Symmetries of graphs and roots of equations

[Stephen Tashi]

Is there a good way to relate the symmetries of the graphs of polynomials to the roots of equations?

There's lots of material on the web about teaching students how to determine if the graph of a function has a symmetry of some sort, but, aside from the task of drawing the graph, I don't find any convincing application of the symmetries.

 

[andrewkirk]

What symmetries did you have in mind? The only one I can think of are where a polynomial function $p:\mathbb R\to\mathbb R$ can be written as $p(x)=q(x-a)+b$ for some real $a,b$ and $q$ an odd or even polynomial function, where an 'odd' ('even') polynomial is one whose terms all have odd (even) powers. Hence the graph is a translation in two dimensions of the graph of an odd or even function.

Then if $q$ is even, or if $q$ is odd and $b=0$, there is a bijection $\theta$ from the set of roots $\geq a$ to the set of roots $\leq a$ such that $\theta(r)=2a-r$.

 

[Stephen Tashi]

What symmetries did you have in mind?

To start with, Euclidean transformations that bring the graph into coincidence with itself.

 

[andrewkirk]

For the graph of a polynomial, are there any of those that are not covered by the above, which covers

• all reflections in vertical lines (the case where $q$ is even and the axis of reflection is $x=a$); and
• rotation by 180 degrees around a point on the $x$ axis (the case where $q$ is odd and the centre of rotation is the point $(a,0)$)?

It seems to me that any other Euclidean transformation of the graph of a polynomial (rotation, other than by 180 degrees around a point on the $x$ axis; reflection in a non-vertical line; translation) would not leave the graph invariant, although I may be missing something.

But I didn't mean to trivialise these symmetries by my use of the word 'only' in post 2. These are enough to be quite useful in finding additional roots of some polynomials.