We have to start with definitions.
Amplitude is the maximum deviation from a neutral point.
For a function #y=cos(x)# it is equal to #1# since it changes the values from minimum #-1# to maximum #+1#.
Hence, the amplitude of a function #y=A*cos(x)# the amplitude is #|A|# since a factor #A# proportionally changes this deviation.
For a function #y=−3cos(2pix−pi)# the amplitude is equal to #3#. It deviates by #3# from its neutral value of #0# from its minimum of #-3# to a maximum of #+3#.
Period of a function #y=f(x)# is a real number #a# such that #f(x)=f(x+a)# for any argument value #x#.
For a function #y=cos(x)# the period equals to #2pi# because the function repeats its values if #2pi# is added to an argument:
#cos(x) = cos(x+2pi)#
If we put a multiplier in front of an argument, periodicity will change. Consider a function #y=cos(p*x)# where #p# - a multiplier (any real number not equal to zero).
Since #cos(x)# has a period #2pi#, #cos(p*x)# has a period #(2pi)/p# since we have to add #(2pi)/p# to an argument #x# to shift the expression inside the #cos()# by #2pi#, which will result in the same value of a function.
Indeed, #cos(p*(x+(2pi)/p)) = cos(px+2pi) = cos(px)#
For a function #y=−3cos(2pix−pi)# with #2pi# multiplier at #x# the period is #(2pi)/(2pi)=1#.
Phase shift for #y=cos(x)# is, by definition, zero.
Phase shift for #y=cos(x-b)# is, by definition, #b# since the graph of #y=cos(x-b)# is shifted by #b# to the right relative to a graph of #y=cos(x)#.
Since #y=−3cos(2pix−pi)=-3cos(2pi(x-1/2))#, the phase shift is #1/2#.
In general, for a function #y=Acos(B(x-C))# (where #B!=0#):
amplitude is #|A|#,
period is #(2pi)/|B|#,
phase shift is #C#.
