Question 1179769
parent function:{{{ f(x) = sin(x)}}}

Periodicity of sin  (x ) is{{{ 2pi }}}.

{{{ graph( 600, 600, -5, 5, -5, 5, sin(x)) }}}

This is the general equation for a sinusoidal function: 

{{{f(x) = a sin (k(x-d)) + c}}}

{{{a}}} determines amplitude:
 {{{a > 1}}} or{{{ a < -1}}}	 →  vertical stretch
 {{{-1 < a < 1}}}	 →  vertical compression
 {{{a < 0}}}→  reflection in {{{x}}}-axis


{{{k}}} determines period:        
 {{{k >1}}} or {{{k < -1}}}	  →  horizontal compression by {{{1/k}}}
 {{{-1 < k < 1}}}	  →  horizontal stretch by {{{1/k}}}
{{{k < 0}}}  	  →  reflection in {{{y}}}-axis

{{{d}}} determines phase shift:
 {{{d > 0}}}→	 shift to the right "{{{d}}}" units.
 {{{d = 0}}}	→ no phase shift
 {{{d < 0}}}→	 shift to the left "{{{d}}}" units

{{{c}}} determines vertical shift:
 {{{c > 0}}}	→ shift up "c" units
 {{{c = 0}}}	→ no vertical shift
 {{{c < 0}}}→	 shift down "{{{c}}}" units

you are given:

{{{f(x) = 3sin (2(x-pi/6)) + 1}}}

so, in this case 

periodicity is{{{ 2pi/k=2pi/2=pi }}} 
{{{a=3}}}=>{{{ a > 1 }}}	 →  vertical stretch
{{{k=2}}}=> {{{k >1}}}   →  horizontal compression by {{{1/k=1/2}}}
{{{d=pi/6}}}=> {{{d > 0}}} →  shift to the right "{{{pi/6}}}" units
{{{c=1}}}=> {{{c > 0}}}→	 shift up "{{{1}}}" unit



{{{ graph( 600, 600, -5, 5, -5, 5, sin(x), 3sin (2(x-pi/6)) + 1) }}}