Question

How to plot the graph of `f(x)=2 sin 4x`?

Answer 1

Draw a standard `sin` curve but "squeeze" the vertical so the period is reduced from `2pi` to `pi/2` and stretch the horizontal so the amplitude is increased from `[-1,1]` to `[-2, 2]`

Explanation:

`sin(theta)` has a period of `2pi`
replacing `theta` by `4x` means that `4x` must have the same period
and therefore each `x` of the `4x` can only take up `(2pi)/4 = pi/2` period length.

`sin(theta)` has an amplitude of `[-1,+1]`
`rarr` for any angle `4x`, `sin(4x)` must be a value `s epsilon [-1,+1]`
and
`rarr 2sin(4x)` must be `2s` giving it a range of `[-2,+2]`