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Chapter 2: Problem 27

Determine the period of each function. $$ y=3 \sec \left(\frac{\pi}{2} x-\pi\right)+3 $$

Short Answer

Expert verified
The period of the function is 4.

Step by step solution

01

- Identify the basic period of the given trigonometric function

The function provided is a secant function. The basic secant function, \(y = \sec(x)\), has a period of \[2\pi.\]
02

- Determine the transformed period

The period of a transformed trigonometric function \(y = 3 \sec\left(\frac{\pi}{2}x - \pi\right)+3\) is calculated by modifying the basic period. The modification factor is the coefficient of \(x\) within the argument of the secant function \(\frac{\pi}{2}x.\)
03

- Calculate the transformed period

To find the new period, use the formula:\[ \text{New period} = \frac{\text{Basic period}}{\text{B}} = \frac{2\pi}{\frac{\pi}{2}} = \frac{2\pi}{\frac{\pi}{2}} =2 \cdot 2 = 4. \] Thus, the period of the function \(y = 3 \sec\left(\frac{\pi}{2}x - \pi\right)+3\) is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function, noted as \(\text{sec}(\theta)\), is a trigonometric function. It is closely related to the cosine function, as \(\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}\). This means the secant function is essentially the reciprocal of the cosine function. One key feature of the secant function is its behavior: it becomes infinite where the cosine function is zero. These points are known as vertical asymptotes, and they occur repeatedly due to the periodic nature of trigonometric functions. The basic secant function has a graph that appears as 'u'-shaped curves facing both upward and downward at regular intervals.
Transformed Period
When dealing with transformed trigonometric functions, it's critical to understand how transformations affect the function's period. Transformations can include horizontal stretching or compressing, as well as shifting the function along the x-axis. For horizontal transformations, the coefficient of x in the function's argument is the key factor. Given the function \( y = 3 \sec \left( \frac{\theta}{2}x - \theta \right) + 3 \), the coefficient of x is \(\frac{\theta}{2}\). This coefficient modifies the function's period, compressing or stretching it. To find the new period, divide the basic period by this coefficient.
Basic Period in Trigonometry
Basic periods are a fundamental characteristic of trigonometric functions. These periods are the intervals after which the function values repeat. For the secant function, \( y = \sec (x) \), the basic period is \(\text{2}\text{蟺}\). Understanding this basic period helps in determining the period of more complex, transformed functions. If given a transformed function such as \( y = 3 \sec \left( \frac{\theta}{2}x - \theta \right) + 3 \), you must adjust the basic period with respect to the given transformations. The formula then becomes \(\text{New period} = \frac{\text{Basic period}}{B} \), where B is the coefficient of x, typically found inside the function鈥檚 argument.

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