**Linear Composition with Core Trigonometric Functions **

[$A \ne 0, B\ne 0$]

$f_{trig}(x) = A \cdot trig (Bx+C) + D$

where $trig$ is $\sin$, $\cos$ or $\tan$.
A general trigonometric function is a composition of core linear functions with a core trigonometric function.

These features include the period [Subsection TRIG.PB] , extreme values [Subsection TRIG.ID] and the related amplitude, as well as phase shift.

To simplify slightly we consider the trigonometric functions with $D = 0$:

$f_s(x) = A\sin(Bx +C) $, $f_c(x)= A\cos(Bx +C) $, and $f_t(x)= A\tan(Bx +C) $.

For $\sin$ and $\cos$ the extremes are $A$ and $-A$, the period is $2\pi/|B|$, the amplitude is $A$, and the phase shift is $-C/B$.

We explore how mapping diagrams for linear compositions with core trigonometric functions help us visualize and understand further these functions.

In these examples we use GeoGebra to visualize the composition $f=h \circ g$ for one linear core function with a core trigonometric function.

You can use this next dynamic example with GeoGebra to investigate further the effects of linear compositions on a core trigonometric function with mapping diagrams.

In the example, visualizing the role of composition with the unit circle mapping diagrams that define these functions also helps make sense of the important connection to the core trigonometric functions.