How do you find the amplitude, period, and phase shift for a sine function?
1 Answer
If you have
#A# is the amplitude;#(2pi)/omega# is the period;#phi# is the shift.
Explanation:
The most general case of a sine function is

Since the sine function is bounded between
#1# and#1# , if we multiply the function by#A# the result will be bounded between#A# and#A# . So, the amplitude is "decided" by the outter coefficient. 
The
#omega# coefficient affects the period: we know that "normally"#x# runs a full period from#0# to#2pi# . Multipling#x# by a factor#w# means to change the speed of the forementioned run. For example, if you change your variable from#x# to#2x# , your variable will be running at twice the speed, and so you will go from#0# to#2pi# as#x# goes from#0# to#pi# .
On the other hand, if you change from#x# to#x/2# , you will go at half the speed, and so#x/2# goes from#0# to#2pi# (i.e, runs a full period) as#x# goes from#0# to#4pi# .
So, we see that the period is#(2pi)/omega# 
Finally, the term
#phi# is the shift, it simply means that instead of starting from zero, you start from#phi# : take for example#sin(x)# and#sin(x+pi/3)# . When#x# is zero,#sin(x+pi/3)# is "already" at#pi/3# , representing what I said before: the function starts from#phi# at not from zero.