What is the phase shift, vertical displacement with respect to $y = sin x$ $y=sinx$ for the graph $y = sin (x + \frac{2 π}{3}) + 5$ $y=sin(x+(2pi)/3)+5$ ?

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Answer 1 · 2018-02-18T13:37:05

See below.

We can represent a trigonometrical function in the following form:

$y = a sin (b x + c) + d$ $y=asin(bx+c)+d$

Where:

$8 a 88 = amplitude$ $color(white)(8)bbacolor(white)(88)= "amplitude"$
$bb((2pi)/b)color(white)(8)= "the period"$ ( note $bb(2pi)$ is the normal period of the sine function )
$bb((-c)/b)color(white)(8)= "the phase shift"$
$color(white)(8)bbdcolor(white)(888)=" the vertical shift"$

From example:

$y=sin(x+(2pi)/3)+5$

Amplitude = $bba = color(blue)(1)$

Period = $bb((2pi)/b)=(2pi)/1=color(blue)(2pi)$

Phase shift = $bb((-c)/b)=((-2pi)/3)/1= color(blue)(-(2pi)/3)$

Vertical shift = $bbd=color(blue)(5)$

So $y=sin(x+(2pi)/3)+5color(white)(88)$ is $color(white)(888)y=sin(x)$ :

Translated 5 units in the positive y direction, and shifted $(2pi)/3$ units in the negative x direction.

GRAPH:

enter image source here

What is the phase shift, vertical displacement with respect to y=sinxy=sinxy=sinx for the graph y=sin(x+(2pi)/3)+5y=sin(x+2π3)+5y=sin(x+(2pi)/3)+5?