Problem 16 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=\frac{1}{2} \csc x$$

Step-by-Step Solution

Answer

Period is \(2\pi\) and graph is a vertically stretched \(\csc x\).

Step 1: Understand the Function Form

The given function is a cosecant function, expressed as \( y = \frac{1}{2} \csc x \). The general form of a cosecant function is \( y = a \csc(bx - c) + d \), where \( a \), \( b \), \( c \), and \( d \) affect the graph's stretch, period, phase shift, and vertical shift, respectively.

Step 2: Determine the Basic Parameters

For the function \( y = \frac{1}{2} \csc x \), the parameters are \( a = \frac{1}{2} \), \( b = 1 \), \( c = 0 \), and \( d = 0 \). This indicates there is a vertical stretch by a factor of \( \frac{1}{2} \), and no horizontal shift or vertical shift.

Step 3: Calculate the Period

The period of a cosecant function is calculated using the formula \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \). This means the function completes one cycle every \( 2\pi \) radians.

Step 4: Graph the Function

To graph \( y = \frac{1}{2} \csc x \), note the cosecant function is undefined where \( \sin x = 0 \), so vertical asymptotes occur at multiples of \( \pi \) (i.e., \( x = n\pi \), where \( n \) is an integer). The function takes minimum and maximum points at \( \frac{1}{2} \) (for \( y = \frac{1}{2}\sin x = \frac{1}{2} \)) or \( -\frac{1}{2} \) (for \( y = -\frac{1}{2}\sin x = -\frac{1}{2} \)). Plot the points and draw the curves towards and away from the asymptotes.

Key Concepts

Cosecant Function

The cosecant function is one of the six primary trigonometric functions and is related to the sine function. Precisely,

• It is denoted by \( \csc x \) and is defined as the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \).
• The function is undefined whenever \( \sin x = 0 \), which occurs at integer multiples of \( \pi \) (i.e., at \( x = n\pi \), where \( n \) is an integer).

The behavior of the cosecant function is such that it has vertical asymptotes at these points where it is undefined. In between these asymptotes, the graph has upward and downward arcs that approach infinity. Cosecant functions are periodic, meaning they repeat their pattern after a certain interval of the input values. The reciprocal property of the cosecant function means that when the sine value is small, its reciprocal, the value of the cosecant, becomes large, which is visually represented by the sharp peaks in its graph.

Period of a Function

The period of a function refers to the length of the smallest interval after which the function repeats its values. For trigonometric functions, this is a fundamental characteristic.

• The general period of the cosecant function is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) in the expression \( \csc(bx) \).
• In the provided function \( y = \frac{1}{2} \csc x \), since \( b = 1 \), the period remains \( 2\pi \).

This signifies that after every \( 2\pi \) radians, the cosecant function \( y = \frac{1}{2} \csc x \) completes a full cycle and begins again. As a result, the points on the graph after \( 2\pi \) radians are identical to those at the graph's start. This periodic nature is critical when sketching or analyzing the function's behavior over its domain.

Graphing Trigonometric Functions

Graphing trigonometric functions requires understanding their unique properties like period, amplitude, and vertical asymptotes. Let's break it down using the example \( y = \frac{1}{2} \csc x \):

• Vertical Stretch: The coefficient of \( \csc x \) impacts its range. Here, \( \frac{1}{2} \) indicates a vertical stretch by half. This means the maximum and minimum of the usual cosecant peaks are reduced to \( \frac{1}{2} \) and \(-\frac{1}{2} \), respectively.
• Periodicity: As calculated, the function's period is \( 2\pi \). The graph repeats every \( 2\pi \).
• Vertical Asymptotes: Occur at every \( n\pi \) (where \( n \) is an integer) since the function is undefined where \( \sin x = 0 \).

To graph:- Identify the asymptotes at the multiples of \( \pi \).- Mark the extrema at \( \frac{1}{2} \) and \(-\frac{1}{2} \) between asymptotes.- Sketch curves that approach the asymptotes from within the interval.Responsive graphing ultimately provides a visual grasp of the function's limits and behaviors around its periodic cycle, making it a vital tool for comprehension.