A nonconstant function f is called periodic if and only if there is a positive real
number
p
for which
f(x+p)=f(x)
for all x in the domain of f. The smallest such value of
p
is called the period of f.
Statement
.
For any nonzero constant
b
the function
f(x) = A· cos(bx
+ c)
has period .
We can verify this by computing. For examle, if b is positive,
then
When
|b|>1
the graph of
y=A· cos(bx
+ c)
is horizontally compressed
compared to the graph of
y=A· cos(x + c)
.
When |b|<1
the graph of
y=A· cos(bx
+ c)
is horizontally stretched compared to the graph of
y=A· cos(x + c).
Definition 2
.
If f(x) has period p, then
is called the frequency of f.
For example, by the Statement,
function f(x)=cos3x
has period
.
Therefore, by the Definition 2, the frequency of
f(x)=cos3x
equals
.