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

|  |  | Definition 1 |  |
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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 |  |
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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 |  |
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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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