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Introduction

My task is to investigate the relationship between the gradients of tangents on the curves of graphs, such as y=x2. To do this, I will first find the gradient of tangents on the graph y=x2 by drawing the graph.

I have labelled the tangents a-g. They go from x=-3 to x=4. Below are the calculations for their gradients. (I am using the formula (y2-y1)/(x2-x1)to calculate the gradient of the line.

a= (12-6)/(-3.5--2.5) = 6/-1 = -6 b= (6-2)/(-2.5--1.5) = 4/-1 = -4

c= (2-0)/(-1.5--0.5) = 2/-1 = -2 d= (2-0)/(1.5-0.5) = 2/1 = 2

e= (6-2)/(2.5-1.5) = 4/-1 = 4 f= (12-6)/(3.5-2.5) = 6/1 = 6

g= (20-12)/(4.5-3.5) = 8/1 = 8

As you can see, the results I have obtained are good round numbers. These results however are not accurate to the tangents I drew on the graph.

Middle

12

3

27

3.0000001

27.0000027

27

4

64

4.0000001

64.0000048

48

5

125

5.0000001

125.0000075

75

6

216

6.0000001

216.0000108

108

The first numbers I saw were the first and third numbers. I noticed these because the first number was 3, the power by which I was multiplying the values. The third number was 27, which is 33. This struck me as strange - only this number was cubed, none of the others. To explain this, I tried cross analysing these results with the results I obtained from the y=x2 graph. I saw that the first value was the value of the power by which I multiplied the values. The third value however was not 32, but the second value was 22. Moving back to the results for y=x3, I tried dividing all the results by 3, and was left with the x1 values squared. Therefore, the formula for the gradient of a tangent on the graph of y=x3 was 3x2.

Conclusion

n class="c5">2

y2

(y2-y1)/(x2-x1)

4x3

1

1

1.0000001

1.0000004

4.0000006

4

2

16

2.0000001

16.0000032

32.00000238

32

3

81

3.0000001

81.0000108

108.0000054

108

4

256

4.0000001

256.0000256

256.0000097

256

5

625

5.0000001

625.00005

500.0000144

500

6

1296

6.0000001

1296.000086

864.0000213

864

 The formula for the gradient of a tangent on the graph of y=xn is nx(n-1)

As I thought, the two sets of results are very close. This shows that the formula is correct. I now need to find a formula which link the different formulae together. As the first number in the formulae is always the power being multiplied by x, and the power in the formulae is always 1 less I can determine the linking formula.

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

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