x5 + x6 + x7 + x8) = 6 – 6 = 0.
Consider y = 2x5 – 6x4 – 2x3 + 7x2 + 2x – 1. There are 10 intersections.
x1 = -0.914 x2 = -0.484 x3 = 0.340 x4 = 1.139 x5 = 2.919 x6 = -1.000
x7 = -0.384 x8 = 0.442 x9 = 1.000 x10 = 2.942
x1, x2, x3, x4, x5 are the x-values of the intersections of the quadric polynomial and the y = x
x6, x7, x8, x9, x10 are the x-values of the intersections of the quadric polynomial and the y = 2x
D = (x1 + x2 + x3 + x4 + x5) - (x6 + x7 + x8 + x9 + x10) = 3 – 3 = 0
The conjecture can be made: “The difference between the sum of all the x-values of the intersections of the polynomial and the line y = x and the sum of all those of the intersections of the polynomial the lines y = 2x equals to 0”
Justification
The justification is same as the justification for cubic polynomial. According to the Viete’s theorem, it can be seen that the sum of all the roots is independent of the coefficient of x. So the sum always equals to Error! Reference source not found. which is a constant for any intersections of the polynomial (with order higher than 3) and any changing lines.
