Problem 51 - Calculus: Early Transcendentals

Question

Slant (oblique) asymptotes Complete the following steps for the given functions. a. Find the slant asymptote of \(f\) b. Find the vertical asymptotes of \(f\) (if any). c. Graph \(f\) and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. $$f(x)=\frac{x^{2}-3}{x+6}$$

Step-by-Step Solution

Answer

Answer: The function has a slant asymptote at \(y = x\) and a vertical asymptote at \(x = -6\). As x approaches -6, the function's graph approaches the vertical asymptote but never crosses it. As x increases or decreases without bound, the graph approaches but never touches the slant asymptote at \(y = x\).

Step 1: Find the slant asymptote of f

The slant asymptote of a rational function occurs when the degree of the numerator is one greater than the degree of the denominator. For the given function \(f(x)=\frac{x^{2}-3}{x+6}\), the degree of the numerator is 2, and the degree of the denominator is 1, so there is a slant asymptote. To find the equation of the slant asymptote, we perform long division or synthetic division: $$ x^{**~2} -3 = (x+6)(x) + (-6x -3)$$ Since the remainder (-6x - 3) has degree less than the denominator, we can discard it when finding the equation for the slant asymptote. The equation for the slant asymptote is \(y = x\).

Step 2: Find the vertical asymptotes of f

To find the vertical asymptotes of a rational function, we need to find the values of x for which the denominator is equal to zero. In the given function, the denominator is \(x+6\): \(x + 6 = 0\) Solving for x, we get \(x = -6\) So, there is a vertical asymptote at \(x = -6\).

Step 3: Graph f and its asymptotes

Using a graphing utility, plot the function \(f(x)=\frac{x^{2}-3}{x+6}\) and its asymptotes (\(y=x\) and \(x=-6\)). The computer-generated graph may show minor errors. Correct these by hand by making sure the graph approaches the asymptotes but never crosses them. Keep in mind that the function will approach each asymptote as x approaches the corresponding x or y value. Your final graph should clearly display the function, slant asymptote, and vertical asymptote.

Key Concepts

Slant Asymptote

A slant asymptote, also known as an oblique asymptote, occurs in rational functions when the degree of the numerator is exactly one more than the degree of the denominator. For instance, in the function given, \( f(x) = \frac{x^{2} - 3}{x + 6} \), the numerator has a degree of 2, which is one more than the degree of the denominator (1). When this condition is met, the graph of the function will have a slant asymptote rather than a horizontal one. To find the equation of a slant asymptote, you perform polynomial long division or synthetic division. This process helps separate the function into two parts: a polynomial and a remainder, with the polynomial acting as the slant asymptote and the remainder having a degree less than the denominator. In our example, after dividing \( x^{2} - 3 \) by \( x + 6 \), the result is \( x \) with a remainder that is discarded for asymptotic purposes. Thus, the equation for the slant asymptote is \( y = x \). When graphing, this line shows the end behavior of the function, illustrating how the graph stretches out to infinity along a path that parallels the slant. This behavior becomes evident as \( x \) grows larger in the positive or negative direction.

Vertical Asymptote

Vertical asymptotes often appear in rational functions and are a result of division by zero in the denominator. They represent edges where the function's value grows indefinitely and the graph approaches vertical lines. For the function \( f(x) = \frac{x^2 - 3}{x + 6} \), finding the vertical asymptotes involves identifying the values of \( x \) that make the denominator undefined. To locate these vertical asymptotes, set the denominator equal to zero:

• \( x + 6 = 0 \)

Solve for \( x \), resulting in \( x = -6 \). Thus, the vertical asymptote is at \( x = -6 \). On the graph, this means that as \( x \) approaches -6 from either direction, the value of \( f(x) \) will shoot up towards positive or negative infinity. Understanding these asymptotes is crucial because they form boundaries for the behavior of rational functions. The function can never cross these vertical lines as they represent points of undefined value. However, the graph will get arbitrarily close to these lines.

Graphing Utility

A graphing utility is a powerful tool that assists in visualizing mathematical functions and their asymptotes. It allows students to quickly plot equations and observe graphical behavior, which is particularly beneficial when dealing with complex functions like \( f(x) = \frac{x^2 - 3}{x + 6} \).To graph this function using a graphing utility, you should input the equation and add the asymptotes:

• The slant asymptote \( y = x \)
• The vertical asymptote \( x = -6 \)

Graphing utilities can sometimes produce graphs with minor inaccuracies due to pixelation or rounding errors. Therefore, it's important to critically evaluate the output and make necessary adjustments by hand. Asymptotes are crucial because they depict the long-range behavior of functions. While the function can approach these lines infinitely close, it will never actually intersect them. Adjust your sketch to ensure the function approaches asymptotes accurately without crossing them. Seeing how the function behaves near the asymptotes provides significant insight, helping to predict behavior in calculus and determine continuity, limits, and bounds of functions.