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f(x) = x^-n : Different values of n and their effect on the shape of the graph

Hi, I've been working on this investigation but I do not think I have done the first task very well. The attached image shows the task outline, and the next attached image is a screenshot of 4 different graphs on one which I made to assist in explaining my problem:

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I graphed various the function with differing values of n, but I do not really know how to explain what the changes are...

So far I have written the following, but I don't know if it's all relevant and if I can include more...:

For all values of n, the graph line intersects at (1,1).

For 0<n<1, all graphs have asymptotes at x = 0 and y = 0.

As n --> 0, between 0<x<1, the slope becomes more negative and therefore the graph line becomes closer to the y axis near to the origin. Also, as n --> 0, when x > 1, the slope decreases, and therefore becomes closer to the x axis near to the origin.

For n>1, as n increases, the graph lines start moving more towards the y axis near to the origin, and become closer to the x axis near the origin.

If graphs of the function with n = 0.01, 0.1, 1, and 10 are graphed together, obvious conclusions can be drawn.

If 0<n<1, the slope of the function for all x is small, and slowly moves towards its horizontal asymptote of y = 0.

If n>1, the slope of the function for x is relatively large, and particularly for 0<x<1, the slope is large (i.e. more negative).

What can I do to improve my answer by changing it/adding to it/removing from it? Thanks

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Re: f(x) = x^-n : Different values of n and their effect on the shape of the graph

note the symmetry w/r to the line $\displaystyle y = x$ in quad I for the functions $\displaystyle y = x^{-n}$ and $\displaystyle y = x^{-1/n}$

what does that say about two such functions?

Re: f(x) = x^-n : Different values of n and their effect on the shape of the graph

Hi there, I came across this forum seeing as it is relevant to the type of maths i'm doing in class right now. As written above, you talk about the line of symmetry passing through the point of intersection of (1,1) with the function y=x for all functions of y=x^(-n). Could you give me a hint to point me in the right direction as to the relationship between the two functions?