Symmetry

Two objects are symmetric to each other with respect to the invariant transformations if one object is obtained from the other by one of the transformations. It is an equivalence relation. In the case of symmetric functions, determining symmetry is as easy as graphing the function or evaluating the function algebraically.  Symmetry of a function can be a simple shift of the graph (transformation) or the function can be symmetric about a point, line or axes.

Symmetric Function Types

Functions and relations can be symmetric about a point, a line, or an axis.  They can have symmetry after a reflection.  To determine if a relation has symmetry, graph the relation or function and see if the original curve is a reflection of itself over a point, line, or axis.  The image below shows an example of a function and its symmetry over the $x$-axis (vertical reflection) and over the $y$-axis (horizontal reflection).  

Symmetry by Reflection

A function can have symmetry by reflecting its graph horizontally or vertically.  This type of symmetry is a translation over an axis.

In the next graph below, quadratic functions have symmetry over a line called the axis of symmetry.  The axis splits the U-shaped curve into two parts of the curve which are reflected over the axis of symmetry.  

Symmetry Parabola

The function $y=x^2+4x+3$ shows an axis of symmetry about the line $x=-2$.  The curve is split into $2$ equivalent halves.  Notice that the $x$-intercepts are reflected points over the axis of symmetry and are equidistant from the axis. 

Determining Symmetry

Example:  Does the function below show symmetry?

Symmetry about a point

The graph above has symmetry since the points labeled are reflected over the origin.  

The graph has symmetry over the origin or point $(0,0)$.  The points given, $(1,3)$ and $(-1,-3)$ are reflected across the origin.