# How do you write the equation of the sine function with an amplitude of 1/9 and a period of 3pi?

Apr 7, 2018

$y = \frac{1}{9} \sin \left(\frac{2}{3} x\right)$

#### Explanation:

The standard form of the sine function with period $p = \frac{2 \pi}{b}$ and amplitude $a$ is given by

$y = a \sin \left(b x\right)$

First, we'll need to solve for $b .$ We're given $p = 3 \pi .$ Moreover, since $p = \frac{2 \pi}{b} ,$

$3 \cancel{\pi} = \frac{2 \cancel{\pi}}{b}$

$b = \frac{2}{3}$

We're given $a = \frac{1}{9.}$ The amplitude must be positive -- this is already positive, so no worries, but were it not positive, we'd need to take the absolute value.

Thus, we have

$y = \frac{1}{9} \sin \left(\frac{2}{3} x\right)$