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

Extracts from this document...

Introduction

Mr. Bailey

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

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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