Question

Graph the function in the interval from 0 to 2Π y = −2 sin(θ − π3) + 2

Answer 1

The graph of the function is shown below.

Explanation:

The given function is

`y=-2\sin(\theta -(\pi)/(3))+2` .... (1)

The general equation of a sine function is

`f(x)=A\sin(Bx+C)+D` .... (2)

Where, A is amplitude,
`(2\pi)/(B)` is period, C is phase shift and 2 is midline or vertical shift.

From (1) and (2), we get

`A=-2, B=1,C=(\pi)/(3),D=2`

The amplitude is -2.

`Period=(2\pi)/(1)=2\pi`

`\text{Phase shift}=-(\pi)/(3)`

It means graph shifts
`(\pi)/(3)` units right.

`Midline:y=2`

The graph shifts 2 units up.

Therefore the graph of the function is shown below.

Answer 2

See the attached graph.

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A graphing utility or spreadsheet program is useful for this.