# Gradient function

Extracts from this document...

Introduction

My coursework on the gradient function is to investigate the gradient on different points on the line and curves.

I will start of my investigation with y = x2. This will be a parabolic curve and the gradient will move from point to point. I will need to start of with a fixed point. I have chosen (2, 4). I will use a table to get close to the point. My table will have five columns. The first column will be x, which will have any numbers between 1and 3. The second column will be y, which will be the result of squaring an x numeral. The third column will be the increase in y, where squared value gets subtracted from 4. The fourth column will be the increase in x, this is where the x values get subtracted from 2. The fifth and last column will be the gradient, where the change in y divided by the change in x, gives me the results. I will do different fixed points so the numbers will vary.

Middle

21.6

2.5

25

-11

-0.5

22

2.6

27.04

-8.96

-0.4

22.4

2.7

29.16

-6.84

-0.3

22.8

2.8

31.36

-4.64

-0.2

23.2

2.9

33.64

-2.36

-0.1

23.6

2.99

35.7604

-0.2396

-0.01

23.96

2.999

35.976

-0.023996

-0.001

23.996

3

36

0

0

0

3.0001

36.0024

0.00240004

0.0001

24.0004

3.001

36.024

0.024004

0.001

24.004

3.01

36.2404

0.2404

0.01

24.04

3.1

38.44

2.44

0.1

24.4

3.2

40.96

4.96

0.2

24.8

3.3

43.56

7.56

0.3

25.2

3.4

46.24

10.24

0.4

25.6

3.5

49

13

0.5

26

3.6

51.84

15.84

0.6

26.4

3.7

54.76

18.76

0.7

26.8

3.8

57.76

21.76

0.8

27.2

3.9

60.84

24.84

0.9

27.6

3.99

63.6804

27.6804

0.99

27.96

3.999

63.968

27.968004

0.999

27.996

4

64

28

1

28

Power: 2

Coefficient: 2

Fixed point: 3

My second fixed point is 5, 100

x | y | increase in y | increase in x | gradient |

4 | 64 | -36 | -1 | 36 |

4.1 | 67.24 | -32.76 | -0.9 | 36.4 |

4.2 | 70.56 | -29.44 | -0.8 | 36.8 |

4.3 | 73.96 | -26.04 | -0.7 | 37.2 |

4.4 | 77.44 | -22.56 | -0.6 | 37.6 |

4.5 | 81 | -19 | -0.5 | 38 |

4.6 | 84.64 | -15.36 | -0.4 | 38.4 |

4.7 | 88.36 | -11.64 | -0.3 | 38.8 |

4.8 | 92.16 | -7.84 | -0.2 | 39.2 |

4.9 | 96.04 | -3.96 | -0.1 | 39.6 |

4.99 | 99.6004 | -0.3996 | -0.01 | 39.96 |

4.999 | 99.96 | -0.039996 | -0.001 | 39.996 |

5 | 100 | 0 | 0 | 0 |

5.0001 | 100.004 | 0.00400004 | 0.00010 | 40.0004 |

5.001 | 100.04 | 0.040004 | 0.001 | 40.004 |

5.01 | 100.4004 | 0.4004 | 0.01 | 40.04 |

5.1 | 104.04 | 4.04 | 0.1 | 40.4 |

5.2 | 108.16 | 8.16 | 0.2 | 40.8 |

5.3 | 112.36 | 12.36 | 0.3 | 41.2 |

5.4 | 116.64 | 16.64 | 0.4 | 41.6 |

5.5 | 121 | 21 | 0.5 | 42 |

5.6 | 125.44 | 25.44 | 0.6 | 42.4 |

5.7 | 129.96 | 29.96 | 0.7 | 42.8 |

5.8 | 134.56 | 34.56 | 0.8 | 43.2 |

5.9 | 139.24 | 39.24 | 0.9 | 43.6 |

5.99 | 143.5204 | 43.5204 | 0.99 | 43.96 |

5.999 | 143.952 | 43.952004 | 0.999 | 43.996 |

6 | 144 | 44 | 1 | 44 |

Power: 2

Coefficient: 2

Fixed point: 5

My third fixed points : 7, 196

x | y | increase in y | increase in x | gradient |

6 | 144 | -52 | -1 | 52 |

6.1 | 148.84 | -47.16 | -0.9 | 52.4 |

6.2 | 153.76 | -42.24 | -0.8 | 52.8 |

6.3 | 158.76 | -37.24 | -0.7 | 53.2 |

6.4 | 163.84 | -32.16 | -0.6 | 53.6 |

6.5 | 169 | -27 | -0.5 | 54 |

6.6 | 174.24 | -21.76 | -0.4 | 54.4 |

6.7 | 179.56 | -16.44 | -0.3 | 54.8 |

6.8 | 184.96 | -11.04 | -0.2 | 55.2 |

6.9 | 190.44 | -5.56 | -0.1 | 55.6 |

6.99 | 195.4404 | -0.5596 | -0.01 | 55.96 |

6.999 | 195.944 | -0.055996 | -0.001 | 55.996 |

7 | 196 | 0 | 0 | 0 |

7.0001 | 196.0056 | 0.00560004 | 0.00010 | 56.0004 |

7.001 | 196.056 | 0.056004 | 0.001 | 56.004 |

7.01 | 196.5604 | 0.5604 | 0.01 | 56.04 |

7.1 | 201.64 | 5.64 | 0.1 | 56.4 |

7.2 | 207.36 | 11.36 | 0.2 | 56.8 |

7.3 | 213.16 | 17.16 | 0.3 | 57.2 |

7.4 | 219.04 | 23.04 | 0.4 | 57.6 |

7.5 | 225 | 29 | 0.5 | 58 |

7.6 | 231.04 | 35.04 | 0.6 | 58.4 |

7.7 | 237.16 | 41.16 | 0.7 | 58.8 |

7.9 | 249.64 | 53.64 | 0.9 | 59.6 |

7.99 | 255.3604 | 59.3604 | 0.99 | 59.96 |

7.999 | 255.936 | 59.936004 | 0.999 | 59.996 |

8 | 256 | 60 | 1 | 60 |

Power: 2

Coefficient: 2

Fixed point: 7

The gradient function for y = 2x2 is definitely m = 4x because the x values are four times smaller than the gradients. E.g. my x numeral is -7 and my gradient is -28, if I divide them I get 4.

Conclusion

y = 4x. abandoned

My gradient function for y = 4x2 is 4x because if you the tables you will see that the gradients are four times bigger than there x values, an example of this x value -3 and gradient -12 as you can see -3 goes into -12 four times making the gradient function y = 4x.

My conclusion for y = x4 is gradient function is y = 3x2 .I got to this answer by noticing that my gradients for my x numerals were double but then I saw that the gradient was also three times larger than the x value also so at the end the slope must be y = 3x2 .

I can also predict the gradient function for y = x 5 .

Using this table I can predict that y = x^5 is gradient function y = 5x^3. As you can see 2 is the x value and the gradient is 40. To get to 40, you’ll need to cube the 2 and times by five and there you have your answer.

This student written piece of work is one of many that can be found in our GCSE Gradient Function section.

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