The document shows examples of linear, quadratic, and cubic equations derived from sets of data. For each type of equation: 1) The differences between corresponding data points are found and a pattern is identified. 2) The equation is constructed by removing terms related to the differences. 3) Constant terms are identified and the equations are used to predict values.

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