Question 1205013
{{{y = A*sin (B(x + C)) + D }}}

where

{{{A}}} is amplitude (The amplitude of the graph is the maximum height the graph reaches from the x-axis.)
{{{(2pi)/B}}} is period (The period is the distance along the x-axis that is required for the function to make one full oscillation.)
{{{C}}} is phase shift (The phase shift is the measure of how far the graph has shifted horizontally.)
{{{D }}}is vertical shift (The vertical shift is the measure of how far the graph has shifted vertically, either up or down, from its initial position.)


here is the graph of parent function {{{y=sin(x)}}}


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

The graph of{{{ y=sin(x)}}} has:

- an amplitude of {{{1}}}

- a period of {{{2pi}}}

- a phase shift of {{{0}}}

- a vertical shift of{{{ 0}}}


you need to compare it to the given graph to determine the values of {{{A}}}, {{{B}}}, {{{C}}}, and{{{ D}}}


as you can see, midline goes through {{{y=-1}}} ( line parallel to {{{x}}}-axis, means the graph is {{{shifted}}}{{{ down}}}{{{ one}}} unit)=> {{{D=-1}}}

the {{{maximum}}} height the graph reaches from the x-axis is from {{{-3 }}}to {{{1}}}, {{{4}}} units, 
the distance from {{{-1}}} to {{{1 }}}and from {{{-1}}} to{{{ -3}}} is {{{2 }}}units, so the amplitude is

{{{A=2}}}

The period of the graph is {{{(2pi)/B}}}.

given function  makes one full oscillation at  {{{(2pi)/3}}} (when you start at point ({{{0}}},{{{-1}}}) and go to point ({{{2pi/3}}},{{{-1}}}))

so, the period is: {{{(2pi)/B=(2pi)/3}}} => {{{B=3}}}

the graph has shifted horizontally {{{pi/3 }}}to the right, so

{{{C = -pi/3}}}

the graph has shifted vertically one unit down, so

{{{D=-1}}}

your equation is:

{{{y = 2sin(3(x -pi/3)) -1}}}


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


Is {{{y = A *sin (B(x + C)) + D}}} the same as {{{y = a*sin(bx - c) + d}}}? => {{{no}}}, it’s not same

reason:

{{{(B(x + C))<>(bx - c)}}}
 when {{{C }}}is positive the graph shifts to the left
 when {{{C}}} is negative the graph shifts to the right