Problem 87 - College Algebra

Question

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}+1}{x^{2}+2 x}$$

Step-by-Step Solution

Answer

The slant asymptote of the function \(f(x)=\frac{x^3 + 1}{x^2 + 2x}\) is \(x-2\)

Step 1: Identify the function and its degree

The given function is \(f(x)=\frac{x^3 + 1}{x^2 + 2x}\). Generally, if the degree of the numerator is exactly one more than the degree of the denominator in a rational function, the function has a slant (or oblique) asymptote. In this case, the degree of the numerator is 3 and the degree of the denominator is 2, so a slant asymptote does exist.

Step 2: Division of Polynomials

The next step involves utilizing polynomial division to divide the numerator by the denominator. This division is best accomplished by using the long division method for polynomials, which looks like this: \(\frac{x^3 + 1}{x^2 + 2x}\). Performing the division, we get \(x-2\) with a remainder of \(5x+1\). We disregard the remainder when finding the slant asymptote.

Step 3: Identify the Slant Asymptote

The quotient we derived in Step 2 is the equation of the slant asymptote. Hence, the slant asymptote of the function \(f(x)=\frac{x^3 + 1}{x^2 + 2x}\) is \(x-2\).

Key Concepts

Understanding Rational Functions

A rational function is expressed as a fraction where both the numerator and the denominator are polynomials. The basic form is \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomial expressions and \( Q(x) \) is not equal to zero. One of the key features of rational functions is the presence of asymptotes, which are invisible lines that the graph of the function approaches but never touches. These include horizontal, vertical, and in cases where the numerator's degree is one higher than that of the denominator, slant or oblique asymptotes.

To determine a slant asymptote, we look at the degrees of the numerator and denominator. If the degree of the numerator is exactly one more than the degree of the denominator, a slant asymptote is typically present. In the example of \( f(x)=\frac{x^3 + 1}{x^2 + 2x} \) from the original exercise, this condition is met, as the degree of the numerator is 3 and the degree of the denominator is 2.

Polynomial Long Division

When dealing with polynomial long division, the goal is to divide the numerator by the denominator as you would with regular numbers, but using the rules for polynomials. This process helps to simplify rational functions and find slant asymptotes. Let's walk through the steps briefly:

• Set up the division, placing the numerator inside the division symbol and the denominator outside, just like a typical long division problem.
• Divide the first term of the numerator by the first term of the denominator to find the first term of the quotient.
• Multiply the entire denominator by this term and subtract the result from the numerator.
• Bring down the next term of the numerator and repeat the process until all terms have been accounted for.

The quotient from this division (ignoring the remainder) is the equation for the slant asymptote if one exists. In our example, dividing \( x^3 + 1 \) by \( x^2 + 2x \) via polynomial long division yields a quotient of \( x-2 \) with a remainder of \( 5x+1 \)—and it is this \( x-2 \) that represents the slant asymptote.

Graphing Rational Functions

The process of graphing rational functions is a bit more complex due to their unique properties, such as asymptotes and possible discontinuities. Here's a simplified guide to graphing these functions:

• Identify any vertical asymptotes by finding where the denominator equals zero. The function will approach but not cross these lines.
• Determine if a horizontal or slant asymptote exists based on the degrees of the numerator and denominator, as we discussed previously.
• Calculate the x- and y-intercepts of the function, which occur where \( f(x) = 0 \) and \( f(0) \) respectively.
• Plot a few additional points to get an accurate shape of the graph.
• Draw the graph, keeping in mind the behavior near the asymptotes and the points plotted.

It's important to sketch the asymptotes onto your graph first as they guide the overall shape of the curve. Using our previous example, the graph of \( f(x) \) would show a curve approaching the line \( y=x-2 \) as \( x \) increases or decreases without ever touching it.