Polynomial Derivation from Data
45
Linear Equation
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Discovering the equation from data alone is possible when using differences and removing shapes [cubes (cubic), squares (quadratic), segs (linear), and ones (the constant)]. Explore the beauty of subQuan and number shapes here.
Transcript of Polynomial Derivation from Data
Linear Equation
B (Base)
Q (Quantity)
6 39
7 45 The
8 51 data
9 57
x
20
100
B (Base)
Q (Quantity)
6 39 Δ Find
7 45 +6 first
8 51 +6 difference
9 57 +6
x
20
100
B (Base)
Q (Quantity)
6 39 Δ First
7 45 +6 difference
8 51 +6 is
9 57 +6 six
x +6x
20 It’s a Seg Equation (linear equation)
100
B (Base)
Q (Quantity) Seg Equation
6 39 Δ 39 -6(6) Remove
7 45 +6 45 -6(7) the
8 51 +6 51 -6(8) six
9 57 +6 57 -6(9) Segs
x +6x
20
100
B (Base)
Q (Quantity) Seg Equation
6 39 Δ 39 -6(6) = +3
7 45 +6 45 -6(7) = +3 Constant
8 51 +6 51 -6(8) = +3 is
9 57 +6 57 -6(9) = +3 three
x +6x +3 = +6x +3
20
100
B (Base)
Q (Quantity) Seg Equation
6 39 Δ 39 -6(6) = +3
7 45 +6 45 -6(7) = +3
8 51 +6 51 -6(8) = +3
9 57 +6 57 -6(9) = +3
x +6x +3 = +6x +3
20 +6(20) +3 = +123 Predict
100 +6(100) +3 = +603 Predict
Square Equation
B (Base)
Q (Quantity)
6 39
7 52 The
8 67 data
9 84
x
20
100
B (Base)
Q (Quantity)
6 39 Δ Find
7 52 +13 first
8 67 +15 difference
9 84 +17
x not the same
20
100
B (Base)
Q (Quantity)
6 39 Δ1 Δ2 Find
7 52 +13 second
8 67 +15 +2 difference
9 84 +17 +2
x
20
100
B (Base)
Q (Quantity)
6 39 Δ1 Δ2 Second
7 52 +13 difference
8 67 +15 +2 is
9 84 +17 +2 two
x +2/2 x2 = +1x2
20 It’s a Square Equation (quadratic equation)
100
B (Base)
Q (Quantity) Square Equation
6 39 Δ1 Δ2 +39 –1(62) Remove
7 52 +13 +52 –1(72) the
8 67 +15 +2 +67 –1(82) one
9 84 +17 +2 +84 –1(92) square
x +1x2
20
100
B (Base)
Q (Quantity) Square Equation
6 39 Δ1 Δ2 +39 –1(62) = +3
7 52 +13 +52 –1(72) = +3 Constant
8 67 +15 +2 +67 –1(82) = +3 is
9 84 +17 +2 +84 –1(92) = +3 three
x +1x2 +3 = +1x2 +3
20
100
B (Base)
Q (Quantity) Square Equation
6 39 Δ1 Δ2 +39 –1(62) = +3
7 52 +13 +52 –1(72) = +3
8 67 +15 +2 +67 –1(82) = +3
9 84 +17 +2 +84 –1(92) = +3
x +1x2 +3 = +1x2 +3
20 +1(20)2 +3 = +403 Predict
100 +1(100)2 +3 = +10003 Predict
Square Equation 2
B (Base)
Q (Quantity)
6 159
7 213
8 275
9 345
x
20
100
B (Base)
Q (Quantity)
6 159 Δ
7 213 +54
8 275 +62
9 345 +70
x
20
100
B (Base)
Q (Quantity)
6 159 Δ1 Δ2
7 213 +54
8 275 +62 +8
9 345 +70 +8
x
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2
7 213 +54
8 275 +62 +8
9 345 +70 +8
x +8x2/2 = 4x2
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15
7 213 +54 213 -4(72) = +17
8 275 +62 +8 275 -4(82) = +19
9 345 +70 +8
x +8x2/2 = 4x2
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ
7 213 +54 213 -4(72) = +17 +2
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ
7 213 +54 213 -4(72) = +17 +2
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2 4x2 +2x
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ +15 -2(6) = +3
7 213 +54 213 -4(72) = +17 +2 +17 -2(7) = +3
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2 4x2 +2x
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ +15 -2(6) = +3
7 213 +54 213 -4(72) = +17 +2 +17 -2(7) = +3
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2 4x2 +2x
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ +15 -2(6) = +3
7 213 +54 213 -4(72) = +17 +2 +17 -2(7) = +3
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2 4x2 +2x 4x2 +2x +3
20
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ +15 -2(6) = +3
7 213 +54 213 -4(72) = +17 +2 +17 -2(7) = +3
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2 4x2 +2x 4x2 +2x +3
20 4(202) +2(20) +3 = 4(400) +40 +3 = 1,600 +43 = 1,643
100
B (Base)
Q (Quantity) Square Equation
6 159 Δ1 Δ2 159 –4(62) = +15 Δ +15 -2(6) = +3
7 213 +54 213 -4(72) = +17 +2 +17 -2(7) = +3
8 275 +62 +8 275 -4(82) = +19 +2
9 345 +70 +8
x +8x2/2 = 4x2 4x2 +2x 4x2 +2x +3
20 4(202) +2(20) +3 = 4(400) +40 +3 = 1,600 +43 = 1,643
100 4(1002) +2(100) +3 = 4(10,000) +200 +3 = 40,000 +203 = 40,203
Cube Equation
B (Base)
Q (Quantity)
6 275
7 425
8 621
9 869
x
20
100
B (Base)
Q (Quantity)
6 275 Δ
7 425 +150
8 621 +196
9 869 +248
x
20
100
B (Base)
Q (Quantity)
6 275 Δ1 Δ2
7 425 +150
8 621 +196 +46
9 869 +248 +52
x
20
100
B (Base)
Q (Quantity)
6 275 Δ1 Δ2 Δ3
7 425 +150
8 621 +196 +46
9 869 +248 +52 +6
x
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3
7 425 +150
8 621 +196 +46
9 869 +248 +52 +6
x +6x3/(2•3) = x3
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59
7 425 +150 425 –(73) = +82
8 621 +196 +46 621 –(83) = +109
9 869 +248 +52 +6
x +6x3/(2•3) = x3
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ
7 425 +150 425 –(73) = +82 +23
8 621 +196 +46 621 –(83) = +109 +27
9 869 +248 +52 +6
x +6x3/(2•3) = x3
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2
7 425 +150 425 –(73) = +82 +23
8 621 +196 +46 621 –(83) = +109 +27 +4
9 869 +248 +52 +6
x +6x3/(2•3) = x3
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2
7 425 +150 425 –(73) = +82 +23
8 621 +196 +46 621 –(83) = +109 +27 +4
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13 Δ
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16 -3
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -3
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13 Δ
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16 -3
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -3
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
x3 +2x2 -3x
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13 Δ -13 +3(6) = +5
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16 -3 -16 +3(7) = +5
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -3
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
x3 +2x2 -3x
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13 Δ -13 +3(6) = +5
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16 -3 -16 +3(7) = +5
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -3
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
x3 +2x2 -3x x3 +2x2 -3x +5
20
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13 Δ -13 +3(6) = +5
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16 -3 -16 +3(7) = +5
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -3
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
x3 +2x2 -3x x3 +2x2 -3x +5
20 (20)3 +2(20)2 -3(20) +5 = +8,000 +800 -60 +5 = +8,805 -60 = +8,745
100
B (Base)
Q (Quantity) Cubic Equation
6 275 Δ1 Δ2 Δ3 275 –(63) = +59 Δ1 Δ2 59 -2(62) = -13 Δ -13 +3(6) = +5
7 425 +150 425 –(73) = +82 +23 82 – 2(72) = -16 -3 -16 +3(7) = +5
8 621 +196 +46 621 –(83) = +109 +27 +4 109 -2(82) = -19 -3
9 869 +248 +52 +6
x +6x3/(2•3) = x3 x3 +4x2/2 = x3 +2x2
x3 +2x2 -3x x3 +2x2 -3x +5
20 (20)3 +2(20)2 -3(20) +5 = +8,000 +800 -60 +5 = +8,805 -60 = +8,745
100 (100)3 +2(100)2 -3(100) +5 = +1,000,000 +20,000 -300 +5 = +1,020,005 -300 = 1,019,705
