Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=-\frac{1}{2} \csc (2 x-\pi) $$

Step-by-Step Solution

Answer

The period is \( \pi \); the asymptotes are at \( x = \frac{(2n+1)\pi}{2} \).

Step 1: Transform the Equation

Recognize the given function: \[ y = -\frac{1}{2} \csc(2x - \pi)\.\] The cosecant function \( \csc \) is the reciprocal of the sine function. Hence the main characteristics are similar to those of \( \sin \) but inverted with undefined values (asymptotes) where sine is zero.

Step 2: Determine the Period

The period of \( \csc(kx) \) (or \( \sin(kx) \)) function is \( \frac{2\pi}{k} \). Here, \( k = 2 \), so the period is \[ \frac{2\pi}{2} = \pi \]. This means that the function will repeat every \( \pi \) units.

Step 3: Locate the Asymptotes

Identify where \( \sin(2x - \pi) = 0 \), since this makes \( \csc \) undefined. Solve for \( 2x - \pi = n\pi \), where \( n \) is an integer:\[ 2x = n\pi + \pi \]\[ x = \frac{n\pi + \pi}{2} \]This gives vertical asymptotes at \( x = \frac{(2n+1)\pi}{2} \).

Step 4: Graph the Function

Since \( y = -\frac{1}{2} \csc(2x - \pi) \), the graph will be the -inverted version of the standard \( \csc \) graph, vertically compressed by \( \frac{1}{2} \). 1. Each period \( \pi \) has two asymptotes at \( x = \frac{(2n+1)\pi}{2} \). 2. The graph oscillates between these asymptotes.3. Reflect the standard \( \csc \) shape downward due to the negative sign.4. Sketch the curves with vertically compressed peaks and troughs.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental to understanding many phenomena in mathematics and science. These functions relate angles of a triangle to the lengths of its sides in a right-angled triangle. Some of the primary trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). From these, secondary functions such as cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)) are derived.

• Sine and Cosecant Relationship: The cosecant function (\( \csc \)) is the reciprocal of sine, meaning \( \csc x = \frac{1}{\sin x} \)
• Function Properties: Trigonometric functions are periodic, meaning they repeat their values at regular intervals. Understanding these properties and relationships is crucial for solving and graphing these functions.

These characteristics enable a deep exploration of wave patterns, oscillations, and cycles, forming the backbone of understanding repetitive patterns in calculus and physics.

Asymptotes

Asymptotes in trigonometric functions describe locations where the function does not have a finite value. For the cosecant function, since it is the reciprocal of the sine function, it becomes undefined wherever the sine function is zero. In the context of our function \( y = -\frac{1}{2} \csc(2x - \pi) \), we need to pinpoint these asymptotic points.- **Calculating Asymptotes:** To locate the asymptotes, set the inside of the sine function equal to zero: \( 2x - \pi = n\pi \), where \( n \) is an integer. Solving gives: \[ x = \frac{n\pi + \pi}{2} \] thus, asymptotes occur at \( x = \frac{(2n+1)\pi}{2} \).Understanding where asymptotes occur helps ensure we correctly sketch and interpret graphs of trigonometric functions, as it indicates points where the graph will shoot off toward infinity and enhances comprehension of the function’s structure.

Periodicity

Periodicity is a core concept when dealing with trigonometric functions because it represents how often the function's values repeat over a certain interval. For the function \( y = -\frac{1}{2} \csc(2x - \pi) \), this periodic behavior helps identify critical patterns in its graph.

• Identifying the period: The function repeats every \( \pi \) interval. Generally, for \( \csc(kx) \), the period is \( \frac{2\pi}{k} \). Here, \( k = 2 \), so the period is \( \pi \).
• Application of Periodicity: By understanding periodicity, we can predict the function's behavior over any interval, establish a framework for graphing, and identify repeating features.

These repeating characteristics serve as consistent checkpoints to predict and understand where a function will oscillate, aiding greatly in sketching accurate graphs and solving equations involving these functions.

Graph Sketching

Graph sketching involves visually interpreting a function to demonstrate its key characteristics, such as its maxima, minima, and asymptotes. For \( y = -\frac{1}{2} \csc(2x - \pi) \), sketching provides a visual grasp of the function's behavior and characteristics visually brought to life.

• Understanding Inversion and Compression: This specific function is an inverted version of the cosecant function due to the negative sign. Furthermore, it is vertically compressed by a factor of \( \frac{1}{2} \).
• Identifying Asymptotes and Periods: Each period (\( \pi \)) should depict two asymptotes, and the graph oscillates between them. Understanding the function's periodicity helps define frames within which to sketch.
• Sketching Approach: Within each period, reflect the standard \( \csc \) shape downward, observe the asymptotes, and trace the function’s curves with compressed peaks and troughs.

Sketching not only solidifies comprehension but also makes abstract function properties concrete through visual representation. It helps identify correct patterns and behaviors, particularly aiding in seeing how the function looks across cyclic intervals.