Problem 96 - Algebra and Trigonometry

Question

In Exercises 95–96, write the equation for a cosecant function satisfying the given conditions. $$ \text { period: } 2 ; \text { range: }(-\infty,-\pi] \cup[\pi, \infty) $$

Step-by-Step Solution

Answer

The equation for the cosecant function with a period of 2 and a range of (-∞, -π] U [π, ∞) is \(y=\csc(\pi x)\).

Step 1: Setting the Amplitude and Vertical Shift

The amplitude for the cosecant function is usually undefined, as it increases or decreases without bound. So, we leave it as 1. The range suggests that the function isn't shifted up or down, so the vertical shift \(K\) is 0.

Step 2: Calculate the Periodic Modifier

We know that the given period is 2 and the period of a cosecant function is calculated as \(\frac{2\pi}{B}\). Now we can solve for B: \(B=\frac{2\pi}{\text{period}} = \frac{2\pi}{2} = \pi\).

Step 3: Calculate the Horizontal Shift

Since there is no information given about any phase shift, we can consider the horizontal shift H to be 0.

Step 4: Write the Equation

Now we can write the equation for a cosecant function, using the modified parameters we have determined. Therefore, the equation for the function is \(y=\csc(\pi x)\).

Key Concepts

Understanding the Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is the reciprocal of the sine function. While the sine function can be described as \( \sin(x) = \frac{1}{\csc(x)} \), the cosecant function is undefined for values where \( \sin(x) = 0 \). This happens at \( x = n\pi \) where \( n \) is an integer.

Some key features of the cosecant function include:

• It has vertical asymptotes where the sine function equals zero.
• It does not have a defined amplitude, as it ranges from negative infinity to positive infinity, skipping the zero.
• Its graph consists of repeating curves sweeping away from the vertical asymptotes.

Understanding these properties helps in sketching its graph and identifying transformations like shifts or stretches.

Decoding the Period of a Function

The period of a trigonometric function is the interval after which the function starts repeating its values. For the cosecant function, this period is influenced by the parameter \( B \) in the function's equation \( y = \csc(Bx) \). Normally, the period of a basic cosecant function is \( 2\pi \).

To determine the new period when modifications are made, the formula \( \frac{2\pi}{B} \) is used. In this exercise, the specified period is 2, leading to the equation:

\[ B = \frac{2\pi}{2} = \pi \]

This calculation shows that each segment of the cosecant graph repeats every 2 units along the x-axis. Understanding how to modify the period can help visualize and write equations for transformed trigonometric functions.

Navigating Vertical Shifts

Vertical shifts move the entire graph of a function up or down the y-axis. For trigonometric functions like the cosecant function, this is typically represented by \( K \) in an equation \( y = \csc(Bx) + K \).

In this example, the range \((-\infty, -\pi] \cup [\pi, \infty)\) suggests the graph isn't shifted vertically. The center line or middle of the graph remains the same, meaning no vertical shift occurs, so \( K = 0 \).

This concept is crucial when customizing the height and position of graph sections, impacting things like amplitude and phase alignment. Ensuring the correct interpretation of \( K \) helps create accurate graph descriptions.