Question

How do you find the amplitude and period of `f(x) = - 8 sin(5*x + pi)`?

Answer 1

Amplitude = `8`
Period = `(2pi)/5`

Explanation:

`f(x) = -8 sin( 5x + pi )`
`=> f(x) = 8 (-sin( 5x + pi ) )`
`=> f(x) = 8 sin( 5x + cancel(pi) - cancel(pi) )` (Since `-sin(x) = sin( x - pi )`
`=> f(x) = 8 sin( 5x )`

Since `| sin(x) | <= 1 forall x`, the amplitude of `f(x)` is 8.

Now, for the period, notice the following:
`f( x + (2pi)/5) = 8 sin( 5 ( x + (2pi)/5 ) ) = 8 sin( 5x + 2pi ) ) = 8 sin( 5x ) = f(x)`

Thus the period of `f(x)` is `(2pi)/5`.

Note: In general, for any sinusoidal function of the form `sin(nu x)` or `cos(nu x)`, the period is `(2pi)/nu`. Here `nu` is the frequency multiplication factor. In this particular case, `nu = 5`.