First, you are only eye-balling the graph. You cannot KNOW where those poinst are unless there is additinoal information.
Second, there often is not a unique answer. It is likely that there are equivalent answers using sine or cosine. Unless the problem statement mentions it, either shoudl do.
That said, let's have a look.
It appears to have f(0) = Greatest Amplitude. This might suggest a Cosine without a phase shift.
The amplitude appears to be 5.
The period appears to be somewhere in the neighborhood of 6. Notice that 2*pi is sowhere in the neighborhood of 6. Don't be too impressed with your eyes. Frankly, it looks like the first period to the right is a little less than 6 and the first period to the right a little more than 6. 2pi/6 = pi/3
So, perhaps
f(x) = 5*cos((pi/3)*x)
OK, that was the easy one. Since the problem statement asks for a sine function, it will take a little more effort.
One thing we can do is take advantage of known identities, like this one, cos(x) = sin(pi/2 - x). This gives
f(x) = 5*sin((pi/2) - (pi/3)*x)
However, that isn't nearly as fun as eye-balling the answer. I REALLY didn't like your "4.5", mostly because it was a really big number when a measly 1.5 the other direction would have been just the same. Then it strucl me that 1.5 was awfully close to pi/2. just guessing, then, produces
f(x) = 5*sin((pi/3)*x + pi/2)[/b]
or
f(x) = 5*sin((pi/3)*(x + 3/2))
and there is your phase shift of 1.5.
Note: Remember that sin((pi/2) - x) = cos(x) is an EVEN function. This will help explain why the two sine answers don't quite look the same.
