The Gradient Function

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The Gradient Function

Part 1

Investigate the gradient function for the set of graphs: -  y=axn where a and n are constant.

For Part 1 of my investigation I will be looking at many different types of y=axn graph    such as:

        y = x2, y = 2x2, y = 3x2, y = 4x2,

        y = x3, y = 2x3, y = 3x3, y = 4x3

        y = x4, y = 2x4, y = 3x4, y = 4x4

        

        y = x2, y = x3, y= x4, y= x5

        y = 2x2 y = 2x3 y =2x4 y = 2x5

I will be trying to find the gradients of certain points on the graph which can lead me to the gradient function for that curve.

Gradient: measures the steepness of  a curve. The gradient of any particular point on                      a curve is defined as the gradient of the tangent drawn to the curve at that                         point.

Tangent: is a straight line which touches but does not cut the given curve at a particular                     point.

The gradient of any tangent is vertical

                                   horizontal

Gradient function: is a common function for the gradients of a set of tangents on a particular curve. When this function is obtained, tangents do not need to be drawn to work out the gradient.

Prediction: I predict that there will be a gradient function to every graph that is y=axn                       then from those gradient function a common function can be found. This                         gradient function will apply to all graphs of y=axn. I also predict that there                       will be different ways to work the gradients or gradient function.

I will be drawing a tangent on the curve y=x2, at a particular point to find the gradient of that point.

You cannot calculate the gradient of the tangent directly. To find the gradient of the tangent at point P on a curve with a given equation (in this case x2) two points need to be known. You can find the gradient by joining chords from point P to other places close to P on the curve. You can find the gradient values of the chords and as the points get closer together they become very close to a limiting value which is the gradient of the tangent as shown in the graph.

To improve the tangent I moved A closer to P so it is at A1. The tangent drawn  then becomes more accurate as does the gradient. The closer A is to P the more accurate the tangent is.

P=(1,1)

                A1           A2                A3                A4                     A5                        A6

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From this we can predict the gradient to be 2 because the numbers behind the decimal point become insignificant.

This shows us the gradient of only one point but to find the gradient function of that curve you need to find a set of gradient from different points on the graphs.

I found the gradients of more points on the curve y=x2 using the same method.

P=(2,4)

                A1           A2                A3                A4                     A5   ...

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