In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec(x + π)

Verified step by step guidance

1. Identify the given function: \(y = 2 \sec(x + \pi)\). Recall that the secant function is the reciprocal of the cosine function, so \(\sec \theta = \frac{1}{\cos \theta}\).
2. Determine the period of the basic secant function. Since \(\sec x\) has the same period as \(\cos x\), which is \(2\pi\), the period of \(y = 2 \sec(x + \pi)\) remains \(2\pi\) because the horizontal shift does not affect the period.
3. Find the horizontal shift caused by the term \((x + \pi)\). This represents a shift to the left by \(\pi\) units. So, the graph of \(\sec x\) is shifted left by \(\pi\).
4. Calculate the amplitude and vertical stretch. The coefficient 2 in front of \(\sec\) stretches the graph vertically by a factor of 2. Note that secant functions do not have a maximum or minimum amplitude, but this factor affects the distance from the midline.
5. To graph two periods, plot the function from \(x = -\pi\) (starting point due to shift) to \(x = -\pi + 4\pi = 3\pi\). Mark key points where \(\cos(x + \pi) = 0\) (vertical asymptotes for \(\sec\)), and plot the corresponding \(y\) values using \(y = 2 \sec(x + \pi)\) between these asymptotes.

Key Concepts

Secant Function and Its Properties

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It has vertical asymptotes where cos(x) = 0, and its graph consists of branches extending to infinity. Understanding its periodicity and behavior near asymptotes is essential for accurate graphing.

Phase Shift in Trigonometric Functions

A phase shift occurs when the input variable x is replaced by (x + c), shifting the graph horizontally. For y = 2 sec(x + π), the graph of sec(x) shifts left by π units. Recognizing this shift helps in correctly positioning the function's key features like peaks, troughs, and asymptotes.

Amplitude and Vertical Stretch

The coefficient 2 in y = 2 sec(x + π) vertically stretches the secant graph by a factor of 2. This affects the distance of the graph's branches from the x-axis, making them twice as far compared to the basic sec(x) graph. Understanding vertical stretch is crucial for accurate graph scaling.