We are all aware of some basic symmetries that a graph of a function can have. For instance , the graph can be symmetric with respect to the $y'y$ axis or the point ${\mathrm O}$ ( that is the intersection of the axis ) . However, it's not uncommon to see a graph be symmetric with respect to a point , call that $\mathrm{A}$.
The definition ( sometimes theorem ) states that
The graph of a function $f:\mathcal{A}\rightarrow \mathbb{R}$ is symmetric with respect to the point $\mathrm{A}(x_0, y_0)$ if :
- Forall $x \in \mathcal{A}$ , $2x_0-x \in \mathcal{A}$ and
- $f(x)+f(2x_0-x)=2y_0$ for all $x\in \mathcal{A}$.
So,if you are given a specific function and a point you can easily check if the graph is symmetric with respect to that point. What if you are not given the point?
Say for example that you want to examine if the following function's graph is symmetric to point.
Exercise: Given the function $$f(x) = \frac{1}{x^2+\sqrt{x^2-2x+2}}$$
find the centre of symmetry.
Things look difficult. Drawing the graph using e.g Geogebra one can easily see that it is symmetric to the point $\mathrm{A}(1, f(1))$. Algebraically, one can see that if he rewrites the function as
$$f(x) = \frac{1}{x+\sqrt{x^2-2x+2}} = \frac{\sqrt{x^2-2x+2} -x}{2-2x}$$
These two formulae for $f$ do not coincide. One suspects now that $1$ may be what we are looking for. Indeed, we check with the definition and we are done.
The main question posed here is where should one look to find the coordinates of the symmetry point if that is not given?
Potential answers could be
- inflection points
- the centre of a vertical asymptote , if the function has one.
Do we have more?