Determine the period and sketch at least one cycle of the graph of each function. State the rang

step 1 the function y equal to the second of the 3x and then we need to find the period of this functio

Determine the period and sketch at least one cycle of the...

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Key Concepts

Period of a Trigonometric Function

The period of a trigonometric function is the interval over which its values repeat. For a basic cosine function, the period is 2?, and when the function is modified by multiplying the variable inside the cosine by a constant, the period is scaled by the reciprocal of the absolute value of that constant.

Secant Function

The secant function is defined as the reciprocal of the cosine function. This means that it is undefined where the cosine equals zero, leading to vertical asymptotes. The shape of the secant function is closely related to that of the cosine function, but with these asymptotic disruptions.

Transformation of Trigonometric Functions

Any constant multiplied by the variable inside a trigonometric function causes a horizontal scaling of the graph. This transformation adjusts the period of the function. Specifically, if the original period is 2? and the function is modified to cos(ax), the new period becomes 2?/|a|.

Range of Secant Function

The range of the secant function is derived from the range of the cosine function. Since cosine outputs values between -1 and 1, its reciprocal, the secant, only outputs values that are less than or equal to -1 or greater than or equal to 1. Thus, the range of the secant function is (-?, -1] U [1, ?).

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