# 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.

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:-

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.

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

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