Analysis Of The Rational Function Y = (6x + 18) / (3x - 18)

Rational functions, represented in the form of a ratio of two polynomials, are a fascinating area of mathematics. Delving into their properties, such as asymptotes, intercepts, and behavior, provides a robust understanding of their graphical representation and practical applications. A key example is the function $y = \frac{6x + 18}{3x - 18}$, which we will analyze in detail. Our exploration will encompass identifying key features, simplifying the function, and examining its graphical behavior. This meticulous analysis aims to provide a comprehensive understanding of how to handle such functions and extract meaningful insights from them. In this context, we will first focus on simplifying the given rational function to a form that is easier to interpret and work with. By identifying common factors and canceling them out, we reduce the complexity of the function, making it more amenable to further analysis. This step is critical for accurately determining the function's asymptotes, intercepts, and overall behavior. Additionally, simplifying the function helps in recognizing any potential discontinuities, which are essential for a complete understanding of the function's domain and range. Through this structured approach, we pave the way for a detailed graphical analysis and a thorough comprehension of the function's characteristics. Rational functions appear frequently in various fields, including physics, engineering, and economics, making their study not only academically valuable but also practically relevant.

Simplifying the Function

To begin our analysis, let's simplify the rational function $y = \frac{6x + 18}{3x - 18}$. Simplifying rational functions involves factoring out common factors from both the numerator and the denominator, and then canceling these common factors to reduce the function to its simplest form. This process often makes it easier to identify key features such as asymptotes and intercepts. In our given function, we can factor out a 6 from the numerator and a 3 from the denominator. This gives us $y = \frac{6(x + 3)}{3(x - 6)}$. Now, we can further simplify by dividing both the numerator and the denominator by their common factor, which is 3. This simplifies the function to $y = \frac{2(x + 3)}{x - 6}$. This simplified form is much easier to work with, and it provides direct insights into the function's behavior. For instance, the denominator $x - 6$ immediately tells us that there is a vertical asymptote at $x = 6$, as the function will be undefined at this point. Similarly, the numerator $2(x + 3)$ helps us find the x-intercept, which occurs when the numerator equals zero. These initial simplifications are crucial for setting the stage for further analysis, such as graphing the function and understanding its transformations. By breaking down the function into its simplest form, we not only make calculations easier but also enhance our understanding of the underlying mathematical structure. This step is fundamental in the study of rational functions and their applications in various mathematical and real-world contexts.

Identifying Asymptotes

Asymptotes play a crucial role in understanding the behavior of rational functions. These are lines that the function approaches but never quite reaches. There are primarily three types of asymptotes: vertical, horizontal, and oblique (or slant). For the function $y = \frac{2(x + 3)}{x - 6}$, we can identify the vertical asymptote by finding the values of $x$ for which the denominator is zero. In this case, the denominator is $x - 6$, which equals zero when $x = 6$. Thus, we have a vertical asymptote at $x = 6$. This means that as $x$ approaches 6 from either side, the value of $y$ will approach infinity or negative infinity. To find the horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator. Here, both the numerator and the denominator are linear (degree 1). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1, so the horizontal asymptote is $y = \frac{2}{1} = 2$. This indicates that as $x$ approaches positive or negative infinity, the function approaches the horizontal line $y = 2$. Finally, an oblique asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator. In our case, the degrees are equal, so there is no oblique asymptote. Identifying these asymptotes provides a framework for sketching the graph of the function. The asymptotes act as guides, helping us understand the overall shape and behavior of the function. This analysis is a cornerstone in the study of rational functions, offering insights into their graphical representation and their role in various applications.

Determining Intercepts

Intercepts are the points where the function's graph intersects the coordinate axes. Identifying these points is essential for accurately sketching the graph of a function and understanding its behavior near the axes. To find the y-intercept, we set $x = 0$ in the function $y = \frac{2(x + 3)}{x - 6}$. Substituting $x = 0$, we get $y = \frac{2(0 + 3)}{0 - 6} = \frac{6}{-6} = -1$. Thus, the y-intercept is at the point (0, -1). This is the point where the graph of the function crosses the y-axis. To find the x-intercept, we set $y = 0$ and solve for $x$. This occurs when the numerator of the rational function is zero, provided the denominator is not also zero at the same value of $x$. Setting the numerator $2(x + 3)$ equal to zero, we have $2(x + 3) = 0$, which implies $x + 3 = 0$, and thus $x = -3$. So, the x-intercept is at the point (-3, 0). This is where the graph of the function crosses the x-axis. Knowing both the x and y intercepts gives us valuable anchor points for sketching the graph. The intercepts, along with the asymptotes, provide a comprehensive framework for visualizing the function's behavior. This information is crucial for applications of rational functions in various fields, such as physics, engineering, and economics, where understanding the intersection points with the axes can provide meaningful insights.

Analyzing the Graph

To analyze the graph of the rational function $y = \frac{2(x + 3)}{x - 6}$, we integrate our understanding of asymptotes and intercepts to sketch the curve. We have already identified a vertical asymptote at $x = 6$, a horizontal asymptote at $y = 2$, a y-intercept at (0, -1), and an x-intercept at (-3, 0). These key features help us piece together the overall shape of the graph. The vertical asymptote at $x = 6$ divides the graph into two regions: one to the left of $x = 6$ and one to the right. As $x$ approaches 6 from the left, the function tends toward negative infinity, and as $x$ approaches 6 from the right, the function tends toward positive infinity. The horizontal asymptote at $y = 2$ indicates that the function approaches this value as $x$ goes to positive or negative infinity. The intercepts provide specific points where the graph crosses the axes, anchoring the curve in these regions. With this information, we can sketch the graph, noting that it will approach the asymptotes without ever crossing them (except for the horizontal asymptote, which can be crossed at some points). The graph will pass through the intercepts (0, -1) and (-3, 0), giving us a clear idea of its behavior in these areas. Analyzing the graph further, we can also discuss its domain and range. The domain consists of all real numbers except $x = 6$ (due to the vertical asymptote), and the range consists of all real numbers except $y = 2$ (due to the horizontal asymptote). This comprehensive graphical analysis is a powerful tool for visualizing and understanding the properties of rational functions. It not only enhances our mathematical intuition but also aids in applying these functions to real-world scenarios.

Domain and Range

The domain and range are fundamental properties that describe the set of possible input and output values for a function. For the rational function $y = \frac{2(x + 3)}{x - 6}$, determining the domain and range involves considering where the function is defined and what values it can take. The domain of a rational function is the set of all real numbers except for the values that make the denominator zero. In this case, the denominator is $x - 6$, which is zero when $x = 6$. Therefore, the domain of the function is all real numbers except $x = 6$. This can be expressed in interval notation as $(-\infty, 6) \cup (6, \infty)$. The range of a rational function is the set of all possible output values (y-values). To find the range, we consider the horizontal asymptote and any potential gaps or restrictions in the function's output. We have already identified a horizontal asymptote at $y = 2$. This means that as $x$ approaches positive or negative infinity, the function approaches 2 but never quite reaches it. To determine if $y = 2$ is actually excluded from the range, we can try to solve the equation $2 = \frac{2(x + 3)}{x - 6}$ for $x$. If there is no solution, then $y = 2$ is not in the range. Solving this equation, we have $2(x - 6) = 2(x + 3)$, which simplifies to $2x - 12 = 2x + 6$. This equation has no solution, indicating that $y = 2$ is indeed excluded from the range. Thus, the range of the function is all real numbers except $y = 2$, which can be expressed in interval notation as $(-\infty, 2) \cup (2, \infty)$. Understanding the domain and range provides a complete picture of the function's behavior. It allows us to know where the function is defined and what output values we can expect, which is crucial for applications in various mathematical and real-world contexts.

In conclusion, our detailed analysis of the rational function $y = \frac{6x + 18}{3x - 18}$ has provided a comprehensive understanding of its properties and behavior. We began by simplifying the function to $y = \frac{2(x + 3)}{x - 6}$, which made it easier to identify key features. We determined the vertical asymptote at $x = 6$ and the horizontal asymptote at $y = 2$. The intercepts were found to be (0, -1) for the y-intercept and (-3, 0) for the x-intercept. By integrating this information, we were able to sketch the graph of the function, showing how it approaches the asymptotes and crosses the intercepts. We also identified the domain as all real numbers except $x = 6$ and the range as all real numbers except $y = 2$. This thorough examination highlights the importance of simplifying rational functions, identifying their asymptotes and intercepts, and understanding their domain and range. These steps are crucial for analyzing and interpreting the behavior of rational functions in various mathematical and real-world contexts. Rational functions are essential tools in mathematics and have applications in numerous fields, including physics, engineering, and economics. This detailed analysis not only enhances our mathematical understanding but also equips us with the skills to apply these concepts in practical scenarios. By mastering the techniques for analyzing rational functions, we can gain valuable insights into complex systems and solve real-world problems more effectively. The ability to break down and understand complex functions is a valuable skill that extends beyond the classroom and into various professional disciplines.