Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a one-period interval. $$y=\sec \left(x+\frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

Graph the shifted and repeated secant function considering asymptotes and range.

Step 1: Identify the Basic Function

The given function is based on the secant function, which is the reciprocal of the cosine function. The formula for the secant function is given by \( y = \sec(x) = \frac{1}{\cos(x)} \).

Step 2: Determine the Period of the Secant Function

The period of the secant function \( \sec(x) \) is the same as the cosine function, which is \( 2\pi \). This means that the function repeats every \( 2\pi \) units.

Step 3: Identify the Phase Shift

In the function \( y = \sec(x + \frac{3\pi}{4}) \), there is a horizontal shift, also known as a phase shift. To find the phase shift, set the inside of the secant function equal to zero: \( x + \frac{3\pi}{4} = 0 \). Solve for \( x \) to get \( x = -\frac{3\pi}{4} \). Thus, the graph is shifted to the left by \( \frac{3\pi}{4} \) units.

Step 4: Identify Asymptotes and Important Points

The secant function has vertical asymptotes where its reciprocal, the cosine function, is zero. For \( \cos(x + \frac{3\pi}{4}) = 0 \) solve: \( x + \frac{3\pi}{4} = \frac{\pi}{2} + n\pi \) where \( n \) is an integer. These values give the locations of the vertical asymptotes within the period.

Step 5: Determine the Range

Since the secant function can never equal zero, its range is \( (-\infty, -1] \cup [1, \infty) \). When plotting \( y = \sec(x + \frac{3\pi}{4}) \), ensure your graph reflects these values.

Step 6: Sketch the Graph

Plot the vertical asymptotes first, then draw the secant function's repetitive U-shaped curves. Take care to position them with respect to the phase shift and period. Note that the function will be undefined and tend towards \( \pm \infty \) at each asymptote.

Key Concepts

Secant Function

The secant function, denoted as \[ y = \sec(x) \]is the reciprocal of the cosine function. This characteristic means that:\[ \sec(x) = \frac{1}{\cos(x)} \]Understanding this relationship is crucial because the secant function will be undefined at any point where the cosine function equals zero. 

This leads to the presence of vertical asymptotes on the graph, which are gaps where the function does not reach finite values.A distinctive feature of the secant function is its repeating pattern, or periodic nature, over intervals of \(2\pi\), similar to the cosine function. The secant function is primarily characterized by its range:

• It never crosses between -1 and 1.
• Always extends to \((-\infty, -1] \text{ or } [1, \infty)\).

These aspects guide how the graph is drawn, typically featuring U-shaped curves that open away from the x-axis between the asymptotes.

Phase Shift

A phase shift in trigonometric functions refers to the horizontal movement of the graph along the x-axis. 

In the function:\[ y = \sec \left(x + \frac{3\pi}{4}\right) \]the term \(\frac{3\pi}{4}\) denotes a phase shift to the left. To determine the exact magnitude of this shift, solve the equation:\[ x + \frac{3\pi}{4} = 0 \]resulting in\[ x = -\frac{3\pi}{4} \]Thus, the entire graph of the function shifts left by \(\frac{3\pi}{4}\) units. 

This movement must be consistently applied to all key points of the graph, including peaks and asymptotes, as it affects where the function starts repeating its cycle.

Graphical Asymptotes

Graphical asymptotes are lines that the graph approaches but never touches. For the secant function, these are vertical lines that occur whenever the base cosine function equals zero. 

In the case of\[ y = \sec \left(x + \frac{3\pi}{4}\right) \]we find asymptotes by solving:\[ \cos \left(x + \frac{3\pi}{4}\right) = 0 \]This happens whenever:\[ x + \frac{3\pi}{4} = \frac{\pi}{2} + n\pi \]where \( n \) is any integer (representing the basic period plus half.) As \( n \) varies, these determine the precise positions of the asymptotes within the function's repeating interval.

• The graph tends to \( \pm \infty \) as it approaches these lines on either side.
• It's crucial to mark these when sketching since they dictate where the secant function jumps from negative to positive sections (and back).

Marking these correctly ensures the graphical representation accurately reflects the trigonometric nature of the secant function.