Trigonometry Transformations of Trigonometric Graphs Transformations of the Cosine Graph
Period
   Concepts 1

Definition 1
.
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 .

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