If the graph of has an oblique asymptote at y = 3x + k, what is the value of k?

In the realm of mathematics, asymptotes serve as critical concepts in understanding the behavior of functions, particularly in the context of limits and infinity. While horizontal and vertical asymptotes are commonly encountered, the presence of an oblique asymptote introduces intriguing dynamics to the function’s behavior. This discussion delves into the intricacies of oblique asymptotes, focusing on their manifestation in the graph of a function and elucidating the determination of the constant $k$ when the graph has an oblique asymptote at $y = 3 x + k$ .

Unraveling the Essence of Asymptotes

Before delving into the specifics of oblique asymptotes, it’s paramount to comprehend the fundamental notion of asymptotes. An asymptote refers to a line that a curve approaches but never quite touches, even as the curve extends infinitely in one or both directions. These lines embody the behavior of the function as it approaches infinity or approaches a certain point in its domain.

Horizontal and Vertical Asymptotes

Horizontal asymptotes, typically denoted as $y = c$ , represent the behavior of a function as $x$ tends towards positive or negative infinity. Conversely, vertical asymptotes, often denoted as $x = a$ , delineate values where the function’s magnitude approaches infinity as $x$ approaches a particular value $a$ . These asymptotes play pivotal roles in understanding the long-term behavior of functions across different domains.

Unveiling Oblique Asymptotes

Unlike horizontal and vertical asymptotes, oblique asymptotes, also known as slant or diagonal asymptotes, manifest as slanted lines on the Cartesian plane. They emerge when the degree of the polynomial in the numerator is one greater than the degree of the polynomial in the denominator of a rational function.

Consider a rational function $f (x) = Q ( x ) P ( x )$ , where both $P (x)$ and $Q (x)$ are polynomials, and the degree of $P (x)$ is exactly one more than the degree of $Q (x)$ . In such scenarios, as $x$ tends toward positive or negative infinity, the function’s graph approximates a straight line with a slope equal to the leading coefficient of $P (x)$ divided by the leading coefficient of $Q (x)$ .

Determining the Value of $k$

Now, let’s delve into the crux of the matter: determining the value of the constant $k$ when the graph of a function exhibits an oblique asymptote at $y = 3 x + k$ . This entails understanding the behavior of the function as it approaches infinity and discerning the interplay between its numerator and denominator.

Consider a rational function $f (x) = Q ( x ) P ( x )$ , where the degree of $P (x)$ is one more than the degree of $Q (x)$ . To find the oblique asymptote, one must perform polynomial long division on $P (x)$ by $Q (x)$ .

Polynomial Long Division

Polynomial long division enables the expression of a rational function as a quotient plus a remainder. Through this process, the polynomial is divided by another polynomial, akin to traditional long division.

Let’s illustrate this through an example:

Suppose we have the rational function $f (x) = x - 1 3 x ^{2} + 2 x + 5$ . To determine the oblique asymptote, we perform polynomial long division as follows:

3x + 5
___________
x – 1 | 3x^2 + 2x + 5
– (3x^2 – 3x)
___________
5x + 5
– (5x – 5)
___________
10

Hence, the quotient of the division is $3 x + 5$ with a remainder of $10$ . Therefore, the oblique asymptote is $y = 3 x + 5$ , where $k = 5$ .

The Role of Remainders

The remainder resulting from the polynomial long division is instrumental in understanding the behavior of the function around the oblique asymptote. As $x$ approaches infinity, the remainder becomes increasingly negligible compared to the quotient, thereby affirming the linearity of the oblique asymptote.

Visual Representation

To solidify our understanding, let’s visualize the concept of oblique asymptotes through graphical representation. Consider the rational function $f (x) = x - 1 3 x ^{2} + 2 x + 5$ .

By plotting the function on a graphing utility, one can observe the asymptotic behavior as $x$ tends towards positive and negative infinity. The oblique asymptote, represented by the line $y = 3 x + 5$ , serves as a guiding principle for the function’s behavior in the long run.

Conclusion

In summary, oblique asymptotes embody fascinating facets of function behavior, particularly in the context of rational functions with a higher-degree numerator than denominator. By understanding the principles of polynomial long division and the interplay between quotient and remainder, one can determine the existence and characteristics of oblique asymptotes, including the determination of the constant $k$ in $y = m x + k$ .

Through thorough analysis and graphical representation, mathematicians and enthusiasts alike can unravel the intricacies of oblique asymptotes, thereby enriching their understanding of function behavior and the underlying principles of calculus and algebra. As such, the exploration of oblique asymptotes underscores the beauty and elegance inherent in mathematical abstraction and analysis.