Question

A spring attached to a ceiling is pulled down `20` cm. After `3` seconds, wherein it completes 6 full periods, the amplitude is `15` cm. Find a sinusoidal function modelling it's position `t` seconds after released?

Answer 1

`h = 15cos(720t) - 20`

Explanation:

We will want the independent variable to be time, `t` here, and the depended variable to be `h`, or height. We will make it a cosine function, because it starts at a maximum (we could also do a sine graph with a phase shift).

The period will be `3/6`, or `0.5` seconds. This means the b-value will be `360/b = 0.5 -> b = 720`

Given the amplitude is `15` cm, we get:

`h = 15cos(720t) - 20`

The vertical transformation will occur because of the 20 cm pulled down. Notice how I let the origin be the starting point for the unstretched spring.

Hopefully this helps!