What is the phase shift, vertical displacement with respect to #y=sinx# for the graph #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=asin(bx+c)+d#

Where:

#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:

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