# The Gradient Function

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Introduction

Richard Q-Ball

## The Gradient Function

I am trying to find a formula that will work out the gradient of any line (the gradient function)

I am going to start with the most simple cases, e.g. y=x, y=x², y=x³ etc. They are probably going to be the easiest equations to solve as they are likely to be less complex, and hopefully the formulas to the more complex equations will be easier to discover by looking at these first formulas.

I am going to look at the line y=x² first.

y=x²

x | 1 | 2 | 3 | 4 |

y | 1 | 4 | 9 | 16 |

Please refer to graph on separate piece of paper

One of the most obvious things I notice is that as the co-ordinates increase so does the gradient. Not only can you see that from the results below, but also on the graph you can see that the line gets steeper and steeper. This makes sense, as the higher the number x is the larger the difference between x and x².

Another thing that I have noticed is that the larger the co-ordinates the smaller the increase in gradient.

Point | Gradient (tangent) | Gradient (Increment Method) |

(1,1) | 2 | 2.01 |

(2,4) | 3.3 | 3.01 |

(3,9) | 6.36 (2dp) | 6.01 |

(3.5,12.3) | 6.4 | 7.001 |

Middle

y

Gradient

1

1

1

2

2

1

3

3

1

4

4

1

There is no need to use the small increment method here, as I know that the gradient is accurate as y=x is always going to equal 1. A formula to work out the gradient function for this equation is:

y/x

Another formula is of course just simple 1.

y=x³

Now I am going to look at the line y=x³.I predict that this line will look similar to y=x² but it will be steeper. I have decided to complete both this stage and all the stages after this stage of my investigation with the use of a computer. I will use the computer to work out and draw the graphs I will be using. This will save time, make it easier to work out any equations or formulas and the graphs and results will be a lot more accurate as computers don’t make as many mistakes as humans and the graphs will be more detailed. This will help me to work out and calculate more precise answers, results, equations and formulas.

x | 1 | 2 | 3 | 4 |

y | 1 | 8 | 27 | 64 |

#### Please refer to graph on separate piece of paper

Point | Gradient (tangent) |

Conclusion

x | y | Gradient |

1 | 1 | 4.060401 |

2 | 16 | 32.240801 |

3 | 81 | 108.541201 |

4 | 256 | 256.96101 |

My prediction was right, the formula for this line is 4 X x3

I am going to put the formulas that I have discovered so far into a table so that they are hopefully easier to interpret.

## Equation | y=x | y=x2 | y=x3 | y=x4 | n |

Gradient Function | 1 | 2x | 3x2 | 4x 3 | nx (n-1) |

Looking at my previous formulas I have come up with a formula that will work for any equation, although I would need to check that it does work for negative, fractions etc. The formula is as follows:

nx (n-1)

This formula has been developed from the other formulas that I have discovered for the equations, as you can see from the table above, it is fairly obvious what the formula is going to have to be. The n stands for the power in the equation

e.g. y=x², n would equal ².

y=x5

I am going to just check that my formula does work with the equation y=x5 although I am sure it will work as it has worked for all the previous equations.

x | y | Gradient |

1 | 1 | 5.10100501 |

2 | 32 | 80.80401001 |

3 | 243 | 407.70902 |

4 | 1024 | 1286.41602 |

5 X 14=5

5 X 24=80

5 X 34=405

5 X 44=1280

Indeed my rule does work, however now that the gradients are such high numbers it is more noticeable that the small increment method is not perfect.

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

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