1. Whiteboardmaths.com © 2007 All rights reserved 5 7 2 1

2. Non-Linear Sequences Points = 2 Points = 3 Points = 4 Points = 5 Points = 6 Lines = 1 Lines = 3 Lines = 6 Lines = 10 Lines = 15 1510631L 65432P

3. Reminder Non-Linear Sequences 5, 8, 11, 14, 17,………………..? 1 st 2 nd 3 rd 4 th 5 th................................n th terms  position  3 3 difference  Remember: From previous work we know that a linear number sequence is a sequence of numbers that has a constant difference between adjacent terms. In the particular case shown below we wanted to obtain a general rule that gave us the value of any term (n th ) in the sequence as a function of the term’s position and by looking at the differences we found this to be 3n + 2. We now want to be able to find the rule when the function in question is not linear. The major part of this work will focus on finding the rule when the function is quadratic although we will briefly look at other functions such as the cubics. Linear Quadratic Cubic Non -Linear

4. Quadratics f(x) = x 2 12345 f(x) = x 2 + 6 12345 f(x) = x 2 + x - 5 12345 f(x) = 2x 2 12345 f(x) = 2x 2 + 3x + 4 12345 f(x) = 2x 2 - x - 1 Calculate f(x) and complete the 1 st and 2 nd difference entries for each of the quadratics functions shown below. Comment on your results. 12345 1491625 3579 222

5. f(x) = x 2 f(x) = x 2 + 6 f(x) = x 2 + x - 5 f(x) = 2x 2 f(x) = 2x 2 + 3x + 4f(x) = 2x 2 - x - 1 The results show that the second differences are constant. They also suggest that when the difference is 2, the co-efficient of x 2 is always 1 and when the difference is 4 the the co-efficient of x 2 is always 2. This is always true and goes a long way in helping us “guess” the rule by trial and improvement provided it is not too complicated.

6. f(x) = ? 1 -2 2345 4142846 f(x) = ? 12345 69142130 12345 310213655 12345 59152333 12345 514274465 12345 014916 f(x) = ? Work out the rule for each table below by looking at the second differences.

7. f(x) = x 2 + 5f(x) = 2x 2 + x 1 -2 2345 4142846 6101418 444 f(x) = 2x 2 - 4 f(x) = x 2 + x + 3f(x) = 2x 2 + 3x 12345 69142130 3579 222 12345 310213655 7111519 444 12345 59152333 46810 222 12345 514274465 9131721 444 12345 014916 1357 222 f(x) = x 2 - 2x + 1 Work out the rule for each table below by looking at the second differences.

8. Mystic Points = 2 Points = 3 Points = 4 Points = 5 Points = 6 Lines = 1 Lines = 3 Lines = 6 Lines = 10 Lines = 15 1510631L 65432P Use the difference method to find the rule connecting the number of lines L, with the number of points P. P(P - 1) 2 L = ½P 2 - ½P = This also generates the triangular numbers

9. f(x) = 3x 2 f(x) = 3x 2 + 6 f(x) = 3x 2 + x - 5 f(x) = 4x 2 - 1f(x) = 4x 2 - 3x + 4f(x) = 4x 2 + x - 9 Calculate f(x) and complete the 1 st and 2 nd difference entries for each of the quadratics functions shown below. Comment on your results.

10. f(x) = 3x 2 f(x) = 3x 2 + 6 f(x) = 3x 2 + x - 5 f(x) = 4x 2 - 1f(x) = 4x 2 - 3x + 4f(x) = 4x 2 + x - 9 The results suggest that the second difference is 6 when the co-efficient of x 2 is 3, and 8 when the co-efficient of x 2 is 4. Combining this with the previous examples it seems clear that the co-efficient of x 2 can be found by simply by dividing the second difference by 2.

11. f(x) = ? 12345 11315995 12202836 888 12345 716315279 9152127 666 12345 310213655 7111519 444 Work out the rule for each table below by looking at the second differences. f(x) = ? 12345 414305280 10162228 666 12345 212305690 10182634 888 12345 -6195499 15253545 10 f(x) = ? -21

12. f(x) = 4x 2 - 5f(x) = 3x 2 + 4 f(x) = 2x 2 + x 12345 11315995 12202836 888 12345 716315279 9152127 666 12345 310213655 7111519 444 Work out the rule for each table below by looking at the second differences. f(x) = 3x 2 + xf(x) = 4x 2 - 2x 12345 414305280 10162228 666 12345 212305690 10182634 888 12345 -6195499 15253545 10 f(x) = 5x 2 - 26 -21 There is a method for working out any quadratic function that does not involve differences. This method is extension work but you can view it below if you wish. Go to General Method Slides Continue

13. Cubic f(x) = x 3 f(x) = 2x 3 + x f(x) = 3x 3 - 1 1234512345 12345 Calculate f(x) and complete the 1 st, 2 nd and 3 rd difference entries for each of the cubic functions shown below. Comment on your results.

14. f(x) = x 3 f(x) = 2x 3 + x f(x) = 3x 3 - 1 12345 182764 7193761 121824 12345 31857 153975 243648 12345 22380 2157 365472 Calculate f(x) and complete the 1 st, 2 nd and 3 rd difference entries for each of the cubic functions shown below. Comment on your results. 6612 18 125 132 255 123 191 374 111 183 In a cubic function the 3 rd differences are constant and you can get the co-efficient of x 3 by dividing this difference by 6.

15. f(x) = ? 12345 062460 6183660 121824 66 120 f(x) = ? 12345 11451 133773 243648 12 124 245 121 f(x) = ? 12345 82986 2157 365472 18 197 380 111 183 In a cubic function the 3 rd differences are constant and you can get the co-efficient of x 3 by dividing this difference by 6. Work out the rule for each table below by looking at the third differences.

16. f(x) = x 3 - x 12345 062460 6183660 121824 66 120 f(x) = 2x 3 - x 12345 11451 133773 243648 12 124 245 121 f(x) = 3x 3 + 5 12345 82986 2157 365472 18 197 380 111 183 In a cubic function the 3 rd differences are constant and you can get the co-efficient of x 3 by dividing this difference by 6. Work out the rule for each table below by looking at the third differences. ax 2  divide the second difference by 2 ax 3  divide the third difference by 6 ax 4  divide the ? difference by ? You may want to investigate differences in higher order functions.

17. Power f(x) = 2 x f(x) = 3 x - 1 f(x) = 4 x 1234512345 12345 Calculate f(x) and complete the 1 st, 2 nd and 3 rd difference entries for each of the Power functions shown below. Comment on your results.

18. f(x) = 2 x f(x) = 3 x - 1 f(x) = 4 x 12345 2481632 24816 248 12345 282680 61854 1236 12345 41664 1248 36 Calculate f(x) and complete the 1 st, 2 nd and 3 rd difference entries for each of the Power functions shown below. Comment on your results. 242472 242 162 108 256 1024 768 192 576144 108 432 Power functions grow very rapidly (exponentially). Use of the difference method fails to produce a constant difference at any point. Simple power functions (like those above) are normally reasonably easy to work out using trial and improvement.

19. f(x) = ? 12345 79132137 12345 5112983 12345 -6654 245 246 1014 Power functions grow very rapidly (exponentially). Use of the difference method fails to produce a constant difference at any point. Simple power functions (like those above) are normally reasonably easy to work out using trial and improvement. Work out the power rule in each case below.

20. f(x) = 2 x + 5f(x) = 3 x + 2 f(x) = 4 x - 10 12345 79132137 12345 5112983 12345 -6654 245 246 1014 Power functions grow very rapidly (exponentially). Use of the difference method fails to produce a constant difference at any point. Simple power functions (like those above) are normally reasonably easy to work out using trial and improvement. Work out the power rule in each case below.

21. Tower of Hanoi The Tower of Hanoi ABC 3115731M 54321D The table below from the Tower of Hanoi game shows the minimum number of moves M, needed to transfer up to 5 Discs D, from one peg to another according to the rules of the game. Find the rule linking M to D. M = 2 D - 1

22. Worksheets 1 - 8 Worksheet 1

23. Worksheet 2

24. Worksheet 3

25. Worksheet 4

26. f(x) = x 3 f(x) = 2x 3 + x f(x) = 3x 3 - 1 1234512345 12345 Calculate f(x) and complete the 1 st, 2 nd and 3 rd difference entries for each of the cubic functions shown below. Comment on your results. Worksheet 5

27. f(x) = 12345 062460 6183660 121824 66 120 f(x) = 12345 11451 133773 243648 12 124 245 121 f(x) = 12345 82986 2157 365472 18 197 380 111 183 Work out the rule for each table below by looking at the third differences. Worksheet 6

28. f(x) = 2 x f(x) = 3 x - 1 f(x) = 4 x 1234512345 12345 Calculate f(x) and complete the 1 st, 2 nd and 3 rd difference entries for each of the Power functions shown below. Comment on your results. Worksheet 7

29. f(x) = 12345 79132137 12345 5112983 12345 -6654 245 246 1014 Work out the power rule in each case below. Worksheet 8

30. Finding Quadratic Rules 01234 -7-272037 f(x) =ax 2 + bx + c When x = 0, f(x) = -7 When x = 1, f(x) = -2 When x = 2, f(x) = 7 -7 =a(0) 2 + b(0) + c  -7 = c -2 =a(1) 2 + b(1) + c  -2 = a + b - 7 7 =a(2) 2 + b(2) + c  7 = 4a + 2b - 7 a + b = 5 4a + 2b = 14 2a + 2b = 10 4a + 2b = 14  2a = 4  a = 2  2 + b = 5  b = 3 Substituting these values gives f(x) = 2x 2 + 3x - 7 Checking for f(4)  2 x 4 2 + 3 x 4 - 7 = 37 Method for finding a Quadratic Rule We will use the output values given in the table to work out the original function. Step 1: Substitute values for f(0), f(1) and f(2), into the general quadratic shown. Step 2: Set up a pair of simultaneous equations in a and b and solve. General Quadratic

31. 01234 -81163764 f(x) =ax 2 + bx + c When x = 0, f(x) = -8 When x = 1, f(x) = 1 When x = 2, f(x) = 16 -8 =a(0) 2 + b(0) + c  - 8 = c 1 =a(1) 2 + b(1) + c  1 = a + b - 8 16 =a(2) 2 + b(2) + c  16 = 4a + 2b - 8 a + b = 9 4a + 2b = 24 2a + 2b = 18 4a + 2b = 24  2a = 6  a = 3  3 + b = 9  b = 6 Substituting these values gives f(x) = 3x 2 + 6x - 8 Checking for f(4)  3 x 4 2 + 6 x 4 - 8 = 64 Method for finding a Quadratic Rule We will use the output values given in the table to work out the original function. Step 1: Substitute values for f(0), f(1) and f(2), into the general quadratic shown. Step 2: Set up a pair of simultaneous equations in a and b and solve. General Quadratic Try this one yourself

32. 01234 1810122446 f(x) =ax 2 + bx + c When x = 0, f(x) = 18 When x = 1, f(x) = 10 When x = 2, f(x) = 12 18 =a(0) 2 + b(0) + c  18 = c 10 =a(1) 2 + b(1) + c  10 = a + b + 18 12 =a(2) 2 + b(2) + c  12 = 4a + 2b + 18 a + b = -8 4a + 2b = -6 2a + 2b = -16 4a + 2b = -6  2a = 10  a = 5  5 + b = -8  b = -13 Substituting these values gives f(x) = 5x 2 - 13x + 18 Checking for f(4)  5 x 4 2 - 13 x 4 + 18 = 46 Method for finding a Quadratic Rule We will use the output values given in the table to work out the original function. Step 1: Substitute values for f(0), f(1) and f(2), into the general quadratic shown. Step 2: Set up a pair of simultaneous equations in a and b and solve. General Quadratic Try a tough one that you’d never guess!