Problem 28 - Precalculus : Building Concepts and Connections

Question

Graph at least two cycles of the given functions. $$f(x)=3 \cos \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=-3 \csc \left(x-\frac{\pi}{3}\right)\) is a reflection of the cosecant function stretched by a factor of 3 and shifted \(\frac{\pi}{3}\) units to the right. The graph has vertical asymptotes at \(x = \frac{\pi}{3} + n\pi\) where \(n\) is an integer, and the function is not defined at these points. The graph has two full periods from \(\frac{\pi}{3} - 2\pi\) to \(\frac{\pi}{3}\) and from \(\frac{\pi}{3}\) to \(\frac{\pi}{3}+2\pi\).

Step 1: Understanding the function

The key to understanding the given function, \(f(x)=-3 \csc \left(x-\frac{\pi}{3}\right)\), is to recognize that it’s a cosecant function which has been transformed. The '-3' is a reflection across the x-axis and vertical stretch by a factor of 3, and \(\left(x-\frac{\pi}{3}\right)\) is a horizontal shift by \(\frac{\pi}{3}\) units to the right. The cosecant function is undefined at x=0, so the function is undefined at \(\frac{\pi}{3}\) due to the shift. The period of the cosecant function is \(2\pi\), so the period of the given function will also be \(2\pi\).

Step 2: Starting the graph

Start the graph at the point where x equals the phase shift, which is \(\frac{\pi}{3}\). On the x-axis, mark a point at \(\frac{\pi}{3}\). Because the function is a cosecant function and it's undefined at this point, draw a vertical dashed line at x = \(\frac{\pi}{3}\). This is called a vertical asymptote. Label this line as x = \(\frac{\pi}{3}\).

Step 3: Drawing the graph

To draw the graph, move \(2\pi\) units to the right from \(\frac{\pi}{3}\) and draw another vertical asymptote. Then, do the same to the left. Now, at the middle between the two adjacent asymptotes, which is the maximum of our function, plot the point (since there is no shift vertically, the maximum is at -3, the value from flipping over x-axis). Following the shape of a cosecant curve, the graph will approach each asymptote but never cross it. Make sure to plot two full periods: one is from \(\frac{\pi}{3} - 2\pi\) to \(\frac{\pi}{3}\) and the second period is from \(\frac{\pi}{3}\) to \(\frac{\pi}{3}+2\pi\).

Step 4: Repeat the pattern

Repeat this process to show at least two full periods (cycles) of the function, including the appropriate reflections and shifts for each cycle. Just replicate the first period to the right and left as many times as needed.

Key Concepts

Transformation of Functions

When working with trigonometric functions, understanding transformations is key to modifying the function's appearance and behavior. Transformations can include translations, reflections, stretches, and compressions. In the given function, \( f(x) = -3 \csc \left( x - \frac{\pi}{3} \right) \), several transformations occur:

• Reflection: The factor of \(-3\) indicates a reflection across the x-axis. This means the graph of the cosecant function will be flipped upside down.
• Vertical Stretch: The coefficient \( -3 \) also scales the function vertically. Instead of rising and falling between included and undefined values, it does so by a factor of 3 times greater than a standard cosecant graph.
• Horizontal Shift: The term \( x - \frac{\pi}{3} \) shifts the entire graph to the right by \( \frac{\pi}{3} \) units. This phase shift affects where each cycle of the graph begins and ends.

Understanding these transformations allows you to accurately draw and interpret trigonometric functions, offering insights into their graphical and analytical behavior.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. In trigonometry, functions like sine, cosine, and cosecant are periodic. The textbook function \( f(x) = -3 \csc\left(x - \frac{\pi}{3}\right) \) inherits this characteristic of periodicity.
The natural period of the cosecant function is \(2\pi\), implying that after each interval of \(2\pi\), the function repeats itself. This means that no matter the transformations applied, the pattern of the cosecant function will recur every \(2\pi\) units.

• Because of the horizontal shift, the period starts at \(\frac{\pi}{3}\) instead of 0.
• The graph cycles through its shape from \(\frac{\pi}{3} - 2\pi\) to \(\frac{\pi}{3}\) and from \(\frac{\pi}{3}\) to \(\frac{\pi}{3} + 2\pi\), continuously replicating.

This defining feature means that once the initial cycle is graphed, subsequent cycles can be drawn by simply repeating the initial period's pattern. Knowing this allows us to anticipate and accurately depict continuous trigonometric behaviors.

Graphing Trigonometric Functions

Graphing trigonometric functions such as the cosecant requires an understanding of certain key points and structures. For the function \( f(x) = -3 \csc \left( x - \frac{\pi}{3} \right) \), the graph reflects the inherent asymptotic nature of the cosecant function.
To successfully graph it, you should:

• Identify intervals where the function is undefined. Due to the phase shift, vertical asymptotes occur at \( x = \frac{\pi}{3}, \frac{\pi}{3} \pm 2\pi, \cdots \)
• Graph the transformation starting at the phase shift of \(\frac{\pi}{3}\), knowing that each asymptote will mark a point where the function shoots towards infinity or negative infinity.
• Plot the maximum or minimum values halfway between each pair of consecutive asymptotes. Due to our function's reflection and stretch, these values will be at \(-3\) or their multiplied equivalents.
• Repeat the cycle at each completed \(2\pi\) interval to show periodicity.

This approach highlights both the structure and pattern the graph follows, giving a visual representation of the equation influenced by its transformations. Understanding how these points align with the asymptotic behavior is crucial for anyone learning to interpret and plot these types of functions.