Problem 74 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\). $$ y=\pi \sec \theta $$

Step-by-Step Solution

Answer

The graph of \(y=\pi \sec \theta\) from 0 to \(2\pi\) has vertical asymptotes at \(\theta = \pi / 2\) and \(\theta = 3\pi / 2\), passes through the points \((0,\pi)\), \((\pi, -\pi)\), and \((2\pi, \pi)\), and has a maximum and minimum y-value of \(\pi\) and -\(\pi\), respectively.

Step 1: Understand the Function

The secant function, \(\sec \theta\), is defined as the reciprocal of the cosine function, or 1/\(\cos \theta\). This means any values of \(\theta\) for which \(\cos \theta = 0\) are not included in the graph of the secant function, since division by zero is undefined.

Step 2: Determine Excluded Values

The cosine function is equal to zero at \(\theta = \pi / 2\) and \(\theta = 3\pi / 2\). Therefore, these values are excluded from the graph of the secant function. This will result in vertical asymptotes at these values of \(\theta\).

Step 3: Graph the Function

The graph of \(y=\pi \sec \theta\) will look like the graph of \(y= \sec \theta\), but multiplied by \(\pi\). It will have vertical asymptotes at \(\theta = \pi / 2\) and \(\theta = 3\pi / 2\), and will pass through the points \((0,\pi)\), \((\pi, -\pi)\), and \((2\pi, \pi)\). The amplitude of this function is \(\pi\), meaning the maximum and minimum y-values are \(\pi\) and -\(\pi\), respectively.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function, which means \( \sec \theta = \frac{1}{\cos \theta} \). This implies that wherever the cosine function equals zero, the secant function will have problems since division by zero is undefined.

In simpler terms, the secant function has values wherever the cosine function is not zero. Think of it as the upside-down cousin of the cosine function, reflecting areas where cosine takes values close to zero into infinitely large or small values on the secant graph. Understanding this reciprocal relationship can help learners grasp when the secant function behaves nicely and when it runs into trouble due to zero values in the cosine denominator.

Graphing Functions

Graphing the function \( y = \pi \sec \theta \) involves understanding its basic components. The secant function itself already presents a unique curve, but multiplying it by \( \pi \) stretches this curve vertically.

When graphing \( \pi \sec \theta \), important points to note include:

• The key passing points such as \((0,\pi)\), \((\pi, -\pi)\), and \((2\pi, \pi)\).
• The effect of \( \pi \) on the curve which elongates the peaks and valleys.
• The periodic nature of the function, repeating every \( 2\pi \).

This way, students can predict where the high and low points of the graph will fall and how the function behaves through its cycle.

Vertical Asymptotes

Vertical asymptotes in trigonometric functions occur because of undefined points in division, due to the function's reciprocal nature. For the secant function, these asymptotes arise wherever the cosine function equals zero.

Specifically for \( \sec \theta \), this happens at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) over each period from 0 to \( 2\pi \). At these points, the graph shoots up or down, never touching the vertical asymptote line. Recognizing these points allows for accurate graph plotting, viewing these vertical lines as barriers the function cannot cross.

Amplitude

While amplitude usually refers to the height of peaks in a standard sine or cosine graph, it's a bit different for the secant function. The amplitude here refers to the maximum and minimum value that \( y = \pi \sec \theta \) can reach.

Given that the regular secant function does not have a strict maximum or minimum (since they shoot off to infinity at vertical asymptotes), the inclusion of \( \pi \) helps redefine this aspect. Here, \( \pi \) acts as a multiplier, indicating that as the absolute maximum or minimum, the function cycles between \( \pi \) and \( -\pi \).

• This means the graph will rise up to \( \pi \) and drop to \( -\pi \), emphasizing amplitude as the vertical stretch factor.

Understanding this allows us to better visualize how the function fits on a graph in both height and depth.