Answer:
[tex]\displaystyle y=-2sin(6t)[/tex]
Step-by-step explanation:
Sinusoid
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[tex]2\pi[/tex] 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
[tex]y=Asin(wt)[/tex]
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 [tex]t=\pi[/tex] radians or 6 cycles in [tex]2\pi[/tex] radians. Or, equivalently
[tex]wt=12\pi[/tex]
[tex]w(2\pi)=12\pi[/tex]
[tex]\displaystyle w=\frac{12}{2}=6\ rad/sec[/tex]
Thus, the function can be expressed as
[tex]\boxed{\displaystyle y=-2sin(6t)}[/tex]
