1. Linear Law “Transformation” of non-linear relationships into linear relationships

2. How it works Quadratic Curve: non-linear!

3. Transforming to linear relationship Linear ? General equation of linear relationship: −3 y x2x2 Plot y vs x 2

4. 3 y 1/x

5. Plot (1/y) vs x 2 m = a, c = b Example 1

6. Plot xy vs x 2 m = a, c = b Example 2

7. m = b, c = a Example 3 Plot xy vs

8. Plot lg y vs x m = lg b, c = lg a Q9

9. Plot xy vs x Plot y vs Grad = b, xy-intercept = aGrad = a, y-intercept = b Q16

10. Plot lg y vs x m = lg b, c = lg a + 3 lg b Q17

11. Plot lg (y – 4) vs x m = lg b, c = lg a Q18

12. Plot lg y vs lg x m = b, c = - lg a Q19

13. m = p, c = - q Q20 Plot vs x

14. Express y in terms of x?y = ?? (0,1) (4,9) a) y x2x2

15. Express y in terms of x? (4,0) (0,2)

16. (5, 9) (2, 3) x + 1 lg y Q1

17. Q (3, 9) P (1, 3) ln (x – 1) ln y Q2

18. The following table gives values of y corresponding to some value of x. x12345 y11.622.282.5 It is known that x and y are related by the equation. (i)Explain how a straight-line graph of against can be drawn to represent the given equation and draw it for the given data. Use this graph to estimate the value of a and of b. (ii) Express the given equation in another form suitable for a straight-line graph to be drawn. State the variables whose values should be plotted.

19. . (i)Explain how a straight-line graph of against can be drawn to represent the given equation and draw it for the given data. Use this graph to estimate the value of a and of b. (i) In order to plot 1/y against 1/x, we need to arrange the equation into (1). b/a represents the gradient and 1/a represents the vertical intercept. (1)

20. x12345 y11.622.282.5 1/x10.50.330.250.2 1/y10.6250.50.440.4 Choose appropriate scales 0.2 0.40.60.8 1.0 0.2 0.4 0.6 0.8 1.0

21. 0.2 0.40.60.8 1.0 0.2 0.4 0.6 0.8 1.0 (0.6, 0.7) (0,0.25)

22. (ii) Express the given equation in another form suitable for a straight-line graph to be drawn. State the variables whose values should be plotted.

23. Q1 The data for x and y given in the table below are related by a law of the form, where p and q are constants. x12345 y41.538.031.522.09.5 By drawing a suitable straight line, find estimates for p and q. Plot (y ─ x) against x 2, p represents the gradient and q represents the (y-x) -intercept.

24. x12345 y41.538.031.522.09.5 x2x2 149162525 y ─ x40.536.028.518.04.5 5 101520 25 5 10 15 20 25 30 35 40 45

25. Q2 The table shows the experimental values of two variables x and y which are known to be related by an equation of the form p(x + y – q) = qx 3, where p and q are constants. x0.51.01.52.02.5 y1.061.001.693.506.81 Draw a suitable straight-line graph to represent the above data. Use your graph to estimate (i)the value of p and of q, (ii)the value of y when x = 2.2. Plot (x + y) against x 3, (q/p) represents the gradient and q represents the (x + y) - intercept.

26. x0.51.01.52.02.5 y1.061.001.693.506.81 x3x3 0.12513.375815.625 x+y1.5623.195.59.31 5 101520 2 4 6 8 10

27. Q3 The table below shows experimental values of two variables, x and y. One value of y has been recorded incorrectly. x12345 y5.716.389.1014.2020.49 It is believed that x and y are related in the form y = x 2 – ax + b, where a and b are constants. Draw a suitable straight-line graph to represent the given data. Use your graph to estimate (i) the value of a and of b, (ii) a value of y to replace the incorrect value. Plot (y ─ x 2 ) against x, ─ a represents the gradient and b represents the (y ─ x 2 ) -intercept.

28. x12345 y5.716.389.1014.2020.49 x12345 y ─ x 2 4.712.380.10-1.80-4.51 1 234 5 1 2 3 4 5 -5 -4 -3 -2

29. Identify the incorrect readings/ outliers!!

30. x12345 y2.653.003.323.713.87 x+234567 y2y2 7.029.0011.0213.7614.98 1 234 5 2 4 6 8 10 6 7 12 14 16 ? One of the values of y is subject to an abnormally large error Identify the abnormal reading and estimate its correct value. abnormal reading: y = 3.71 Correct value should be

31. 1 234 5 2 4 6 8 10 6 7 12 14 16 Estimate the value of x when y = 2

32. Q4 The table below shows the experimental values of two variables x and y. It is known that one value of y has been recorded incorrectly x0.511.52.02.5 y1.201.000.860.700.66 It is known that x and y are related by an equation of the form, where a and b are constants. By plotting against x, obtain a straight-line graph to represent the above data. Use your graph to estimate the value of a and of b. (i) Use your graph to estimate a value of y to replace the incorrect value. (ii) Find the value of x when y =. (iii) By inserting another straight line to your graph, find the value of x and of y which satisfy the simultaneous equations and

33. x0.511.52.02.5 y1.201.000.860.700.66 x0.511.52.02.5 1/y0.8311.161.431.52 0.5 1.01.52.0 2.5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

34. 0.5 1.01.52.0 2.5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 abnormal reading: y = 0.70 Correct value should be x0.511.52.02.5 y1.201.000.860.700.66 x0.511.52.02.5 1/y0.8311.161.431.52

35. 0.5 1.01.52.0 2.5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Estimate the value of x when y =

36. and Need to draw this and find the point of intersection of the 2 lines Bear in mind: need to use the same axes as first line! Vertical intercept  (0, -1.2) Horizontal intercept  (0.8, 0)

37. 0.5 1.01.52.0 2.5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Vertical intercept  (0, -1.2) Horizontal intercept  (0.8, 0) -0.2 -0.4 -0.6 -0.8 -1.2

38. Q5 The variables x and y are known to be connected by the equation An experiment gave pairs of values of x and y as shown in the table. One of the values of y is subject to an abnormally large error. x1234567 y56.2029.9025.108.916.313.351.78 Plot lg y against x and use the graph to (i) identify the abnormal reading and estimate its correct value. (ii) estimate the value of C and of a. (iii) estimate the value of x when y = 1.

39. x1234567 y56.2029.9025.108.916.313.351.78 lg y1.751.481.400.950.800.530.25 1 234 5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 6 7 1.8 2.0 (i) abnormal reading: y = 25.10 Correct value should be

40. 1 234 5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 6 7 1.8 2.0 89 estimate the value of x when y = 1.