Symmetry of Trigonometric Functions
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.
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 ($x = -C/B$).