Linear Equations

Linear Group Function Numbers

1. y=a0+a1*x
2. y=a0+a1/x
3. y=a0+a1*sqrt(x)
4. y=a0+a1*sqrt(x)
5. y=a0+a1/x^2
6. y=a0+a1*exp(x)
7. y=a0+a1*exp(–x)
8. y =a0+a1*ln(x)
9. y=a0+a1/ln(x)
10. y=a0+a1*(ln(x))^2
11. 1/y=a0+a1*x
12. 1/y=a0+a1/x
13. 1/y=a0+a1*sqrt(x)
14. 1/y=a0+a1/sqrt(x)
15. 1/y=a0+a1/x^2
16. 1/y=a0+a1*exp(x)
17. 1/y=a0+a1*exp(–x)
18. 1/y=a0+a1*ln(x) 
19. 1/y=a0+a1/ln(x)
20. 1/y=a0+a1*(ln(x))^2
21. y^2=a0+a1*x
22. y^2=a0+a1/x
23. y^2=a0+a1*sqrt(x)
24. y^2=a0+a1/sqrt(x)
25. y^2=a0+a1/x^2
26. y^2=a0+a1*exp(x)
27. y^2=a0+a1*exp(–x)
28. y^2=a0+a1*ln(x)
29. y^2=a0+a1/ln(x)
30. y^2=a0+a1*(ln(x))^2
31. sqrt(y)=a0+a1*x
32. sqrt(y)=a0+a1/x
33. sqrt(y)=a0+a1*sqrt(x)
34. sqrt(y)=a0+a1/sqrt(x)
35. sqrt(y)=a0+a1/x^2
36. sqrt(y)=a0+a1*exp(x)
37. sqrt(y)=a0+a1*exp(–x)
38. sqrt(y)=a0+a1*ln(x)
39. sqrt(y)=a0+a1/ln(x)
40. sqrt(y)=a0+a1*(ln(x))^2

Exponential Group Function Numbers

41. ln(y)=a0+a1*x
42. ln(y)=a0+a1/x 
43. ln(y)=a0+a1*sqrt(x)
44. ln(y)=a0+a1/sqrt(x)
45. ln(y)=a0+a1/x^2
46. ln(y)=a0+a1*exp(x)
47. ln(y)=a0+a1*exp(–x)
48. ln(y)=a0+a1*ln(x)
49. ln(y)=a0+a1/ln(x)
50. ln(y)=a0+a1*(ln(x))^2

Power Group Function Numbers

51. y=a0+a1*x^2
52. y=a0+a1*x^3
53. y=a0+a1*x^4
54. y=a0+a1*x^5
55. y=a0+a1*x^6
56. 1/y=a0+a1*x^2
57. 1/y=a0+a1*x^3
58. 1/y=a0+a1*x^4
59. 1/y=a0+a1*x^5
60. 1/y=a0+a1*x^6
61. y^2=a0+a1*x^2
62. y^2=a0+a1*x^3
63. y^2=a0+a1*x^4
64. y^2=a0+a1*x^5
65. y^2=a0+a1*x^6
66. sqrt(y)=a0+a1*x^2
67. sqrt(y)=a0+a1*x^3
68. sqrt(y)=a0+a1*x^4
69. sqrt(y)=a0+a1*x^5
70. sqrt(y)=a0+a1*x^6
71. ln(y)=a0+a1*x^2
72. ln(y)=a0+a1*x^3
73. ln(y)=a0+a1*x^4
74. ln(y)=a0+a1*x^5
75. ln(y)=a0+a1*x^6   

Polynomial Group Function Numbers

76. y=a0+a1*x+a2*x^2
77. y=a0+a1*x+a2*x^2+a3*x^3
78. y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
79. y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
80. y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
81. 1/y=a0+a1*x+a2*x^2
82. 1/y=a0+a1*x+a2*x^2+a3*x^3
83. 1/y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
84. 1/y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
85. 1/y=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
86. y^2=a0+a1*x+a2*x^2
87. y^2=a0+a1*x+a2*x^2+a3*x^3
88. y^2=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
89. y^2=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
90. y^2=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
91. sqrt(y)=a0+a1*x+a2*x^2
92. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3
93. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
94. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
95. sqrt(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6
96. ln(y)=a0+a1*x+a2*x^2
97. ln(y)=a0+a1*x+a2*x^2+a3*x^3
98. ln(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4
99. ln(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5
100. ln(y)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6

Notes:

• Equations 11-20, 56-60, 81-85: 1/y=f(x) corresponds to fitting data to y=1/(f(x))
  • For example, 1/y = a0+a1ln(x) is the linearized version of the nonlinear equation y = 1/(a0+a1exp(x)).
• Equations 11-20, 56-60, 81-85: 1/y=f(x) corresponds to fitting data to y=1/(f(x))
  • For example, 1/y = a0+a1ln(x) is the linearized version of the nonlinear equation y = 1/(a0+a1exp(x)).
• Equations 21-30, 61-65, 86-90: y^2=f(x) corresponds to fitting data to y = sqrt(f(x))
  • For example, y^2 = a0+a1x^2 is the linearized version of the nonlinear equation y = sqrt(a0+a1x^2).
• Equations 31-40, 66-70, 91-95: sqrt(y)=f(x) corresponds to fitting data to y=(f(x))^2
  • For example, sqrt(y) = a0+a1x+a2x^2+a3x^3 is the linearized version of the nonlinear fitting equation y = (a0+a1x+a2x^2+a3x^3)^2
• Equations 41-50, 71-75, 96-100: ln(y)=f(x) corresponds to fitting data to y = exp(f(x))
  • For example, ln(y) = a0+a1x is the linearized form of nonlinear equation y = exp(a0+a1x)
  • Note this corresponds to the familiar exponential equation y = a0′exp(a1′x) where a0′ = exp(a0) and a1′ = a1.

Nonlinear Equations

Standard Functions

Power
y=a0+a1*x^a2

Exponential
y=a0+a1*exp(-x/a2)

Peak Functions

Gaussian
y=a0+a1exp(-0.5((x–a2)/a3)^2)

Lorentzian
y=a0+a1/(1+((x–a2)/a3)^2)

Logistic Peak
y=a0+a14n/(1+n)^2 n = exp(-(x–a2)/a3)

Erfc Peak
y=a0+a1*erfc(((x–a2)/a3)^2)

Log-Normal
y=a0+a1exp(-0.5(ln(x/a2)/a3)^2)

Transition Functions

Sigmoidal
y=a0+a1/(1+exp(-(x–a2)/a3))

Cumulative
y=a0+a10.5(1+erf((x–a2)/(sqrt(2)*a3)))

DoseRspLgstc
y=a0+a1/(1+(x/a2)^a3)

pH Activity
y=(a0+a1*10^(x–a2))/(1+10^(x–a2))

1-Site Ligand
y=a0*x/(a1+x)

Photosynthesis Rate
y=a0a1x/(a0+a1*x)

Waveform Functions

SineWave
y=a0+a1sin(2pi*x/a3+a2)

SineWaveSquared
y=a0+a1(sin(2piπ*x/a3+a2))^2