The question is as follows:
Find a formula for a function f whose graph $y = f(x)$ has period 12 and y-values that vary between the extremes 2 and 8. How many such examples are there?
If I am able to understand extremes well, then I think that it means the maximum value of the sinusoidal graph. I was able to find two equations with that had the maximum of 2 -- they are $f(x) = 2\cos(30x)$ and $f(x) = 2\sin(30x)$. If it continues like this for all of the numbers between 2 and 8 where there would be two equations each (one cosine and the other sine) then wouldn't there be 14 examples (starting from 2 and going to 8 there's seven numbers). But I was also thinking that if the graphs were shifted and the amplitudes were changed to get to extremes between 2 and 8 then there would be so many more examples. Any help will be greatly appreciated.
I think "between the extremes of $2$ and $8$" means the minimum is $2$ and the maximum is $8.$
In any case, unless there is some limitation you did not mention in the problem, there are infinitely many functions that satisfy the requirements. For comparison, consider the functions that have minimum $-1$ and maximum $1$ with period $360.$ Assuming $\sin(x)$ means the sine of $x$ in degrees (which it almost never does on this site, but your textbook or teacher evidently thinks you should learn it this way), the function $\sin(x)$ qualifies, but so do $\sin(x+1),$ $\sin(x+\frac12),$ $\sin(x+\frac13),$ $\sin(x+\frac14),$ $\sin(x+\frac15),$ and so forth forever.
And that's only considering sinusoidal functions. How about $f(x) = 2(\sin(x))^4 - 1$ as a function with minimum $-1$ and maximum $1$?