We were asked to graph some of the Legendre polynomials and look at them.

Here are the 1st ten (11 actually, n=[0,10])

 Kind of interesting.  It looks like some of them go through the origin, some dont.  Also, some seem to hit 1 at x = 1, but some hit 1 at x=-1 while others hit -1.  Not only that, but it appears that all of them are either even or odd.

Heres a graph of the n=even ones from above.

Well, It appears that the n=even functions are the ones that are even and don't pass through the origin.
Here are the odd ones from the 1st picture.  Also there appear to be the same number of zeros as the "n" of the polynomial, which isnt all that surprising since n is the highest power in the polynomial.  Note, the polynomials start with y=1 at n = 0 and get progressively "bumpier", so you can tell them apart on the graph.

As expected, the n=odd functions go through the origin, and appear to be odd. There appear to be the same number of zeroes as the n value of the polynomial here as well.

At theta =0, cos theta =1, so x=1 and the polynomial Pn(cos(theta))=1 for all n.
At theta =pi, cos theta = -1, so x=-1 and the polynomial Pn(cos(theta))=(-1)^n, that is 1 for n even, -1 for n odd.
 
 
 

The behavior of the polynomials outside this region is not nearly as interesting, as the next graphs attest. I graphed these over [0,2pi] (since they are all even or odd the negative half is unneccessary).  They all just seem to blow up rather quickly, as you would expect for positive powered (?) polynomials.

 
                                    n-0                                                                n=1

                                    n-2                                                                n=3
 

                                    n-4                                                                n=5