Problem 19 - Precalculus: A Concise Course

Question

Sketch the graph of the function. Include two full periods. $$ y=-\frac{1}{2} \sec x $$

Step-by-Step Solution

Answer

The plot of the function \( -\frac{1}{2} \sec x \) is a vertically shrunk and reflected graph of the standard secant function with a period of \( 2\pi \). It is infinitely increasing and decreasing wave which has vertical asymptotes and never crosses the x-axis.

Step 1: Understand the basic graph of sec x

First, try to understand the basic secant function. By definition, \( \sec x = \frac{1}{\cos x} \). The secant function graph is a wave that is infinitely increasing and decreasing, and it never crosses the x-axis. Also, there are vertical asymptotes where \( \cos x = 0 \). The period of \( \sec x \) function is \( 2\pi \).

Step 2: Derive transformation from the given function.

Given function is \( -\frac{1}{2} \sec x \). The negative sign makes the original secant graph reflect over the x-axis, and the prefactor \( \frac{1}{2} \) shrinks the graph vertically by half, making the values of the function half of the original values. The period of function remains \( 2\pi \) as there is no change in the x value.

Step 3: Plot the graph

Having derived the transformation from the given function in Step 2, now plot the graph using these transformations. First mark the points for a normal secant function, then reflect it over the x-axis and shrink its value by half. Also, highlight vertical asymptotes and mark the period \( 2\pi \). Repeat this process to get two complete periods.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are foundational in mathematics, especially when it comes to waves, oscillations, and circles. Among these functions is the secant (sec), which is related to the cosine function. The secant of an angle in a right-angled triangle is defined as the ratio of the hypotenuse to the adjacent side. In terms of a unit circle, or a circle with a radius of one, secant is the reciprocal of the cosine function, meaning it is expressed as \( \sec x = \frac{1}{\cos x} \).

The graph of the secant function is characterized by its repeating wave-like pattern, known as a periodic function, which continues indefinitely in both the positive and negative directions on the y-axis. Unlike the sine and cosine functions, the secant function does not oscillate between -1 and 1 but rather extends to infinity whenever the cosine function equals zero, creating what is known as asymptotes.

Exploring Graph Transformations

Graph transformations allow us to modify the basic secant graph into more complex forms. The general form of the secant function is \( y=a\sec(bx+c)+d \), where each parameter 'a', 'b', 'c', and 'd' transforms the graph in a different way. For the function \( y=-\frac{1}{2} \sec x \), there are two key transformations:

• The coefficient \( -\frac{1}{2} \) indicates a vertical compression by a factor of 1/2 as well as a reflection across the x-axis due to the negative sign.
• Because there are no 'b' or 'c' values altering 'x', and no 'd' affecting the vertical position, the period and horizontal position remain unchanged.

During graphing, you always start with these transformations on the parent graph. This means we plot the basic secant graph first then apply the vertical compression and reflection. Understand that these transformations do not impact the locations of asymptotes since they arise where the cosine function equals zero, regardless of stretching or flipping the graph.

Identifying Asymptotes in Trigonometric Graphs

Asymptotes are essential in understanding trigonometric graphs. They are lines where the function approaches but never reaches, indicating points of discontinuity in the graph. For the secant function, vertical asymptotes occur at the same x-values where the cosine function is zero, because secant is the reciprocal of cosine.

For the standard secant function \( \sec x \), the asymptotes occur at \( x = \frac{1}{2}\pi + n\pi \) where 'n' is any integer. When you introduce transformations such as reflection or stretching, the positions of the asymptotes remain the same, since these transformations do not affect the x-values where \( \cos x = 0 \).

Recognizing the position of these asymptotes is crucial when sketching the graph of a transformed secant function, as they define the bounds within which each segment of the wave-like graph exists, guiding you towards an accurate depiction of the function's behavior.