Graph each function over a one-period interval.
y = 2 + 3 sec (2x - π)

Verified step by step guidance

1. Identify the standard form of the secant function: \( y = a + b \sec(c(x - d)) \). In this case, \( a = 2 \), \( b = 3 \), \( c = 2 \), and \( d = \frac{\pi}{2} \).
2. Determine the period of the secant function. The period of \( \sec(cx) \) is \( \frac{2\pi}{c} \). Here, \( c = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).
3. Find the phase shift by solving \( c(x - d) = 0 \). This gives \( 2(x - \frac{\pi}{2}) = 0 \), leading to \( x = \frac{\pi}{2} \). The graph is shifted to the right by \( \frac{\pi}{2} \).
4. Determine the vertical shift and amplitude. The graph is shifted vertically by \( a = 2 \), and the amplitude is \( |b| = 3 \), affecting the vertical stretch of the secant function.
5. Graph the function over one period \([\frac{\pi}{2}, \frac{\pi}{2} + \pi] = [\frac{\pi}{2}, \frac{3\pi}{2}]\), considering the vertical and phase shifts, and plot the asymptotes where the cosine function (the reciprocal of secant) is zero.

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 secant is crucial for graphing, as it influences the shape and position of the graph.

Transformations of Functions

Transformations involve shifting, stretching, or compressing the graph of a function. In the given function y = 2 + 3 sec(2x - π), the '+2' indicates a vertical shift upwards by 2 units, while the '3' scales the secant function vertically. The '2x - π' represents a horizontal compression by a factor of 1/2 and a phase shift to the right by π/2. Understanding these transformations is essential for accurately graphing the function.

Period of Trigonometric Functions

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 it can change based on the coefficient of x. In the function y = 2 + 3 sec(2x - π), the period is adjusted to π due to the coefficient '2' in front of x. Recognizing the period is vital for determining the intervals over which to graph the function.