91Ó°ÊÓ

The secant function, denoted as \( \sec(x) \), is a vital concept when graphing trigonometric functions. It is the reciprocal of the cosine function, defined as \( \sec(x) = \frac{1}{\cos(x)} \). Since cosine can be zero, resulting in division by zero, the secant function has undefined points, often resulting in vertical asymptotes. These points occur at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. The typical graph of a secant function includes these asymptotes which create a repeating pattern of curves extending to infinity both positively and negatively.

• Consider the domain and range: \( x \) cannot be any value that makes \( \cos(x) = 0 \).
• The fundamental period of \( \sec(x) \) is \( 2\pi \).

Knowing these basics helps in modifying the secant function for graphing purposes, such as scaling and shifting.

Every trigonometric function, including the secant, has a specific period, which is the interval after which the function starts repeating itself. In its basic form, the secant function \( \sec(x) \) has a period of \( 2\pi \). However, when transformations are applied, such as changes in frequency, the period can change. In the exercise, the period of \( f(x) = 0.5 \sec(0.5x) \) becomes \( 4\pi \) due to the multiplier \( 0.5 \) applied to \( x \). This means the graph repeats every \( 4\pi \) units.

Vertical asymptotes of the secant function also shift according to these transformations. Originally found at \( x = \frac{\pi}{2} + k\pi \), for \( \sec(0.5x) \), they move to \( x = \pi + 2k\pi \).

• Period determined by frequency: \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \).
• Locate asymptotes within the viewing window for accurate graphing.

When modifying a function such as \( g(x) = 0.5 \sec\left[0.5\left(x - \frac{\pi}{2}\right)\right] - 1 \), transformations change the graph's appearance. These include:

• Horizontal Shifts: Moving the graph left or right by changing \( x \) values. Here, \( x - \frac{\pi}{2} \) results in a rightward shift of \( \frac{\pi}{2} \) units.
• Vertical Shifts: Moving the graph up or down through additions or subtractions outside the function. In this case, \(-1\) shifts the entire structure downward by one unit.
• Scaling: Multiplying the function by a constant affects its height, as seen with the factor \( 0.5 \) creating a vertically compressed secant graph.

Understanding these transformations allows one to accurately predict changes in the graph, including movements of peaks, valleys, and asymptotes.

When graphing functions, especially trigonometric ones, selecting an appropriate viewing rectangle is essential for capturing the function's behavior. A viewing rectangle is defined by the range of \( x \) (horizontal) and \( y \) (vertical) values you choose to display.

In our problem, the specified viewing rectangle is \([-2\pi, 2\pi, \pi / 2]\) by \([-4,4]\). This means:

• The graph will display \( x \) values from \(-2\pi\) to \( 2\pi \), with tick marks at intervals of \( \pi/2 \).
• The \( y \)-axis will stretch from \(-4\) to \( 4 \) to accommodate the maximum and minimum values of this particular function.

Choosing the right viewing rectangle ensures that significant features like key points, asymptotic behavior, and amplitude of the function are visible within the graph, allowing for thorough analysis and prediction of function characteristics.