Question 763669
{{{y=2sin(3x-pi/2)-1}}} --> {{{y=2sin(3(x-pi/6))-1}}}
 
Let's start with the simplest related function.
The graph for {{{y=sin(x)}}} is {{{graph(400,200,-1,7,-2,2,sin(x))}}}. Its period is {{{2pi}}}.
{{{graph(400,200,-1,7,-2,2,sin(3x))}}} is the graph of {{{y=sin(3x)}}}, whose period is {{{highlight(period=2pi/3)}}}.
Substituting {{{x-pi/6}}} for {{{x}}} translates the function {{{highlight(pi/6)}}} units to the {{{highlight(right)}}}. That's the phase shift.
So, {{{graph(400,200,-1,7,-2,2,sin(3(x-pi/6)))}}} is the graph of {{{y=sin(3(x-pi/6))}}} (with the same period).
A factor of {{{2}}} in front dilates the graph vertically to get {{{highlight(amplitude=2)}}}, and the {{{-1}}} added at the end translates the function {{{highlight(down)}}} by {{{highlight(1)}}} unit:
{{{graph(400,300,-1,7,-4,2,2sin(3(x-pi/6))-1,-1)}}}