The Gradient Function Coursework

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The Gradient Function Coursework


In this piece of coursework I am going to do research on the gradient of various graphs at various points, in order to find a function, which will determine the gradient of these points without drawing or using approximations. I will only need to know the coordinates of the point as well as the type of graph I am considering, to submit them into the gradient function and determine the gradient at this point. The formulae I will use and produce will have particular parameters. Now I am going to explain them.

a: this letter will stand for the coefficient of x in the function y=ax^n and

    determines how steep the graph will be.

n: this letter will be the power to which x is raised in the function y=ax^n and

    determines the shape of the graph.

m: this letter will stand for the gradient at any point of any graph. I can say for

     example the gradient at the point P(1;1) of the graph y=x is 1. Therefore here m=1.

The first range of graphs I am going to investigate will have the function y=ax. I will draw three graphs on the next pages and hope to see a pattern between the gradient and the function of the graphs. I do not need to consider the coordinates of the points at which I will determine the gradient, as the gradient is the same at any point on the graph y=ax.

From these three graphs I clearly recognise a pattern. I will show how I noticed it, by presenting the graphs and their gradients:

y=x        m=1

y=2x      m=2

y=3x      m=3

It is obvious that the gradient of every linear function is equal to the coefficient of x. So for every function y=ax, the gradient will be m=a.

The next range of graphs I am going to investigate will be parabolas. These have the function y=ax². Here the gradient is different at different points of the graph. So I will have to use a different method. I will draw a tangent as near as possible to a point at which I would like to know the gradient, as the gradient of a point on this graph is defined as the gradient of the tangent that touches this point.

I will draw two graphs on the next two pages. From each of these two graphs I will choose three points at which I will use this method to determine the gradient and hope to find a pattern.

I marked the points on the graphs with a cross, presented their coordinates in brackets, and wrote the gradient near the points which I found by using the tangent method. I will only investigate one half of the curve, because the negative half will have the same gradients at the negative points. The gradients only differ as the positive half will have positive gradients and the negative graph negative gradients.

I drew the graphs as accurately as I could to avoid inaccurate gradients. But the numbers were still not accurate enough. When I compared my result with the results of other students, whose graphs were very accurate, too, I noticed differences, which reached about 1.3 units. This method was thus too inaccurate to detect a pattern and also took a great amount of time, so I will have to use another method for finding the gradients of these curves more efficiently.

I will use the chord method for every graph that I will investigate from now on.

With this method I will approximate the gradient of a curve at a given point by finding the gradient of a chord that joins this point with another point sufficiently close by:

Join now!

This method is more accurate as it only involves calculations. On the next pages I used Microsoft Excel to find the gradient of three points on the graph y=x² and y=2x². I will present the coefficient of x² below the actual calculation.

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Here's what a star student thought of this essay

This piece of coursework is structured well, and spelling and grammar are strongly utilised. Although pieces of formal coursework are becoming less common in mathematics at GCSE, I would note that this piece could be made more sophisticated by removing the frequent use of the first person. Other than that, it should be admired!

The analysis in this piece is strong. I liked how they begin by trying to work out the gradient of a function at a point, and then showing the understanding that this isn't accurate enough. They use plenty of excel screenshots to give evidence for their hypothesis, but what I really liked was the proof by first principles (shown under the more about calculus section). Using and understanding limits at GCSE level shows high level analysis and I imagine showing this technique will help secure top marks in any piece of coursework. This piece of coursework even goes further to understanding how differential equations and integration may be useful in a real-life application. If I were to make one suggestion, it would be to explore the notation a bit further.

This piece of coursework superbly explores the gradient of basic polynomial curves. The argument posed is logical, and diagrams and screenshots of excel are used to support this argument.