Horizontal Asymptotes And Intersections Of Q(x) = (3x - 2) / (3x^2 - 4x - 5)
Introduction
In the realm of mathematical functions, rational functions hold a significant place. They are defined as the ratio of two polynomials, and their graphs often exhibit interesting behaviors, including asymptotes and intersections with these asymptotes. In this article, we will delve into the analysis of a specific rational function, q(x) = (3x - 2) / (3x^2 - 4x - 5), focusing on identifying its horizontal asymptotes and determining any points where the graph of the function crosses these asymptotes. Understanding these characteristics provides valuable insights into the overall behavior and graphical representation of the function.
(a) Identifying Horizontal Asymptotes
Horizontal asymptotes are horizontal lines that the graph of a function approaches as x tends towards positive or negative infinity. They represent the long-term behavior of the function. To determine the horizontal asymptotes of a rational function, we compare the degrees of the polynomials in the numerator and denominator.
In our case, the function is q(x) = (3x - 2) / (3x^2 - 4x - 5). The numerator is a polynomial of degree 1 (the highest power of x is 1), and the denominator is a polynomial of degree 2 (the highest power of x is 2). When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0. This is because as x becomes very large (either positive or negative), the denominator grows much faster than the numerator, causing the fraction to approach zero.
To further illustrate this, we can consider the limits as x approaches infinity and negative infinity:
- lim (x→∞) q(x) = lim (x→∞) (3x - 2) / (3x^2 - 4x - 5) = 0
- lim (x→-∞) q(x) = lim (x→-∞) (3x - 2) / (3x^2 - 4x - 5) = 0
Both limits confirm that as x approaches infinity or negative infinity, the function q(x) approaches 0. Therefore, the horizontal asymptote of the function is the line y = 0, which is the x-axis.
Understanding the concept of horizontal asymptotes is crucial for sketching the graph of a rational function. It provides a visual guide for the function's behavior as x moves towards extreme values. In this specific case, the presence of the horizontal asymptote y = 0 tells us that the graph of q(x) will get closer and closer to the x-axis as we move further away from the origin along the x-axis.
Furthermore, the horizontal asymptote can also provide insights into the function's overall shape and behavior. For instance, if the graph crosses the horizontal asymptote, it indicates that the function's value can be zero for some finite value of x. This leads us to the next part of our analysis, where we investigate whether the graph of q(x) intersects its horizontal asymptote.
(b) Determining Intersection Points with the Horizontal Asymptote
Now that we have identified the horizontal asymptote as y = 0, we need to determine if the graph of the function q(x) = (3x - 2) / (3x^2 - 4x - 5) intersects this asymptote. To find the points of intersection, we set the function equal to the value of the horizontal asymptote and solve for x.
In this case, we set q(x) = 0:
(3x - 2) / (3x^2 - 4x - 5) = 0
A fraction is equal to zero if and only if its numerator is equal to zero, and its denominator is non-zero. Therefore, we need to solve the equation:
3x - 2 = 0
Adding 2 to both sides and then dividing by 3, we get:
x = 2/3
Now, we need to check if the denominator is non-zero at x = 2/3:
3(2/3)^2 - 4(2/3) - 5 = 3(4/9) - 8/3 - 5 = 4/3 - 8/3 - 15/3 = -19/3
Since the denominator is not zero at x = 2/3, this value of x corresponds to a point where the graph intersects the horizontal asymptote. To find the y-coordinate of the intersection point, we substitute x = 2/3 into the function q(x). However, since we already set q(x) = 0 to find the x-coordinate, we know that the y-coordinate is 0.
Therefore, the graph of the function q(x) intersects its horizontal asymptote at the point (2/3, 0). This means that the function's value is zero when x = 2/3, and the graph crosses the x-axis at this point.
Finding the points of intersection with the horizontal asymptote provides additional information about the behavior of the function. It tells us where the function's value is equal to the limiting value as x approaches infinity or negative infinity. In this case, the intersection point (2/3, 0) indicates that the function changes sign at x = 2/3, crossing from negative values to positive values (or vice versa) as x increases.
In summary, by identifying the horizontal asymptote and finding the intersection point, we have gained a deeper understanding of the function's behavior. We know that the graph approaches the x-axis as x moves towards infinity or negative infinity, and we know that it crosses the x-axis at the point (2/3, 0).
Conclusion
In this article, we have thoroughly analyzed the rational function q(x) = (3x - 2) / (3x^2 - 4x - 5). We successfully identified the horizontal asymptote as y = 0 by comparing the degrees of the numerator and denominator polynomials. We also determined that the graph of the function intersects its horizontal asymptote at the point (2/3, 0) by setting the function equal to zero and solving for x.
This analysis demonstrates the importance of understanding the concepts of horizontal asymptotes and intersections in the context of rational functions. These characteristics provide crucial information about the function's long-term behavior, its graphical representation, and its overall properties. By identifying these features, we can gain a deeper understanding of the function and its applications in various mathematical and real-world contexts.
The process of finding horizontal asymptotes and intersection points is a fundamental technique in the analysis of rational functions. It allows us to visualize the graph of the function, predict its behavior, and solve related problems. The knowledge gained from this analysis can be applied to various fields, including physics, engineering, and economics, where rational functions are used to model real-world phenomena.
In conclusion, the analysis of the rational function q(x) = (3x - 2) / (3x^2 - 4x - 5) has provided a valuable exercise in understanding horizontal asymptotes and their intersections. By applying the principles and techniques discussed in this article, we can confidently analyze other rational functions and gain insights into their behavior and properties.
