# What functions have symmetric graphs?

Sep 9, 2014

There are several "families" of functions that have different types of symmetry, so this is a very fun question to answer!

First, y-axis symmetry, which is sometimes called an "even" function: The absolute value graphs shown are each symmetric to the y-axis, or have "vertical paper fold symmetry". Any vertical stretch or shrink or translation will maintain this symmetry. Any kind of right/left translation horizontally will remove the vertex from its position on the y-axis and thus destroy the symmetry. I performed the same type of transformations on the quadratic parabolas shown. They also have y-axis symmetry, or can be called "even" functions.

Some other even functions include $y = \frac{1}{{x}^{2}}$ , y = cos(x), and $y = {x}^{4}$ and similar transformations where the new function is not removed from its position at the y-axis. Next, there is origin symmetry, or rotational symmetry. One can call these the "odd" functions. You can include functions like y = x, $y = {x}^{3}$, y = sin(x) and $y = \frac{1}{x}$. An interesting trig graph would be the tangent graph: Stretches and shrinks may be applied to the odd functions, but translations in any direction will ruin the rotation that occurs around the origin! Here are some nice examples of stretches and shrinks of the sine graph. 