Investigate the gradients of the graphs Y=AXN

The above table shows that both methods agree for my formula for cases in which A=1 and N= a positive integer, so I shall now see what happens if N= a negative integer and whether the formula still works.  I shall not use the graph method any more because the increment method is a faster way of working out the gradients.

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Y=X-2

This table shows that my earlier formula ANXN-1 still works for negative powers.  To back up this evidence I am now going to work out the gradient function for two more examples were N= a negative integer.

Y=X-3

Y=X-4

In both of the above cases the formula is almost identical to that of the increment method, this shows that my formula works of all cases for when A=1 and N= an integer. I am now going to experiment for when N= a faction.

Y=X1/2

This table shows that my formula work when X is a fraction and to prove this I will now look at the graph Y=X1/3 and see if the formula still works.

Y=X1/3

The formula works for when n= a fraction as well.  This is good for I now know that my formula works for any cases of N.  I shall now see if the A in the formula ANXN-1 is correct.  I shall do this by keeping N as 2 as it is the lowest number and make A 2 then 3.

 Y=2X2

Y=3X2

I have now proven that the formula works with any positive whole number values of A.  Next I am going to see if it works when A is negative by using values for A as –2 and –3, N will still be 2.

Y=-2X2

Y=-3X2

I have now shown that the formula works for all whole number values for A and I am now going to investigate for A as a fraction.

Y= 1/2X2

Y=1/3X2

I have now shown that the formula works for all values of A and N in the set of graphs Y=AXN Where a and n are constants.  Now that I have proven this I shall try it out in other sets of graphs and see if it still works.  I am now going to work out the gradient of the graph Y=2X3+4X+9.  I am going to spilt the formula up into smaller fragments to make it easier to apply the formula so Y=(2X3)+(4X)+9.  The numbers before the X (Y=(2X3)+(4X)+9) are the A’s and the number after the X (Y=(2X3)+(4X)+9) is N, this means that the formula should be Gradient= (ANXN-1)+(AX).  The 9 at the end of the equation will make no difference to the gradient at all, it just tells us were the line cuts the Y axis.

Y=2X3+4X+9

The formula still seems to be working for different curves and I feel that it always work for all graph that includeY=AXN