In this problem we deal with a type of function #F(x)# that is called sinusoidal.
These functions have a period, that is there exists a real value #T# such that #F(x+T)=F(x)# for any #x#.
These functions also have a characteristic called an amplitude - the maximum deviation from zero occurred on any interval of at least a size of a period #T#.
If function #F(x)# has a period #T#, then a function #F(k*x)# has a period of #T/k#.
This fact follows from a simple chain of equalities:
#F(k*(x+T/k))=F(k*x+k*T/k)=#
#=F(k*x+T)=F(k*x)#
(since #T# is a period of #F(x)# for any value of an argument, including #k*x#).
We have proven that
#F(k*(x+T/k))=F(k*x)#,
which means that #T/k# is a period.
If a function #F(x)# has an amplitude #A>0#, then a function #C*F(x)# has an amplitude #|C|*A#.
Indeed, let's consider that the maximum deviation of #F(x)# on a segment #[0,T]# equals #A# and it is reached at #x=x_0#, that is #F(x_0)=|A|#.
Then #C*F(x_0)=C*|A|#, from which follows that an amplitude of #C*F(x)# is, at least, equal to #|C*A|=|C|*A# (as #A>0#). It also cannot be greater that #|C|*A# since at that point of an #x# the original function #F(x)# would be greater than #A#.
Obviously, changes in period and altitude of a sinusoidal function are independent of each other and can be evaluated separately.
Using the above theoretical consideration, we see in our case for a function #y=-0.3cos(2x)# that its period equals to #T/2# (where #T# is a period of a function #y=cos(x)#) and its altitude equals to #|-0.3|*A# (where #A# is an altitude of a function #y=cos(2x)#.
We know that the period of a function #y=cos(x)# equals to #2pi#. Therefore, the period of a function #y=cos(2x)# equals to #(2pi)/2=pi#.
The amplitude of a function #y=cos(2x)# equals to #1#. Therefore, the amplitude of #y=-0.3cos(2x)# equals to #|-0.3|*1=0.3#
