Example TRIG.SYM.1 : $A=B=1; trig(x) = \cos(x); C = -2; D=0; f(x)=cos(x-2).$
This is best understood as the composition of the linear function $ f_{+C}(x) = x-2$ followed by the core trigonometric function $trig(x) =\cos(x) $, so $f(x) =\cos(x-2)$.
Draw a mapping diagram showing this composition  or  use the diagram created with GeoGebra to explore the diagram further.
Compare the mapping diagram with the graphs of $trig(x)$ and $f(x)$

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For any $a \gt 0$ the even symmetry with respect to $x=2$ of $f_c$ gives $f_c(2+a) = f_c(2-a)$