My aim is to investigate the gradient function for all kinds of curves.

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Aim: to investigate the gradient function for all kinds of curves.

Research: What does “gradient” mean? Generally, the “steepness” of a curve is measured by its gradient. We can look at the figure below:

This is the curve of y=x2. The point P (3, 9) has been marked and the tangent QPM drawn. The gradient of the tangent is QN/MN.

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

First Step: I am going to investigate the gradient of y=x, y=x2, y=x3 first because they are likely to be the simplest equations to solve, and after getting these results easily, by looking at them, the more complex equations will seem easier to discover.

I am going to look at y=x first because it is the easiest.

Please see graph on separate pieces of paper

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

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

Then let us try the graph of y=x2.

By drawing the tangents of each point, we can calculate the gradients. However, as the graph is not always accurately drawn, there must be some error between the results. In order to avoid this, I am going to use another method to calculate the gradients: increment method.

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Increment method works like this: you need 4 numbers: x1, x2, y1, y2. If you want to get the gradient of the co-ordination (1, 1), namely (x1, y1) in the graph y= x2, then look for a very close number as x2 to x1, usually 1.01, then use the equation of the line to work out y2, which can be got by square x2 and gives you the value 1.0201. And then put the numbers in the formula below:

(y2-y1)/(x2-x1)

Which gives you the value of the gradient you want, 2.01

The advantage of this method is to get a very accurate ...

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