Problem 42 - Precalculus : Building Concepts and Connections

Question

Graph the given pair of functions in the same window. Graph at least two cycles of each function, and describe the similarities and differences between the graphs. $$f(x)=\frac{1}{2} \csc (x) ; f(x)=\csc \left(\frac{1}{2} x\right)$$

Step-by-Step Solution

Answer

Both functions are variations of the cosecant function. The first function is a vertically compressed version of the cosecant function by a factor of 1/2, while the second function is a horizontally stretched version of the cosecant function by a factor of 2. The amplitude of the first function is smaller, and the period of the second function is longer.

Step 1: Recognize the functions

Recognize that both functions are variations of the cosecant function. In the first function \(f(x)=\frac{1}{2} \csc (x)\), \(\csc(x)\) is being scaled by a factor of 1/2. In the second function \(f(x)=\csc \left(\frac{1}{2} x\right)\), the input x is being scaled by a factor of 1/2.

Step 2: Graph the functions

To sketch the graph of these functions, plot some key points and basic shape of a \(\csc\) graph. Remember the shape of \(\csc(x)\) is 'U' shaped or an inverted 'U', depending where it is defined. The function \(\csc(x)\) is undefined for \(x=nπ\), where n is an integer, this is because \(\csc(x)= 1/\sin(x)\), and \(\sin(nπ)=0\). So mark these undefined points on x-axis with a dotted line. The amplitude of function \(f(x)=\frac{1}{2} \csc (x)\) will be half as it is scaled by a factor of 1/2, and period of \(f(x)=\csc \left(\frac{1}{2} x\right)\) will be 2π instead of π as it is scaled by a factor of 1/2 in input.

Step 3: Observe and compare

Describe the similarities and differences between the two graphs. The functions have similar shapes, as both are variations of the cosecant function. The amplitude of the first function is smaller, as it is scaled by 1/2. The period of the second function is longer, as the input x is scaled by 1/2.

Key Concepts

Cosecant Function

Understanding the cosecant function is crucial before diving into any transformations. The cosecant function, denoted as \( \csc(x) \), is the reciprocal of the sine function. This implies that \( \csc(x) = \frac{1}{\sin(x)} \). It's important to note that whenever the sine function equals zero, the cosecant function becomes undefined. This occurs at integer multiples of \( \pi \) (i.e., \( x = n\pi \) where \( n \) is an integer).
The typical graph of the cosecant function forms 'U' shapes or inverted 'U' shapes between these undefined points. These sections of the graph extend from \( -\infty \) to \( \infty \) above and below each undefined point. Remembering these basic characteristics will help in graphing and understanding any transformations of the cosecant function.

Amplitude

In trigonometry, the amplitude of a function typically refers to the height of its peaks from the centerline to the top of a peak, however, for the cosecant function, this concept is slightly different. The amplitude determines how stretched or compressed the 'U' shapes of the cosecant graph are.
For example, in the function \( f(x) = \frac{1}{2} \csc(x) \), the graph is 'scaled' vertically by a factor of 1/2. This means that the 'U' shapes are compressed towards the x-axis, thereby reducing the vertical extent between the peaks and valleys of the graph. Essentially, while sine functions have a clear amplitude, with cosecant, the amplitude plays the role in determining how spread out the graph is vertically.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle. This is essential in determining how the graph repeats over the x-axis.
For a standard \( \csc(x) \) function, the period is \( 2\pi \), which means every \( 2\pi \) units along the x-axis, the graph repeats itself. However, transformations can alter this period. In the function \( f(x) = \csc\left(\frac{1}{2}x\right) \), the period is doubled because the x-values are "stretched" by a factor of 2. This means the graph now completes a cycle every \( 4\pi \) units instead of \( 2\pi \).
Understanding how the period affects the graph helps in predicting and sketching the behavior of the function over different intervals.

Graphing Functions

Graphing trigonometric functions like the cosecant involves understanding the basic shape of the graph and how it is affected by different transformations. Before plotting, it's helpful to list down key points, including undefined points where the cosecant is not defined (at \( x = n\pi \)). Use these key points as guidelines.
Here are steps to aid in graphing:

• Identify the vertical asymptotes at points where \( \sin(x) = 0 \).
• Determine how transformations like scaling affect the graph: scaling affects either amplitude or period.
• Start by plotting critical points and asymptotes, then sketch the upward and downward 'U' shapes between these asymptotes.

The function \( f(x) = \frac{1}{2} \csc(x) \) will resemble the original \( \csc(x) \), but with a reduced vertical span. On the other hand, \( f(x) = \csc\left(\frac{1}{2}x\right) \) will look stretched horizontally, with its repeated pattern extending over a wider x-interval. By keeping these guidelines in mind, you can graph these functions effectively.