Rational Function With Asymptotes At Y=3, X=5, And X=-2
In the realm of rational functions, understanding asymptotes is crucial for analyzing their behavior. Asymptotes, the lines that the function approaches but never quite touches, provide valuable insights into the function's limits and overall structure. In this article, we will delve into the process of identifying a rational function that exhibits specific horizontal and vertical asymptotes. Specifically, we aim to determine which rational function among the given options has a horizontal asymptote at y = 3 and vertical asymptotes at x = 5 and x = -2. To achieve this, we will explore the relationship between the function's equation and its asymptotic behavior. Understanding how the degrees of the numerator and denominator influence horizontal asymptotes, and how the roots of the denominator dictate vertical asymptotes, is paramount. This exploration will not only help us solve the given problem but also enhance our comprehension of rational functions in general. By meticulously analyzing each option, we will pinpoint the function that satisfies the specified asymptote conditions, solidifying our grasp of these fundamental concepts.
Understanding Asymptotes of Rational Functions
Before diving into the specific problem, let's solidify our understanding of asymptotes in the context of rational functions. A rational function is essentially a fraction where both the numerator and the denominator are polynomials. The asymptotes of a rational function are lines that the graph of the function approaches as x or y approaches infinity (horizontal asymptotes) or as x approaches a specific value (vertical asymptotes). The location and nature of these asymptotes are directly determined by the structure of the polynomials in the numerator and denominator.
Vertical asymptotes occur at the x-values where the denominator of the rational function equals zero, provided the numerator does not equal zero at the same x-value. These x-values represent points where the function becomes undefined, causing the graph to approach infinity (or negative infinity). For example, if the denominator has a factor of (x - a), then there will be a vertical asymptote at x = a. Conversely, horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. The presence and location of a horizontal asymptote depend on the degrees of the numerator and denominator polynomials.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be a slant or oblique asymptote). Understanding these rules is fundamental to identifying rational functions with specific asymptotic behaviors. In the context of our problem, we need a function with a horizontal asymptote at y = 3, indicating that the degrees of the numerator and denominator must be equal and the ratio of their leading coefficients must be 3. We also need vertical asymptotes at x = 5 and x = -2, implying that the denominator must have factors of (x - 5) and (x + 2).
Analyzing the Given Options
Now, let's methodically analyze the given options to pinpoint the rational function that satisfies our conditions: a horizontal asymptote at y = 3 and vertical asymptotes at x = 5 and x = -2. This involves examining each function's structure, specifically the degrees and coefficients of the polynomials in the numerator and denominator, as well as the roots of the denominator. We will assess each option against the criteria we established in the previous section, focusing on how the function behaves as x approaches infinity (for horizontal asymptotes) and the values of x that make the denominator zero (for vertical asymptotes).
Option A: y = x2 / (x2 - 3x - 10)
First, let's consider Option A: y = x2 / (x2 - 3x - 10). To determine the vertical asymptotes, we need to find the values of x that make the denominator zero. The denominator can be factored as (x - 5)(x + 2). Thus, the vertical asymptotes are at x = 5 and x = -2, which matches our requirement. Now, let's examine the horizontal asymptote. The degree of the numerator is 2, and the degree of the denominator is also 2. Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, which is 1/1 = 1. This does not match our required horizontal asymptote of y = 3. Therefore, Option A is not the correct answer.
Option B: y = 3x2 / (x2 - 3x - 10)
Next, let's analyze Option B: y = 3x2 / (x2 - 3x - 10). The denominator, as before, factors to (x - 5)(x + 2), so the vertical asymptotes are at x = 5 and x = -2. For the horizontal asymptote, the degrees of the numerator and denominator are both 2. The ratio of the leading coefficients is 3/1 = 3. This matches our required horizontal asymptote of y = 3. Therefore, Option B satisfies both the horizontal and vertical asymptote conditions. This strongly suggests that Option B is the correct answer, but we will still examine the remaining options to ensure we have made the correct choice.
Option C: y = x2 / (x2 + 3x - 10)
Now, let's analyze Option C: y = x2 / (x2 + 3x - 10). We first find the vertical asymptotes by setting the denominator equal to zero. Factoring the denominator, we get (x + 5)(x - 2). This means the vertical asymptotes are at x = -5 and x = 2, which do not match our required vertical asymptotes of x = 5 and x = -2. Therefore, Option C is not the correct answer, as it does not fulfill the vertical asymptote condition.
Option D: y = 3x2 / (x2 + 3x - 10)
Finally, let's consider Option D: y = 3x2 / (x2 + 3x - 10). The denominator is the same as in Option C, (x2 + 3x - 10), which factors to (x + 5)(x - 2). As we determined in the analysis of Option C, the vertical asymptotes are at x = -5 and x = 2. These do not match our required vertical asymptotes of x = 5 and x = -2. Thus, Option D is not the correct answer, as it fails to satisfy the vertical asymptote condition.
Conclusion: Identifying the Correct Rational Function
After a thorough analysis of all the options, we can confidently conclude that Option B is the correct answer. Option B, y = 3x2 / (x2 - 3x - 10), is the only rational function that meets both the required conditions: a horizontal asymptote at y = 3 and vertical asymptotes at x = 5 and x = -2. The vertical asymptotes are confirmed by factoring the denominator as (x - 5)(x + 2), while the horizontal asymptote is verified by the ratio of the leading coefficients being 3/1 = 3.
The systematic approach we employed, starting with a clear understanding of asymptotes and then meticulously evaluating each option, highlights the importance of a strong conceptual foundation in mathematics. This problem not only reinforces the understanding of rational functions but also showcases the power of analytical techniques in solving mathematical problems. By breaking down the problem into smaller, manageable steps, we were able to confidently identify the correct answer. This method can be applied to a wide range of mathematical challenges, emphasizing the significance of a structured and thoughtful approach.
In summary, identifying rational functions with specific asymptotic behaviors involves a blend of algebraic manipulation and conceptual understanding. This exercise underscores the vital role that asymptotes play in the graphical representation and analysis of functions. Mastering these concepts provides a solid base for tackling more advanced topics in calculus and mathematical analysis. The ability to discern and interpret asymptotes empowers us to predict the long-term behavior of functions, a skill that has far-reaching implications in various scientific and engineering disciplines.
