In this piece of coursework I will investigate the gradient function.

Victoria Jennings- 11SEG

GCSE Mathematics Coursework 2003

The Gradient Function

The “steepness” of a curve is measured by its gradient.

The gradient of a curve at a particular point is defined as the gradient of the tangent drawn to the curve at that point.

The gradient at a point on any curve is defined as the gradient function.

I will carry out this investigation using the equation:

y = axn

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

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We can use the “tangent method” to obtain the gradients of graphs of different functions.

I am going to look at y=x1 first because it is the easiest. See graph A

As we see, the gradients of y=x is very simple, g=1. We even do not need to draw any tangents to obtain the gradients. So the relationship between g and x can be shown in the table below:

So it is obvious that in the graph of y=x, whatever x is, the gradient, g stays 1.

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I will now look at the graph of y=x2. See graph B

By drawing the tangents of each point, we can calculate the gradients. However, as the graph is not always completely accurately drawn, there is likely to be some error in the results.

As you can see, as the tangent method was producing results that were not very accurate and it is also difficult and time consuming to do. To avoid this I used another method called the small increment method (see calculations, page C).

This is an alternative method that gives you a more accurate gradient. You ‘zoom in’ and look at the graph in more detail. You take two points very close together and join them with a straight line. Because the graph is a much larger scale, the line should almost follow the path of the curve. The more you zoom in, the more accurate the gradient will be.

The next graph I will draw is y=2x². See graph D (see calculations, page C).

In both these graphs, one of the most obvious thing is notice is as the co-ordinates increase, so does the gradient and as the co-ordinates decrease, so does the gradient.

As the results show, the small increment method is a very accurate method. Drawing the tangent onto the graph means that both the tangent and the graph have to be exact.

From these results I can say that the formula for the gradient is g=anx

I will see if this works for my next graphs.

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The next graph I will look at is y=x³. See graph E.

To find out the gradient for this graph I will once again use the small increment method as I have found that it is more accurate (see calculations, page F).

When using the small increment method you don’t actually need to sketch the magnified area, you can just use the numbers and put them in the formula- ...