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  • Level: GCSE
  • Subject: Maths
  • Word count: 2459

The Gradient Function

Extracts from this document...

Introduction

The Gradient Function

The aim of this investigation is to discover the gradient function for the graphs y = ax where a and n are constants.  

I will do this by beginning with the simplest cases, as I believe that these will be the most simple equations to solve.  I am doing this in the hope that discovering the equations for these simple cases will aid me in discovering the more complex formulas.  

Firstly I will construct the graphs of: y=x, y=2x, y=3x, y=4x.  And attempt to find a general equation.

Y=x

X

1

2

3

4

Y

1

2

3

4

Y=2x

X

1

2

3

4

Y

2

4

6

8

Y=3x

X

1

2

3

4

Y

3

6

9

12

Y=4x

X

1

2

3

4

Y

4

8

12

16

(Graph2)

Using the rule stated on the candidate sheet I can calculated the gradient for each of the above straight lines.

Equation of line

Gradient Function

Y=x

1

Y=2x

2

Y=3x

3

Y=4x

4

Doing this I have discovered that the co-efficient of x is the gradient of the line.

I will now go on to construct the graph of y=x2.  This is because I know that the graph of x2 will be a curve and it is curves that I am investigating

X

1

2

3

4

Y

1

4

9

16

To find the gradient of this line, I will use the tangent method.  If I have a point on the curve, I will draw a tangent so only the point on the curve is touching the tangent.  I will then draw a right-angled triangle with the tangent.

...read more.

Middle

4

3

9

6

4

16

8

Y = 2x2

X=1, Y=2

Gradient = 2.42-2/1.1-1

= 4.2

Gradient = 2.0402-2/1.01-1

            = 4.02

Gradient = 2.004002-2/1.001-1

= 4.002

X=2, Y=8

Gradient = 8.82-2/2.1-2

= 8.2

Gradient = 8.0802-8/2.01-2

= 8.02

Gradient = 8.008002-8/2.001-2

= 8.002

X=3, Y=18

Gradient = 19.22-18/3.1-3

= 12.2

Gradient = 18.1202-18/3.01-3

= 12.02

Gradient = 18.012002-18/3.001-3

= 12.002

X=4, Y=32

Gradient = 33.62-32/4.1-4

= 16.2

Gradient =32.1602-32/4.01-4

= 16.02

Gradient = 32.016002-32/4.001-4

= 16.002

x

Y

Small increment

1

2

4

2

8

8

3

18

12

4

32

16

Y = 3x2

X=1, Y=3

Gradient = 3.63-3/1.1-1

= 6.3

Gradient = 3.0603-3/1.01-1

= 6.03

Gradient = 3.006003-3/1.001-1

= 6.003

X=2, =12

Gradient = 13.23-12/2.1-2

= 12.3

Gradient = 12.1203-12/2.01-2

= 12.03

Gradient = 12.012003-12/2.001-2

= 12.003

X=3, Y=27

Gradient = 28.83–27/3.1-3

= 18.3

Gradient =27.1803-27/3.01-3

= 18.03

Gradient = 27.018003-27/3.001-3

=18.003

X=4, Y=48

Gradient = 50.43-48/4.1-4

= 24.3

Gradient = 48.2403-48/4.01-4

=24.03

Gradient = 48.024003-48/4.001-4

=24.003

X

Y

Small increment

1

3

6

2

12

12

3

27

18

4

48

24

Y = x3

X = 1, Y=1

Gradient = 1.331-1/1.1-1

= 3.31

Gradient = 1.030301-1/1.01-1

= 3.0301

Gradient = 1.003003001-1/1.001-1

= 3.003001

X = 2, Y=8

Gradient = 9.261-8/2.1-2

= 12.61

Gradient = 8.120601-8/2.01-2

=12.0601

Gradient = 8.012006001-8/2.001-2

= 12.006001

X=3, Y=27

Gradient = 29.791-27/3.1-3

=27.91

Gradient = 27.270901-27/3.01-3

= 27.0901

Gradient = 27.027009-27/3.001-3

= 27.009001

X=4, Y=64

Gradient = 68.921-64/4.1-4

= 49.21

Gradient = 64.481201-64/4.01-4

= 48.1201

Gradient =64.048012-64/4.001-4

=48.012001

x

Y

Small increment

1

1

3

2

8

12

3

27

27

4

64

48

Y = 2x3

X =1, Y=2

Gradient = 2.662-2/1.1-1

= 6.62

Gradient = 2060602-2/1.1-1

= 6.0602

Gradient = 2.006006002-2/1.001-1

= 6.006002

X=2, Y=16

Gradient = 18.522-16/2.1-2

= 25.22

Gradient = 16.241202-16/2.01-2

=24.1202

Gradient = 16.024012-16/2.001-2

= 24.012002

X=3, Y=54

Gradient =59.582-54/3.1-3

= 55.82

Gradient = 54.541802-54/3.01-3

= 54.1802

Gradient = 54.054018-54/3.001-3

=54.018002

x

y

Small increment

1

2

6

2

16

24

3

54

54

4

128

96

Y= 3x3

X = 1, Y=3

...read more.

Conclusion

Small increment method to find the graph of y=x1/2

Where x=4, y=2

Gradient =2.0248457-2/4.1-4

           =0.248457

Gradient =2.002498439-2/4.01-4

           =0.2498439

Gradient =2.00025-2/4.001-4

           =0.25

Gradient using small increment method of curve y=x1/2 at point x=4, y=2

=0.25

I will then check this using my equation:

½ x (41/2-1)

=0.25

I can now conclude that this formula is correct for graphs where the power of x is a fraction.

Finally I will check if the formula works for a compound graph.

I will take the graph y=x2+4x.  To calculate the gradient of this graph I will use algebra in the hope of

y=x2+4x

Using the formula I have discovered I can prediction that the gradient function of this curve will be= 2x+4

Gradient = y-y/x-x

(x+h)2+4(x+h)-(x2+4x)/x+h-x

(x+H)(x+H)+4x+4h-(x2+4x)/h

x2+2hx+h2+4x+4h-x2-4x/h

2hx +h2+4h/h

2x+4+h

Therefore h=0

Gradient =2x+4

I can therefore say that my prediction is correct and can conclude that the equation works for compound graphs.

Overall I can say that for all the different types of graphs I have tried, my equation has been correct.  I can put forward the suggestion that the equation naxn-1 gives the gradient of any curve.  However I would need to extend this investigation to insure this is conclusive and try it with many more graphs to check there are no anomalies.

...read more.

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