**Symmetry of Trigonometric Functions**

Other symmetries of the trigonometric functions are related to

Consider the trigonometric functions $f(x) = A trig(Bx +C) $ where $trig = \sin, \cos,$ or $\tan$.

We explore how these forms are similar to the Vertex Form for quadratic functions. In particular they can
be used to understand the symmetry of these functions as a composition
of core functions.

In these examples we use GeoGebra to visualize symmetry with the composition for one linear core function with a core trigonometric function.

Compositions are a key to understanding the symmetry of trigonometric functions. The following key result on trigonometric functions as compositions is very useful.

This result says certain **trigonometric functions are built by composing
three core linear functions with a core trigonometric function:**
$trig(x)$ which can be any core trigonometric function with linear functions, $f_{+C}, f_B$, and $f_{*A}$.

Here is an example from TRIG.FORM that visualizes the theorem.

** FIX Example** TRIG.SYM.4
Suppose $f$
is a trigonometric function with leading coefficient $ =A=2$
and extreme value $f(1)=3$. Find the vertex form of the
trigonometric
function. Visualize $f$ with a mapping diagram that illustrates
the function as the composition of the four core functions
$f_{+3}∘f_{∗2}∘q∘f_{−1}$.

Understanding $f$ with compositions and mapping diagrams explains why trigonometric cosine functions of this form always have even symmetry with respect to an axis ($x = -C/B$).

You can use this next dynamic example with GeoGebra to investigate further the symmetry of a trigonometric functions in a mapping diagram of $f$.