Question

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

Answer 1

See below.

Explanation:

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: