For this investigation, I have to find the relationship between a point of any non-linear graph and the gradient of the tangent, which is the gradient function.

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For this investigation, I have to find the relationship between a point of any non-linear graph and the gradient of the tangent, which is the gradient function. First of all, I have to define the word, ‘Gradient’. Gradient means the slope of a line or a tangent at any point on a curve. A tangent is basically a line, curve, or surface that touches another curve but does not cross or intersect it. To find a gradient, observe the graph below:

All you have to do to find the gradient is to divide the change in X with the change in Y. In this case, on the graph above, AB and you would have gotten the

                                          BC                                        

gradient for that particular point of the graph.                          

 I am going start by finding the gradient function of y=x², y=2x², and then y=ax². I will move on finding the gradient function of y=x³, y=2x³, and finally y=ax³. I will then find the similarities and generalise y=axⁿ where ‘a’ and ‘n’ are constants, and then investigate the Gradient function for any curves of my choice.

I will first find the gradient of tangents on the graph y=x² by drawing the graph (page 3), and then find the gradient for a number of selected points on the graph:

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As you can see, the gradient is always twice the value of its original X value Where y=x². So the gradient function has to be f `(x)=2x for y=x².

I will now move on to the y=2x² graph and find its gradient function. I will draw the graph (page 4) and put my findings in a table. I will now find the gradients for a selected number of points in the graph y=2x² so as to find the gradient function:

From the table, you can see that the gradient of the point is always 4 ...

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