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  • Level: GCSE
  • Subject: Maths
  • Word count: 3665

Gradient Function

Extracts from this document...

Introduction

Gradient Function

Gradient Function

Mr. Bailey

Adnan Sarayqum

CA9

Plan

I am aiming to find the relationship between the X values and the slopes/gradients from given points on a curve.

The first graph I will use is for the equation Y=X2. I will choose a fixed point e.g. (2, 4), (4, 16). I can choose the values of X which can be negative numbers or positive numbers. I will find gradient from different points on the curve using my fixed points. I will choose a higher value and a lower value than my fixed point and move towards my fixed point. I will see how the gradient (change in y/change in x) changes as I move closer to my fixed point. I will see the relationship between my fixed point and the gradient and try to look for a formula. I will also use other equations, for example, Y=3X2, Y=X2 - 4 and Y=X3. Then I will look at my results tables and graphs and try to formulate a general formula for gradients.

Y=X2

My first fixed point is 2, 4

x

y

change in y

change in x

gradient

1

1

3

1

3

1.1

1.21

2.79

0.9

3.1

1.2

1.44

2.56

0.8

3.2

1.3

1.69

2.31

0.7

3.3

1.4

1.96

2.04

0.6

3.4

1.5

2.25

1.75

0.5

3.5

1.6

2.56

1.44

0.4

3.6

1.7

2.89

1.11

0.3

3.7

1.8

3.24

0.76

0.2

3.8

1.9

3.61

0.39

0.1

3.9

1.99

3.9601

0.0399

0.01

3.99

1.999

3.996001

0.003999

0.001

3.999

2

4

2.001

4.004001

-0.004001

-0.001

4.001

2.01

4.0401

-0.0401

-0.01

4.01

2.1

4.41

-0.41

-0.1

4.1

2.2

4.84

-0.84

-0.2

4.2

2.3

5.29

-1.29

-0.3

4.3

2.4

5.76

-1.76

-0.4

4.4

2.5

6.25

-2.25

-0.5

4.5

2.6

6.76

-2.76

-0.6

4.6

2.7

7.29

-3.29

-0.7

4.7

2.8

7.84

-3.84

-0.8

4.8

2.9

8.41

-4.41

-0.9

4.9

3

9

-5

-1

5


Power: 2

Coefficient: 1

Fixed point: 2

My second fixed point is 5, 25

x

y

change in y

change in x

gradient

4

16

9

1

9

4.1

16.81

8.19

0.9

9.1

4.2

17.64

7.36

0.8

9.2

4.3

18.49

6.51

0.7

9.3

4.4

19.36

5.64

0.6

9.4

4.5

20.25

4.75

0.5

9.5

4.6

21.16

3.84

0.4

9.6

4.7

22.09

2.91

0.3

9.7

4.8

23.04

1.96

0.2

9.8

4.9

24.01

0.99

0.1

9.9

4.99

24.9001

0.0999

0.01

9.99

4.999

24.99

0.009999

0.001

9.999

5

25

5.001

25.01

-0.010001

-0.001

10.001

5.01

25.1001

-0.1001

-0.01

10.01

5.1

26.01

-1.01

-0.1

10.1

5.2

27.04

-2.04

-0.2

10.2

5.3

28.09

-3.09

-0.3

10.3

5.4

29.16

-4.16

-0.4

10.4

5.5

30.25

-5.25

-0.5

10.5

5.6

31.36

-6.36

-0.6

10.6

5.7

32.49

-7.49

-0.7

10.7

5.8

33.64

-8.64

-0.8

10.8

5.9

34.81

-9.81

-0.9

10.9

6

36

-11

-1

11

Power: 2

Coefficient: 1

Fixed Point: 5

My third fixed point is -3, 9

x

y

change in y

change in x

gradient

-4

16

-7

1

-7

-3.9

15.21

-6.21

0.9

-6.9

-3.8

14.44

-5.44

0.8

-6.8

-3.7

13.69

-4.69

0.7

-6.7

-3.6

12.96

-3.96

0.6

-6.6

-3.5

12.25

-3.25

0.5

-6.5

-3.4

11.56

-2.56

0.4

-6.4

-3.3

10.89

-1.89

0.3

-6.3

-3.2

10.24

-1.24

0.2

-6.2

-3.1

9.61

-0.61

0.1

-6.1

-3.01

9.0601

-0.0601

0.01

-6.01

-3.001

9.006001

-0.006001

0.001

-6.001

-3

9

-2.999

8.994001

0.005999

-0.001

-5.999

-2.99

8.9401

0.0599

-0.01

-5.99

-2.9

8.41

0.59

-0.1

-5.9

-2.8

7.84

1.16

-0.2

-5.8

-2.7

7.29

1.71

-0.3

-5.7

-2.6

6.76

2.24

-0.4

-5.6

-2.5

6.25

2.75

-0.5

-5.5

-2.4

5.76

3.24

-0.6

-5.4

-2.3

5.29

3.71

-0.7

-5.3

-2.2

4.84

4.16

-0.8

-5.2

-2.1

4.41

4.59

-0.9

-5.1

-2

4

5

-1

-5

...read more.

Middle

3.8

43.32

-16.32

-0.8

20.4

3.9

45.63

-18.63

-0.9

20.7

4

48

-21

-1

21

Power: 2

Coefficient: 3

Fixed point: 3

My Second fixed point 6,108

x

y

Change in y

Change in x

Gradient

5

75

33

1

33

5.1

78.03

29.97

0.9

33.3

5.2

81.12

26.88

0.8

33.6

5.3

84.27

23.73

0.7

33.9

5.4

87.48

20.52

0.6

34.2

5.5

90.75

17.25

0.5

34.5

5.6

94.08

13.92

0.4

34.8

5.7

97.47

10.53

0.3

35.1

5.8

100.92

7.08

0.2

35.4

5.9

104.43

3.57

0.1

35.7

5.99

107.6403

0.3597

0.01

35.97

5.999

107.964

0.035997

0.001

35.997

6

108

0

6.001

108.036

-0.036003

-0.001

36.003

6.01

108.3603

-0.3603

-0.01

36.03

6.1

111.63

-3.63

-0.1

36.3

6.2

115.32

-7.32

-0.2

36.6

6.3

119.07

-11.07

-0.3

36.9

6.4

122.88

-14.88

-0.4

37.2

6.5

126.75

-18.75

-0.5

37.5

6.6

130.68

-22.68

-0.6

37.8

6.7

134.67

-26.67

-0.7

38.1

6.8

138.72

-30.72

-0.8

38.4

6.9

142.83

-34.83

-0.9

38.7

7

147

-39

-1

39

Power: 2

Coefficient: 3

Fixed point: 6

My third fixed point is -5, 75

x

y

Change in y

Change in x

Gradient

-6

108

0

12

0

-5.9

104.43

3.57

11.9

0.3

-5.8

100.92

7.08

11.8

0.6

-5.7

97.47

10.53

11.7

0.9

-5.6

94.08

13.92

11.6

1.2

-5.5

90.75

17.25

11.5

1.5

-5.4

87.48

20.52

11.4

1.8

-5.3

84.27

23.73

11.3

2.1

-5.2

81.12

26.88

11.2

2.4

-5.1

78.03

29.97

11.1

2.7

-5.01

75.3003

32.6997

11.01

2.97

-5.001

75.03

32.969997

11.001

2.997

-5

75

33

-4.999

74.97

33.029997

10.999

3.003

-4.99

74.7003

33.2997

10.99

3.03

-4.9

72.03

35.97

10.9

3.3

-4.8

69.12

38.88

10.8

3.6

-4.7

66.27

41.73

10.7

3.9

-4.6

63.48

44.52

10.6

4.2

-4.5

60.75

47.25

10.5

4.5

-4.4

58.08

49.92

10.4

4.8

-4.3

55.47

52.53

10.3

5.1

-4.2

52.92

55.08

10.2

5.4

-4.1

50.43

57.57

10.1

5.7

-4

48

60

10

6

Power: 2

Coefficient: 3

Fixed point: -5

image18.png

Observation

For the equation Y=3X2, the coefficient is 3 the power is 2.

I have used 3 different fixed points which are

(3, 27), (6, 108) and (-5, 75). From my results I can see that the closer I get to the fixed points, the gradient gets closer to 3 times the squared value of X. Using the information I can conclude that the gradient on any point on that curve will be three times the squared value of X co-ordinate. Therefore, the gradient on this curve is

m=3x2Ximage03.pngimage04.png

image05.pngimage06.png

Using differentiation for Y=3X2

image07.png

dy

dximage08.png

=3x2X

The graph for this equation and the graph for Y=X2 looks similar, because I have only changed the coefficient.

Y=X3

My first fixed point is 3, 27

x

y

change in y

change in x

gradient

2

8

19

1

19

2.1

9.261

17.739

0.9

19.71

2.2

10.648

16.352

0.8

20.44

2.3

12.167

14.833

0.7

21.19

2.4

13.824

13.176

0.6

21.96

2.5

15.625

11.375

0.5

22.75

2.6

17.576

9.424

0.4

23.56

2.7

19.683

7.317

0.3

24.39

2.8

21.952

5.048

0.2

25.24

2.9

24.389

2.611

0.1

26.11

2.99

26.7309

0.269101

0.01

26.9101

2.999

26.97301

0.026991001

0.001

26.991

3

27

3.001

27.02701

-0.027009

-0.001

27.009

3.01

27.2709

-0.270901

-0.01

27.0901

3.1

29.791

-2.791

-0.1

27.91

3.2

32.768

-5.768

-0.2

28.84

3.3

35.937

-8.937

-0.3

29.79

3.4

39.304

-12.304

-0.4

30.76

3.5

42.875

-15.875

-0.5

31.75

3.6

46.656

-19.656

-0.6

32.76

3.7

50.653

-23.653

-0.7

33.79

3.8

54.872

-27.872

-0.8

34.84

3.9

59.319

-32.319

-0.9

35.91

4

64

-37

-1

37

...read more.

Conclusion

m=3X 2

Using differentiation for Y=X3, I can write this gradient as

dyimage10.pngimage09.png

dximage08.png

image11.png

Y=3X2 +5x

My first fixed point is 3, 32

x

y

change in y

change in x

gradient

2

17

15

1

15

2.1

18.23

13.77

0.9

15.3

2.2

19.52

12.48

0.8

15.6

2.3

20.87

11.13

0.7

15.9

2.4

22.28

9.72

0.6

16.2

2.5

23.75

8.25

0.5

16.5

2.6

25.28

6.72

0.4

16.8

2.7

26.87

5.13

0.3

17.1

2.8

28.52

3.48

0.2

17.4

2.9

30.23

1.77

0.1

17.7

2.99

31.8203

0.1797

0.01

17.97

2.999

31.982

0.017997

0.001

17.997

3

32

3.001

32.018

-0.018003

-0.001

18.003

3.01

32.1803

-0.1803

-0.01

18.03

3.1

33.83

-1.83

-0.1

18.3

3.2

35.72

-3.72

-0.2

18.6

3.3

37.67

-5.67

-0.3

18.9

3.4

39.68

-7.68

-0.4

19.2

3.5

41.75

-9.75

-0.5

19.5

3.6

43.88

-11.88

-0.6

19.8

3.7

46.07

-14.07

-0.7

20.1

3.8

48.32

-16.32

-0.8

20.4

3.9

50.63

-18.63

-0.9

20.7

4

53

-21

-1

21

My second fixed point 4, 53

x

y

change in y

change in x

gradient

3

32

21

1

21

3.1

33.83

19.17

0.9

21.3

3.2

35.72

17.28

0.8

21.6

3.3

37.67

15.33

0.7

21.9

3.4

39.68

13.32

0.6

22.2

3.5

41.75

11.25

0.5

22.5

3.6

43.88

9.12

0.4

22.8

3.7

46.07

6.93

0.3

23.1

3.8

48.32

4.68

0.2

23.4

3.9

50.63

2.37

0.1

23.7

3.99

52.7603

0.2397

0.01

23.97

3.999

52.976

0.023997

0.001

23.997

4

53

4.001

53.024

-0.024003

-0.001

24.003

4.01

53.2403

-0.2403

-0.01

24.03

4.1

55.43

-2.43

-0.1

24.3

4.2

57.92

-4.92

-0.2

24.6

4.3

60.47

-7.47

-0.3

24.9

4.4

63.08

-10.08

-0.4

25.2

4.5

65.75

-12.75

-0.5

25.5

4.6

68.48

-15.48

-0.6

25.8

4.7

71.27

-18.27

-0.7

26.1

4.8

74.12

-21.12

-0.8

26.4

4.9

77.03

-24.03

-0.9

26.7

5

80

-27

-1

27

My third fixed point 5, 80

x

y

change in y

change in x

gradient

4

53

27

1

27

4.1

55.43

24.57

0.9

27.3

4.2

57.92

22.08

0.8

27.6

4.3

60.47

19.53

0.7

27.9

4.4

63.08

16.92

0.6

28.2

4.5

65.75

14.25

0.5

28.5

4.6

68.48

11.52

0.4

28.8

4.7

71.27

8.73

0.3

29.1

4.8

74.12

5.88

0.2

29.4

4.9

77.03

2.97

0.1

29.7

4.99

79.7003

0.2997

0.01

29.97

4.999

79.97

0.029997

0.001

29.997

5

80

5.001

80.03

-0.030003

-0.001

30.003

5.01

80.3003

-0.3003

-0.01

30.03

5.1

83.03

-3.03

-0.1

30.3

5.2

86.12

-6.12

-0.2

30.6

5.3

89.27

-9.27

-0.3

30.9

5.4

92.48

-12.48

-0.4

31.2

5.5

95.75

-15.75

-0.5

31.5

5.6

99.08

-19.08

-0.6

31.8

5.7

102.47

-22.47

-0.7

32.1

5.8

105.92

-25.92

-0.8

32.4

5.9

109.43

-29.43

-0.9

32.7

6

113

-33

-1

33

Observation image20.png

I have used 3 different fixed points which is (3, 32), (4, 58) and (5, 80) from my results I can see that the closer I get to the fixed points, the gradient gets closer to

Conclusion

For the equation y=x2, the gradient is m=2x.

For the equation y=x3,the gradient is m=3x.

Therefore, for a general equation of type y=xn the gradient will be m=xn

In the above equation the coefficient is 1. Changing the coefficient to ‘a’ the general equation for gradient will be m=axn

Adnan Sarayqum

CA9

...read more.

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