The Gradient Function

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The Gradient Function

The aim of this investigation is to discover the gradient function for the graphs y = ax where a and n are constants.  

I will do this by beginning with the simplest cases, as I believe that these will be the most simple equations to solve.  I am doing this in the hope that discovering the equations for these simple cases will aid me in discovering the more complex formulas.  

Firstly I will construct the graphs of: y=x, y=2x, y=3x, y=4x.  And attempt to find a general equation.

Y=x

Y=2x

                                        

Y=3x

Y=4x

                                        

(Graph1)

Using the rule stated on the candidate sheet I can calculated the gradient for each of the above straight lines.

Doing this I have discovered that the co-efficient of x is the gradient of the line.

                                

I will now go on to construct the graph of y=x2.  This is because I know that the graph of x2 will be a curve and it is curves that I am investigating

To find the gradient of this line, I will use the tangent method.  If I have a point on the curve, I will draw a tangent so only the point on the curve is touching the tangent.  I will then draw a right-angled triangle with the tangent.  I will then take the values of the opposite line y and the adjacent line, x.  If I then divide the x value by the y value then I will have the gradient.

(Graph 2)

The immediately noticeable thing about this is that as the co-ordinates increase, so does the gradient.  Another visible thing is that as the difference between the co-ordinates increases the smaller the increase in the gradient.

This method I have used for attaining these results is inaccurate.  This is because I am drawing tangents to the line by eye and therefore there is a large margin for error.  As a result of this I will construct the gradient using the small increment method.  This method will hopefully be more accurate and enable me to discover trends within the data more easily.

The small increment methods works on the theory that if you draw a tangent to a curve and zoom in on the curve, there will be co-ordinates on the line that will have the same gradient as the point being investigated.  

For example, say the graph above is the graph of 2x2 .   I know the x value of the point is 2, therefore to find the y value I would square the x value and then multiply it by 2.  This means y=8.  I then take a point very close to the x value, for example 2.01, I will call this xt.  I will then apply the point 2.01 to the equation.  This tells me 2x2.012 = 8.0802   I can then use the y2-y1/x2-x1 method:

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8.0802-8/2.01-2 = 8.02

From this answer I could speculate that the gradient of the curve at this point could be 8.  Just to ensure this, I could ‘zoom in’ to the graph further and apply the same method above to the point 2.001:

8.008002-8/2.001-2 = 8.002

Bearing in mind that as you ‘zoom in’ to the graph further you get a more accurate estimate as to what the gradient is, you can see that from the above example, as you zoom in further, the .2 gets smaller and smaller.  If I zoomed into the graph further, ...

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