Example TRIG.LCOMP.4 :Suppose $f$ is a trigonometric function of the form  $f_s(x)=2\sin(Bx+C)$ with period $ \pi$, and  maximum value $f(1)=2$. Find $B$ and one (of the many) possible  value(s) for $C$. Visualize $f$ with a mapping diagram that illustrates the function as the composition of the three core functions  $f_{*2}\circ \sin \circ f_{+C} \circ f_{\times B}$.
Solution: Since the period, $2\pi/B = \pi$, simple algebra provides the solution that $B = 2$. Since a maximum value for the sine occurs when $Bb+C = B(1)+C =\pi/2$ and $B=2$, we have $2+C = \pi/2$ and thus $C =\pi/2 -2$. So $f_s(x) =2\sin( 2x -2+\pi/2)$ will satisfy the required conditions.

Here is the desired mapping diagram:

Martin Flashman, Feb. 19, 2017. Created with GeoGebra

Notice how the arrows on the mapping diagrams are paired with the points on the graph of the functions.
You can move the point for $x$ on the mapping diagram to see how the function value for the functions change both on the diagram and on the graph.
The arrow on the point on the mapping diagram target axis indicates whether the value of the function is increasing (pointing up) or decreasing (pointing down).