Answer:
![\displaystyle y=-2sin(6t)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%3D-2sin%286t%29)
Step-by-step explanation:
<u>Sinusoid</u>
The shape of a sinusoid is well-know because it describes a curve with a smooth and periodic oscillation. The sine and cosine are the two trigonometric functions in which the graph matches the description of a sinusoid. The sine can be identified because its value is zero at time zero.
The graph shown in the figure corresponds to a sine. The other characteristics of the sine function are
* It completes a cycle in
radians
* It has a maximum of 1 and a minimum of -1
* It's increasing for a quarter of the cycle, decreasing for half of the cycle, and increasing for the remaining quarter of the cycle
* The equation is
![y=Asin(wt)](https://tex.z-dn.net/?f=y%3DAsin%28wt%29)
The function starts decreasing for the first quarter, which only is possible if the amplitude A is negative. We can also see the maximum and minimum values are 2 and -2 respectively. This means the amplitude is A=-2
We can also see the function completes 3 cycles in
radians or 6 cycles in
radians. Or, equivalently
![wt=12\pi](https://tex.z-dn.net/?f=wt%3D12%5Cpi)
![w(2\pi)=12\pi](https://tex.z-dn.net/?f=w%282%5Cpi%29%3D12%5Cpi)
![\displaystyle w=\frac{12}{2}=6\ rad/sec](https://tex.z-dn.net/?f=%5Cdisplaystyle%20w%3D%5Cfrac%7B12%7D%7B2%7D%3D6%5C%20rad%2Fsec)
Thus, the function can be expressed as
![\boxed{\displaystyle y=-2sin(6t)}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20y%3D-2sin%286t%29%7D)