Finding The X-intercept Of Y = (6x + 18) / (3x - 18)
This article delves into the function y = (6x + 18) / (3x - 18), exploring its properties and, most importantly, determining its x-intercept, represented by the coordinate point ([?], 0). We will analyze the function's structure, identify any restrictions on its domain, and then systematically solve for the x-value that corresponds to y = 0. This exploration will provide a comprehensive understanding of the function and its behavior within the coordinate plane.
Analyzing the Function y = (6x + 18) / (3x - 18)
To effectively analyze the function y = (6x + 18) / (3x - 18), we must first understand its structure. This is a rational function, which is defined as a function that can be expressed as the quotient of two polynomials. In this case, the numerator is the linear expression (6x + 18), and the denominator is also a linear expression (3x - 18). The key characteristic of rational functions is that they can have vertical asymptotes, points where the function approaches infinity or negative infinity. These asymptotes occur when the denominator of the function equals zero, leading to an undefined value for the function. Therefore, to fully grasp the behavior of this function, we need to identify any potential vertical asymptotes.
To find the vertical asymptotes, we set the denominator equal to zero and solve for x:
3x - 18 = 0
Adding 18 to both sides:
3x = 18
Dividing both sides by 3:
x = 6
This tells us that there is a vertical asymptote at x = 6. This means the function will approach infinity or negative infinity as x gets closer to 6 from either the left or the right. This is a crucial piece of information as we continue to analyze the function and search for the x-intercept. In addition to vertical asymptotes, rational functions can also have horizontal asymptotes, which describe the function's behavior as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the polynomials in the numerator and the denominator. In this case, both the numerator and the denominator are linear expressions (degree 1). When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient in the numerator is 6, and the leading coefficient in the denominator is 3. Therefore, the horizontal asymptote is y = 6/3 = 2. This means that as x becomes very large (positive or negative), the function will approach the value of 2.
Before we proceed to finding the x-intercept, let's further simplify the function. Notice that we can factor out common factors from both the numerator and the denominator:
y = (6x + 18) / (3x - 18) = 6(x + 3) / 3(x - 6)
We can then simplify further by dividing both the numerator and the denominator by their common factor, which is 3:
y = 2(x + 3) / (x - 6)
This simplified form of the function makes it easier to analyze and identify key features. The simplified form clearly shows the vertical asymptote at x = 6 and the horizontal asymptote at y = 2. It also makes the process of finding the x-intercept more straightforward. Understanding the asymptotes and the simplified form of the function is essential for a complete analysis. Now, we can move on to the crucial step of determining the x-intercept of the function, which is the point where the graph crosses the x-axis.
Determining the X-intercept
The x-intercept of a function is the point where the graph intersects the x-axis. At this point, the y-coordinate is always zero. Therefore, to find the x-intercept, we set y = 0 in the function and solve for x. Using the simplified form of our function, y = 2(x + 3) / (x - 6), we have:
0 = 2(x + 3) / (x - 6)
To solve for x, we can multiply both sides of the equation by the denominator (x - 6), but we must keep in mind that x cannot be equal to 6, as this would make the denominator zero and the function undefined. Multiplying both sides by (x - 6) gives:
0 * (x - 6) = 2(x + 3)
This simplifies to:
0 = 2(x + 3)
Now, we can divide both sides by 2:
0 = x + 3
Finally, we subtract 3 from both sides to isolate x:
x = -3
Therefore, the x-intercept of the function is x = -3. This means the graph of the function crosses the x-axis at the point (-3, 0). This is a significant point on the graph, as it gives us a clear indication of the function's behavior. The x-intercept, along with the asymptotes we previously identified, helps us sketch the graph of the function and understand its overall shape and behavior. Knowing the x-intercept and the asymptotes allows us to accurately plot key points and lines on the coordinate plane, which in turn enables us to visualize the function's trajectory. The function approaches the vertical asymptote (x = 6) but never touches it, and it approaches the horizontal asymptote (y = 2) as x moves towards positive or negative infinity. The x-intercept (-3, 0) is the point where the graph crosses the x-axis, providing a crucial reference point for understanding the function's behavior in the vicinity of the x-axis. By combining this information, we can develop a comprehensive understanding of the function's behavior across its domain.
Conclusion
In conclusion, we have thoroughly analyzed the function y = (6x + 18) / (3x - 18). We identified the vertical asymptote at x = 6 and the horizontal asymptote at y = 2. We also successfully determined the x-intercept to be x = -3, which corresponds to the coordinate point (-3, 0). This comprehensive analysis provides a clear understanding of the function's behavior and its graphical representation. Understanding rational functions, including identifying asymptotes and intercepts, is fundamental in mathematics and has applications in various fields, including physics, engineering, and economics. The ability to analyze such functions allows us to model and understand real-world phenomena that exhibit similar mathematical relationships. By finding the x-intercept, we've located a key point where the function's output is zero, which can be significant in many practical applications. For instance, in a scenario where this function represents a cost or profit model, the x-intercept would represent the break-even point. Similarly, in a physics context, it might represent a point of equilibrium. Therefore, the process of determining the x-intercept is not merely a mathematical exercise but a tool that can provide valuable insights in a range of real-world situations. The techniques used in this analysis, such as identifying asymptotes and solving for intercepts, are transferable and can be applied to other rational functions and mathematical problems, reinforcing the importance of mastering these skills.
