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# Polynomial Derivation from Data

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.
Published on: Mar 4, 2016
Published in: Education

#### Transcripts - Polynomial Derivation from Data

• 1. Linear Equation
• 2. B (Base) Q (Quantity) 6 39 7 45 The 8 51 data 9 57 x 20 100
• 3. B (Base) Q (Quantity) 6 39 Δ Find 7 45 +6 first 8 51 +6 difference 9 57 +6 x 20 100
• 4. 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
• 5. 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
• 6. 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
• 7. 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
• 8. Square Equation
• 9. B (Base) Q (Quantity) 6 39 7 52 The 8 67 data 9 84 x 20 100
• 10. 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
• 11. 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
• 12. 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
• 13. 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
• 14. 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
• 15. 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
• 16. Square Equation 2
• 17. B (Base) Q (Quantity) 6 159 7 213 8 275 9 345 x 20 100
• 18. B (Base) Q (Quantity) 6 159 Δ 7 213 +54 8 275 +62 9 345 +70 x 20 100
• 19. B (Base) Q (Quantity) 6 159 Δ1 Δ2 7 213 +54 8 275 +62 +8 9 345 +70 +8 x 20 100
• 20. 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
• 21. 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
• 22. 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
• 23. 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
• 24. 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
• 25. 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
• 26. 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
• 27. 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
• 28. 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
• 29. Cube Equation
• 30. B (Base) Q (Quantity) 6 275 7 425 8 621 9 869 x 20 100
• 31. B (Base) Q (Quantity) 6 275 Δ 7 425 +150 8 621 +196 9 869 +248 x 20 100
• 32. B (Base) Q (Quantity) 6 275 Δ1 Δ2 7 425 +150 8 621 +196 +46 9 869 +248 +52 x 20 100
• 33. B (Base) Q (Quantity) 6 275 Δ1 Δ2 Δ3 7 425 +150 8 621 +196 +46 9 869 +248 +52 +6 x 20 100
• 34. 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
• 35. 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
• 36. 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
• 37. 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
• 38. 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
• 39. 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
• 40. 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
• 41. 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
• 42. 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
• 43. 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
• 44. 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
• 45. 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