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The concept of the period of a function is vital when dealing with trigonometric functions. A function's period is the length of the smallest interval over which the function repeats its values. Essentially, for a function like sine, cosine, or secant, you'll see the same wave pattern repeated over and over. Knowing the period helps us understand how frequently these repetitions occur.
In the case of the standard secant function, its period is similar to the cosine function, from which it is derived. The cosine function repeats its pattern every \(2\pi\). Therefore, the secant function also has a standard period of \(2\pi\). But this period can be altered if there's a multiplication factor involved in the function's argument, as we'll see with horizontal compression.

The secant function, denoted as \(y = \sec x\), is a trigonometric function that is the reciprocal of the cosine function. In simpler terms, \(\sec x = \frac{1}{\cos x}\). Its graph is characterized by its vertical asymptotes, occurring at points where the cosine function is zero, due to division by zero being undefined.
Understanding its fundamental properties, such as the period, helps predict the repeating nature of its graph. For the standard secant function, this period is \(2\pi\), matching the cosine function. Thus, the same pattern in the graph occurs every \(2\pi\) along the x-axis.
In the example given, \(y = \sec 5x\), the introduction of the factor 5 affects the period, thanks to horizontal compression. This multiplication factor alters how quickly the secant function completes its cycles.

Horizontal compression involves adjusting the length of a function's period by multiplying the variable by a constant. This constant effectively compresses or extends the graph along the x-axis, depending on its value.
Let's consider \(y = \sec 5x\). Here, the variable \(x\) is multiplied by 5, which means the graph of the secant function is compressed horizontally by this factor. Instead of taking the usual \(2\pi\) to complete one cycle, it completes a cycle more rapidly because the period is divided by 5.

• This results in the new period being \(\frac{2\pi}{5}\).
• It implies that the graph repeats its cycles more frequently along the x-axis.

This transformation is crucial in trigonometric functions, allowing us to predict and graph their behavior precisely when these compressions or stretches are applied.