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  • Level: GCSE
  • Subject: Maths
  • Word count: 2783

The Gradient Function

Extracts from this document...

Introduction

The Gradient Function

Part 1

Investigate the gradient function for the set of graphs: -  y=axn where a and n are constant.

For Part 1 of my investigation I will be looking at many different types of y=axn graph    such as:

        y = x2, y = 2x2, y = 3x2, y = 4x2,

        y = x3, y = 2x3, y = 3x3, y = 4x3

        y = x4, y = 2x4, y = 3x4, y = 4x4

        y = x2, y = x3, y= x4, y= x5

        y = 2x2 y = 2x3 y =2x4 y = 2x5

I will be trying to find the gradients of certain points on the graph which can lead me to the gradient function for that curve.

Gradient: measures the steepness of  a curve. The gradient of any particular point on                      a curve is defined as the gradient of the tangent drawn to the curve at that                         point.

Tangent: is a straight line which touches but does not cut the given curve at a particular                     point.

The gradient of any tangent is vertical

                                   horizontal

Gradient function: is a common function for the gradients of a set of tangents on a particular curve. When this function is obtained, tangents do not need to be drawn to work out the gradient.

Prediction: I predict that there will be a gradient function to every graph that is y=axn                       then from those gradient function a common function can be found.

...read more.

Middle

1.5

2

3

-2

Gradient

8

12

16

24

-16

Now we can predict the gradient function to be 8x as the gradient is always 8 times ‘x’

I have found that there is a link between the gradient function for these graphs.

Graph

Gradient Function

y=x2

2x

y=2x2

4x

y=3x2

6x

Using my gradient functions for these curves I can predict the gradient function for y=ax2

y=ax2

2ax

For the curve y=10x2, using the gradient function 20x I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

20

30

40

60

-40

Method (2ax)

a=10, x=1

2 x 10 x 1

= 20

a=10, x=1.5

2 x 10 x 1.5

= 30

a=10, x=2

2 x 10 x 2

= 40

a=10, x=3

2 x 10 x 3

= 60

a=10, x=-2

2 x 10 x -2

= -40

I checked this on Omnigraph and found that the gradients were the same as I thought they would be so this gradient function 20x clearly works showing that gradient function for ax2 is right.

For the curve y=x3, the gradient function would be 2x so the gradients of certain points are:-

Tangent at x

1

1.5

2

3

-2

Gradient

2

3

4

6

-4

I checked this on Omnigraph but these gradients weren’t right. These were the correct ones below.

For the curve y=x³, the gradient of the tangent at x:-

Tangent at x

1

1.5

2

3

-2

x2

1

2.25

4

9

4

Gradient

3

6.75

12

27

12

Therefore I think the gradient function for this curve is 3x2 because the gradient is 3 times x2

For the curve y=2x³, the gradient of the tangent at x:-

...read more.

Conclusion

For the curve y=x5, using the gradient function 5x4 I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

5

25.3

80

405

80

For the curve y=2x5, using the gradient function 10x4 I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

10

50.6

160

810

160

For the curve y=3x5, using the gradient function 15x4 I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

15

76

240

1220

240

I checked the gradients of these graphs on Omnigraph and found that they were the same as I thought that they would be which shows that I the gradient function for all y=anx(n-1) clearly works. I have only tested this on graphs where ‘n’ is constant and ‘a’ varies so now I will test it on graphs of - 10x2, 10x3, 10x4, 10x5 using the same method as before.

For the curve y=10x2, using the gradient function 20x I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

20

30

40

60

-40

For the curve y=10x3, using the gradient function 30x2 I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

30

67.5

120

270

120

For the curve y=10x4, using the gradient function 40x3 I think the gradients will be:-

Tangent at x

1

1.5

2

3

-2

Gradient

40

135

320

1080

-320

Again I checked this on Omnigraph and found that the gradients I thought it would be was correct. This means the gradient function anx(n-1) really does work for all y=axn graphs. It worked on graphs with ‘n’ constant and graphs with ‘a’ constant.

...read more.

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