Problem 16 - Algebra and Trigonometry

Question

7–52 Find the period and graph the function. $$y=-3 \sec x$$

Step-by-Step Solution

Answer

The period of \( y = -3 \sec x \) is \( 2\pi \).

Step 1: Identify the Basic Function

The function given is \( y = -3 \sec x \). The function \( \sec x \) is the secant function, which is the reciprocal of the cosine function, i.e., \( \sec x = \frac{1}{\cos x} \). The basic function \( \sec x \) has a period of \( 2\pi \).

Step 2: Determine the Period

The period of the function \( y = -3 \sec x \) is determined by the coefficient of \( x \) inside the secant function. Since there is no coefficient other than 1 inside the function, the period remains \( 2\pi \).

Step 3: Apply the Amplitude and Reflection

The coefficient \(-3\) affects the function by stretching it vertically by a factor of 3 and reflecting it across the x-axis. This changes the 'height' of the secant's peaks and valleys but does not alter the period.

Step 4: Graph the Function

To graph \( y = -3 \sec x \), start by plotting the cosine graph for reference. Draw vertical asymptotes at the points where \( \cos x = 0 \), which are at odd multiples of \( \frac{\pi}{2} \). Then, sketch the secant graph which will have upward and downward curves stretching towards positive and negative infinity at these asymptotes. Because of the \(-3\), these curves reflect downward and stretch vertically.

Step 5: Conclusion

The function \( y = -3 \sec x \) has a period of \( 2\pi \). The graph can be drawn using the reflection and stretching rule applied to the secant graph, keeping all vertical asymptotes and points consistent with the transformed cosine function.

Key Concepts

Secant Function

The secant function, represented as \( \sec x \), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function; therefore, \( \sec x = \frac{1}{\cos x} \).
The secant function is important because it helps us understand the behavior of another similarly essential trigonometric function, the tangent, through its reciprocal relationship. With this function, the understanding of trigonometric relationships can deepen, making it possible to solve more complex problems.
A few key points to remember about the secant function:

• It has vertical asymptotes at points where the cosine of \( x \) is equal to zero, specifically at \( x = \frac{(2n+1)\pi}{2} \), where \( n \) is an integer.
• Because of these asymptotes, the secant function features distinctive upward and downward curves.
• The basic function, \( \sec x \), is not defined at these points, which leads to the undefined nature of secant for input values that zero-out cosine.

These characteristics make the secant function quite unique among trigonometric functions, creating its characteristic "waves" above and below the x-axis, respectively.

Periodic Function

A periodic function is remarkable because it repeats its values at regular intervals over its domain. In the world of trigonometry, this property is crucial as it helps predict and model cyclic phenomena such as waves and vibrations.
For the function \( y = -3 \sec x \), we observe that it retains the periodic nature of the basic secant function, \( \sec x \). The period is the length of the smallest interval after which the function repeats itself. For \( \sec x \), this period is \( 2\pi \).
Even with the transformation of the secant function, including amplitude changes and reflection (as seen with \( -3 \sec x \)), the period remains unaffected, at \( 2\pi \). This is important because it allows us to anticipate the regularity in the repeating intervals of the graph.
To summarize the properties of periodic functions:

• The graph of the function repeats at regular intervals.
• Transformations such as vertical stretching or reflections do not change the period of the function.
• For the secant and cosine functions, the repeated interval remains \( 2\pi \) in the absence of additional changes to the variable \( x \) inside the function.

Understanding periodic functions teaches us about predictability in complex patterns, a fundamental concept across advanced mathematics and physics.

Graphing Transformations

Graphing transformations involve altering a function's graph in various ways, such as stretching, shifting, or reflecting. They play a critical role in understanding how changes in a trigonometric function's equation affect its overall shape.
When graphing the function \( y = -3 \sec x \), we apply specific transformations to the graph of \( \sec x \).
**Amplitude and Vertical Scaling**
The coefficient \(-3\) in front of \( \sec x \) scales the graph vertically. This means that every point on the basic secant graph is stretched away from the x-axis by a factor of 3. This effectively influences the 'height' of the graphed curves.
**Reflection**
Additionally, the negative sign before the 3 indicates reflection across the x-axis. The upward curves of the standard secant are flipped downward, and vice versa. This flipping does not impact the period of the function but rather inverts the direction of peaks and troughs.
To effectively graph \( y = -3 \sec x \):

• First, visualize or draw the parent function, \( \sec x \).
• Identify and keep the period \( 2\pi \) unchanged.
• Apply the vertical stretch and reflection according to the \(-3\) factor.
• Mark vertical asymptotes at \( x = \frac{(2n+1)\pi}{2} \).

Through these transformations, you will see how altering specific coefficients and constants in a function's equation reveals different graphical presentations while preserving essential properties like the period.