Question

1) State the amplitude, period, phase shift, vertical shift and graph

Answer 1

See answers and explanations below (along with a visual graph)

Explanation:

For the function
`y=acos(b(x+c))+d`:

Amplitude:
`|a|`

Period:
`(2\pi)/(|b|)`

Phase shift:
`(-c)/(b)`

Vertical shift:
`d`

Therefore, for
`y=2cos((2)/(3)(\theta+\pi))-1`:

Amplitude:
`|a|=|2|=2`, which is 2 units above and below the midline (see d)

Period:
`(2\pi)/(|b|)=(2\pi)/(|(2)/(3)|)=(2\pi)/((2)/(3))=3\pi`, so the cycle will repeat every 3π units

Phase shift:
`(-c)/(b)=(-(2\pi)/(3) )/((2)/(3) )=-\pi`, or π units to the left

Vertical shift:
`d=-1`, or 1 unit down