Question

Find the amplitude, the period, and the phase shift please show all work

Answer 1

A) Amplitude: 8

Period: 2π/3

Phase shift: Right π/12

Step-by-step explanation

A sine function can be written in the form,

`y=A\sin (B(x+C))+D`

Where A is the amplitude, C is the phase shift left and D is the vertical shift. B is related to the period as follows,

`T=(2\pi)/(B)`

In this case, the given function is,

`y=8\sin \mleft(3\theta-(\pi)/(4)\mright)-5`

We can see that A = 8, D = -5 and B = 3. Hence, the amplitude is 8 and we have to use the other data to find the phase shift and the period.

If B is 3, then the period is,

`T=(2\pi)/(3)`

To find the phase shift, we have to factor out B,

`3\theta-(\pi)/(4)=3\mleft(\theta-(\pi)/(3\cdot4)\mright)=3\mleft(\theta-(\pi)/(12)\mright)`

C is negative, so the phase shift is to the right.

In summary, the amplitude is 8, the period is 2π/3 and the phase shift is π/12 to the right.