Question

How do you find the amplitude and period of `y=4sin3x`?

Answer 1

Amplitude of `4sin(3x)` is `4`
Period of `4sin(3x)` is `6pi`

Explanation:

`y=sin(theta)` has an amplitude of `1`
`color(white)("XXXX")` `y` varies between `+1` and `-1`
If `y=4sin(theta)`
`color(white)("XXXX")`then `y` will vary between `+4` and `-4`
`color(white)("XXXX")` `rarr y=4sin(theta)` has an amplitude of `4` (for any `theta`)

`y=k*sin(theta)` has a period of `2pi`
`color(white)("XXXX")` `y` completes one cycle while `theta` increases by `2pi`
if `3x =theta`
`color(white)("XXXX")` `y` completes one cycle while `3x` increases by `2pi`
`color(white)("XXXX")`or (expressed in another way)
`color(white)("XXXX")` `y` completes `1/3` of a cycle while `x` increase by `2pi`

`color(white)("XXXX")` `rArr y` completes 1 cycle while `x` increases by `3xx2pi = 6pi`