Graph each function over a one-period interval.


y = 3 sec [(1/4)x]

Verified step by step guidance

1. Identify the basic function: The given function is \( y = 3 \sec\left(\frac{1}{4}x\right) \). The basic function here is \( \sec(x) \), which is the reciprocal of \( \cos(x) \).
2. Determine the period of the function: The period of \( \sec(x) \) is \( 2\pi \). For \( \sec\left(\frac{1}{4}x\right) \), the period is adjusted by the coefficient \( \frac{1}{4} \) inside the function. The new period is \( 2\pi \times 4 = 8\pi \).
3. Identify the vertical stretch: The coefficient 3 in front of the secant function indicates a vertical stretch. This means the graph will be stretched vertically by a factor of 3.
4. Determine the asymptotes: The secant function has vertical asymptotes where the cosine function is zero. For \( \sec\left(\frac{1}{4}x\right) \), these occur at \( \frac{1}{4}x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Solve for \( x \) to find the asymptotes.
5. Graph the function: Plot the function over one period \([0, 8\pi]\). Mark the vertical asymptotes, and sketch the secant curve, considering the vertical stretch and the period.

Key Concepts

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function. It is defined as sec(x) = 1/cos(x). The secant function has vertical asymptotes where the cosine function is zero, leading to undefined values. Understanding the behavior of the secant function is crucial for graphing it accurately, especially in identifying its periodic nature and asymptotic behavior.

Period of a Trigonometric Function

The period of a trigonometric function is the length of one complete cycle of the function. For the secant function, the standard period is 2π, but this can change with transformations. In the given function y = 3 sec[(1/4)x], the coefficient (1/4) affects the period, stretching it to 8π. Recognizing how to determine and adjust the period is essential for accurate graphing.

Vertical Stretch

A vertical stretch occurs when a function is multiplied by a constant factor greater than one, affecting the amplitude of the graph. In the function y = 3 sec[(1/4)x], the factor of 3 indicates that the graph of the secant function will be stretched vertically by this factor. This means that the peaks and troughs of the graph will be three times higher or lower than the standard secant function, which is important for visualizing the graph's shape.