Problem 43 - College Algebra with Modeling and Visualization

Question

Graph \(f\) and identify any asymptotes. $$ f(x)=-\frac{1}{2 x} $$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = 0\); horizontal asymptote at \(y = 0\).

Step 1: Identify the Function Type

The function given is \( f(x) = -\frac{1}{2x} \). This is a rational function, where the numerator is \(-1\) and the denominator is \(2x\).

Step 2: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function is equal to zero. This happens when \(2x = 0\). Solving this, we find \(x = 0\), so there is a vertical asymptote at \(x = 0\).

Step 3: Determine Horizontal Asymptotes

For rational functions, horizontal asymptotes are found by comparing the degrees of the numerator and the denominator. Here, the degree of the numerator is 0, and the degree of the denominator is 1. Since the degree of the denominator is greater, the horizontal asymptote is at \(y = 0\).

Step 4: Graph the Function

To graph \(f(x) = -\frac{1}{2x}\), note the following: for values of \(x > 0\), \(f(x)\) will be negative and approach the vertical asymptote from below; for values of \(x < 0\), \(f(x)\) will also be negative and again approach the vertical asymptote. The graph is hyperbolic, opening downwards in both sections, with \(x = 0\) as a vertical asymptote and \(y = 0\) as a horizontal asymptote.

Key Concepts

Graphing Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. These functions often create interesting graphs that include asymptotic behavior and discontinuities. To graph a rational function like \[f(x) = -\frac{1}{2x}\], you first need to get a sense of its general shape by considering its key features: intercepts, asymptotes, and behavior as \(x\) approaches infinity or negative infinity.
Here are some steps to keep in mind when graphing such functions:

• Identify intercepts by setting \(f(x)\) to zero or solving for \(x\).
• Pinpoint asymptotes; these lines will indicate where the function heads as it stretches to infinity.
• Analyze the function's behavior in different intervals, especially around discontinuities like asymptotes.

This approach helps visualize how components of a rational function interact across its domain. Always begin with asymptotes, since they greatly influence the graph's shape.

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. For the rational function \(f(x) = -\frac{1}{2x}\), locating vertical asymptotes involves finding where the function becomes undefined.
A function becomes undefined when its denominator equals zero. By setting the denominator of \(2x\) to zero, we solve for \(x\), yielding \(x = 0\). This provides us the vertical asymptote. In the context of graphing, as \(x\) approaches zero, \(f(x)\) heads towards positive or negative infinity, indicating that the graph will rise or fall sharply near \(x = 0\).
It's crucial to remember that vertical asymptotes heavily impact the domain of rational functions. These points of non-existence indicate behavior where the function drastically changes, often affecting its overall shape and continuity.

Horizontal Asymptotes

Horizontal asymptotes represent the value that a function approaches as its input grows very large or very small. In \(f(x) = -\frac{1}{2x}\), the degrees of the numerator and the denominator help us find the horizontal asymptote.
Here, the numerator has a degree of 0, while the denominator's degree is 1. Since the degree of the denominator is higher, the horizontal asymptote sets at \(y = 0\).
This implies that as \(x\) becomes infinitely large or infinitely small, \(f(x)\) approaches zero but never truly reaches it. Hence, in this case, the curve will get closer and closer to the x-axis (\(y = 0\)), effectively making the x-axis itself the horizontal asymptote.