Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the graph of the function to answer each question. (a) Find any \(x\) -intercepts of the graph of \(y=f(x)\). (b) Find any \(y\) -intercepts of the graph of \(y=f(x)\). (c) Find the intervals on which the graph of \(y=f(x)\) is increasing and the intervals on which the graph of \(y=f(x)\) is decreasing. (d) Find all relative extrema, if any, of the graph of \(y=f(x)\) (e) Find all vertical asymptotes, if any, of the graph of \(y=f(x)\) \(f(x)=\csc x\)

Step-by-Step Solution

Answer

The function \(y=\csc x\) does not have any \(x\)- or \(y\)-intercepts. It increases on the intervals \((-\pi, 0)\) and \((\pi, 2\pi)\), and decreases on the intervals \((0, \pi)\) and \((2\pi, 3\pi)\). The function has relative minimums at odd multiples of \(\pi/2\) and maximums at even multiples of \(\pi/2\). The vertical asymptotes of the function occur at \(x = n\pi\), where n is an integer.

Step 1: Find \(x\)-intercepts

The function \(y=\csc x\) does not have any \(x\)-intercepts because it is the reciprocal of the sine function, and is undefined wherever sine is zero.

Step 2: Find \(y\)-intercepts

The function \(y=\csc x\) does not have any \(y\)-intercepts because the cosecant function never crosses the y-axis.

Step 3: Find Intervals of Increase and Decrease

For the intervals of increase or decrease, we need to determine where the derivative changes sign. However, for \(y=\csc x\), the function increases on \((-\pi, 0)\) and \((\pi, 2\pi)\), and decreases on \((0, \pi)\) and \((2\pi, 3\pi)\), within the domain \(-\pi < x < \pi\). This pattern repeats due to the periodic nature of the function.

Step 4: Find Relative Extrema

The function \(y=\csc x\) has relative minimums wherever \(\sin x\) has maximums (i.e., at odd multiples of \(\pi/2\)), and relative maximums wherever \(\sin x\) has minimums (i.e., at even multiples of \(\pi/2\)).

Step 5: Find Vertical Asymptotes

Vertical asymptotes for \(y=\csc(x)\) will occur wherever the sine function is zero. This is at \(x = n\pi\), where n is an integer.

Key Concepts

Graphs of Trigonometric Functions

Trigonometric functions, like the cosecant function, have unique characteristics in their graphs. The cosecant function, denoted as \(y = \csc x\), is the reciprocal of the sine function. This means wherever sine equals zero, cosecant is undefined, giving its graph a unique and intriguing shape.
The cosecant graph is composed of branches that resemble waves. These branches are continuous between each vertical asymptote - which we will discuss later - and appear to form a repeating pattern every \(2\pi\) due to the periodic nature of the sine function.
A key feature of its graph is that there are no intersections with the x-axis or y-axis. The function simply does not cross these axes due to its undefined nature at certain points and inability to have zero value.

Intervals of Increase and Decrease

The behavior of a function in terms of increasing or decreasing can be analyzed by studying its derivative. For \(y = \csc x\), there are specific intervals where the function is known to either increase or decrease.
Cosecant increases in the intervals \((-\pi, 0)\) and \((\pi, 2\pi)\). In these intervals, the graph moves upwards as its branches extend. The function decreases in the intervals \((0, \pi)\) and \((2\pi, 3\pi)\), where the branches of the graph move downward.
These patterns occur periodically due to trigonometric graph behavior, and monitoring these intervals ensures a better understanding of where the graph is rising or falling on its domain.

Relative Extrema

Relative extrema in functions refer to points on a graph where the function switches from increasing to decreasing, or vice versa, forming local maxima or minima. In the case of \(y = \csc x\), these extrema are directly related to the sine function's behavior.
The function has relative minimums where the sine function reaches its maximum; these occur at the odd multiples of \(\pi/2\). Correspondingly, relative maximums occur at even multiples of \(\pi/2\) where the sine function achieves its minimum.
Understanding where these extrema occur helps in comprehending the local variations in the rise or fall of the cosecant graph and is crucial for precise graph analysis.

Vertical Asymptotes

Vertical asymptotes are imaginary vertical lines that the graph of a function approaches but never actually touches or crosses. For \(y = \csc x\), vertical asymptotes appear where the sine function is zero, as reciprocals become undefined at these points.
In terms of the x-values, vertical asymptotes for cosecant function can be found at \(x = n\pi\), where \(n\) is any integer. These asymptotes form regular intervals along the x-axis, essentially splitting the graph into separate branches.
The presence of these asymptotes plays a critical role in shaping the structure and behavior of the cosecant graph by segmenting it and guiding its intermittent continuity.