# Gradient Function

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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 |

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

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=3x2X

Using differentiation for Y=3X2

dy

dx

=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 |

Conclusion

m=3X 2

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

dy

dx

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

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

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

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