• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  • 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.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Gradient Function essays

  1. Peer reviewed

    The Gradient Function Coursework

    5 star(s)

    m=0.5 The function appears to be correct so far, but I will have to use many more examples in order to show that the gradient function I developed is generally correct. I will show working out the gradient for the calculation below by using the formula: m=nax^n-1 => m=0.5*3.31*1^-0.5 =>

  2. Gradient function

    16 -20 -1 20 2.1 17.64 -18.36 -0.9 20.4 2.2 19.36 -16.64 -0.8 20.8 2.3 21.16 -14.84 -0.7 21.2 2.4 23.04 -12.96 -0.6 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

  1. Gradient Function

    19.5 3.6 38.88 -11.88 -0.6 19.8 3.7 41.07 -14.07 -0.7 20.1 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

  2. The Gradient Function

    Another way to find the formula for the gradient function is by : The point B is the point with co-ordinates (x,x2) and the point C is the point near to B with co-ordinates (x+h,(x+h)2). For the curve y=2x�, the gradient of the tangent at x is four times the

  1. The Gradient Function

    This proves that the gradient is a measure of steepness, and since these lines are parallel to the X-axis they have no gradient. Thus the final conclusion that I have come to is that for all equations of Y=axn where n = 0, the gradient is 0.

  2. The Gradient Function

    G = y1-y2 x1-x2 G = 10.001 G = 10 For point x=6: y1 = 62 y1 = 36 x2 = 6.001 y2 = 6.0012 y2 = 36.010001 G = y1-y2 x1-x2 G = 12.001 G = 12 To work out the pattern of these numbers, I can look for

  1. Investigate the gradients of the graphs Y=AXN

    Next I am going to see if it works when A is negative by using values for A as -2 and -3, N will still be 2. Y=-2X2 X Predicted gradient with formula X Co-ordinates Y Co-ordinates Increment formula Increment gradient 1 -2*2*11 = -4 1,1.001 -2, -2.004002 -0.004002/0.001 -4.002

  2. The Gradient Function.

    And because the graph is to a much larger scale the line should follow almost the same path as the curve. Of course the more you zoom in the more accurate the gradient will be. You then use the same formula to work out the gradient.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work