Finding The X-intercept Of Y=(4x+24)/(x-6) A Step-by-Step Guide

In the realm of mathematics, particularly when dealing with functions, finding the $x$-intercept is a fundamental skill. The $x$-intercept, also known as the root or zero of a function, is the point where the graph of the function intersects the $x$-axis. At this point, the $y$-coordinate is always zero. This guide will provide a detailed explanation of how to find the $x$-intercept of a rational function, using the example provided: $y = \frac{4x + 24}{x - 6}$. We will delve into the underlying concepts, step-by-step procedures, and potential pitfalls to avoid. Understanding how to find $x$-intercepts is crucial for analyzing the behavior of functions, solving equations, and applying mathematical concepts in real-world scenarios. For instance, in physics, finding the $x$-intercept of a projectile's trajectory can determine the landing point. In economics, it can represent the break-even point where costs equal revenue. Therefore, mastering this skill is essential for anyone pursuing studies or careers in STEM fields.

Understanding $x$-intercepts

Before we dive into the specific example, let's solidify our understanding of $x$-intercepts. The $x$-intercept is the point where a graph crosses the $x$-axis. This is a critical concept in algebra and calculus, as it helps us understand the behavior of functions and their relationship to the coordinate plane. To find the $x$-intercept, we set $y = 0$ and solve for $x$. This is because any point on the $x$-axis has a $y$-coordinate of 0. Graphically, the $x$-intercept is where the function's line or curve intersects the horizontal axis. This point provides valuable information about the function, such as where it changes sign (from positive to negative or vice versa) or where it has a value of zero. Understanding this concept is crucial for analyzing functions, solving equations, and applying mathematical models to real-world problems. For example, in physics, the $x$-intercept might represent the time when an object hits the ground, or in economics, it could represent the quantity of goods that need to be sold to break even. Therefore, a firm grasp of $x$-intercepts is essential for anyone working with mathematical models.

Why are $x$-intercepts Important?

$X$-intercepts are not just abstract mathematical concepts; they hold significant practical importance in various fields. In the context of a graph, the $x$-intercept represents the point where the function's value is zero. This can be a crucial piece of information, depending on the context of the problem. For example, in physics, the $x$-intercept of a projectile's trajectory can indicate the landing point, a vital piece of data for calculations and predictions. In business and economics, the $x$-intercept of a cost-revenue function might represent the break-even point, where the company neither makes a profit nor incurs a loss. Understanding the $x$-intercept in this context is critical for making informed business decisions. Furthermore, in engineering, the $x$-intercept can represent a critical point in a system's performance, such as the point of resonance or instability. The ability to find and interpret $x$-intercepts is therefore a fundamental skill for anyone working with mathematical models in real-world applications. By understanding the significance of $x$-intercepts, we can better appreciate their role in solving problems across various disciplines.

Finding the $x$-intercept of $y = \frac{4x + 24}{x - 6}$

Now, let's apply this concept to our specific example: $y = \frac{4x + 24}{x - 6}$. To find the $x$-intercept, we need to set $y = 0$ and solve for $x$. This is because the $x$-intercept is the point where the graph of the function crosses the $x$-axis, and on the $x$-axis, the $y$-coordinate is always 0. Therefore, setting $y$ to 0 allows us to find the $x$-value at which the function intersects the $x$-axis. This process involves manipulating the equation to isolate $x$, which will give us the $x$-coordinate of the intercept. Understanding this process is crucial for solving a variety of mathematical problems, including those involving rational functions, quadratic equations, and other types of functions. It is a fundamental skill that is applied in many areas of mathematics and science. By mastering this technique, we can gain a deeper understanding of the behavior of functions and their graphical representations. This knowledge is essential for further studies in mathematics and for applying mathematical concepts to real-world scenarios.

Step 1: Set $y = 0$

The first step in finding the $x$-intercept is to set the function's output, $y$, equal to zero. This is because the $x$-intercept is the point where the graph of the function intersects the $x$-axis, and at any point on the $x$-axis, the $y$-coordinate is always zero. By setting $y = 0$, we are essentially finding the $x$-value(s) that make the function equal to zero, which corresponds to the point(s) where the graph crosses the $x$-axis. This is a fundamental step in finding roots or zeros of any function, whether it's a linear, quadratic, polynomial, or rational function. The logic behind this step is consistent across different types of functions, making it a core concept in algebra and calculus. Understanding why we set $y = 0$ is just as important as knowing how to do it, as it provides a deeper understanding of the relationship between the function, its graph, and its roots. Therefore, this initial step is crucial for solving for the $x$-intercept and understanding the function's behavior.

So, we have:

$0 = \frac{4x + 24}{x - 6}$

Step 2: Solve for $x$

Now that we've set $y = 0$, we need to solve the equation $0 = \frac{4x + 24}{x - 6}$ for $x$. To do this, we can start by multiplying both sides of the equation by the denominator, $(x - 6)$. This will eliminate the fraction and simplify the equation, making it easier to solve. However, it's crucial to remember that we must consider the possibility of extraneous solutions. An extraneous solution is a value of $x$ that satisfies the transformed equation but not the original equation. This can occur when we multiply both sides of an equation by an expression that can be zero. In this case, we must make sure that the solution we find does not make the denominator, $(x - 6)$, equal to zero. If it does, then that solution is extraneous and must be discarded. This step is essential for ensuring the accuracy of our solution and for understanding the limitations of algebraic manipulations. By carefully considering the possibility of extraneous solutions, we can avoid errors and gain a deeper understanding of the equation we are solving.

Multiplying both sides by $(x - 6)$, we get:

$0(x - 6) = \frac{4x + 24}{x - 6}(x - 6)$

$0 = 4x + 24$

Next, we isolate $x$ by subtracting 24 from both sides:

$-24 = 4x$

Finally, divide both sides by 4:

$x = -6$

Step 3: Verify the Solution

After finding a potential solution, it's crucial to verify that it is indeed a valid $x$-intercept. This is particularly important when dealing with rational functions, as we need to ensure that our solution doesn't make the denominator of the original function equal to zero. If the solution makes the denominator zero, it would result in an undefined expression, and thus, the solution would be extraneous. Extraneous solutions can arise from the algebraic manipulations we perform while solving the equation, such as multiplying both sides by an expression that contains a variable. Therefore, verifying the solution is not just a formality, but a necessary step to ensure the accuracy of our answer. This step demonstrates a thorough understanding of the function and its domain, and it helps prevent errors that can occur when dealing with rational expressions. By verifying our solution, we can have confidence in our answer and its interpretation within the context of the problem.

Let's plug $x = -6$ back into the original equation:

$y = \frac{4(-6) + 24}{-6 - 6} = \frac{-24 + 24}{-12} = \frac{0}{-12} = 0$

Since $y = 0$ when $x = -6$, this confirms that $x = -6$ is indeed the $x$-intercept.

The $x$-intercept

Therefore, the $x$-intercept of the function $y = \frac{4x + 24}{x - 6}$ is $(-6, 0)$. This point represents where the graph of the function crosses the $x$-axis. Understanding how to find this point is crucial for analyzing the function's behavior and its graph. The $x$-intercept, along with other key features such as the $y$-intercept, vertical asymptotes, and horizontal asymptotes, provides a comprehensive picture of the function's characteristics. This information is valuable for various applications, including modeling real-world phenomena, solving equations, and making predictions based on the function's behavior. By accurately determining the $x$-intercept, we gain a valuable insight into the function's properties and its relationship to the coordinate plane. This knowledge is essential for further studies in mathematics and for applying mathematical concepts to practical problems.

Common Mistakes to Avoid

When finding $x$-intercepts, there are several common mistakes that students often make. Avoiding these mistakes can significantly improve accuracy and understanding. One of the most frequent errors is forgetting to check for extraneous solutions, especially when dealing with rational functions. As we discussed earlier, extraneous solutions can arise when multiplying both sides of an equation by an expression that contains a variable. Failing to verify the solution can lead to incorrect answers and a misunderstanding of the function's behavior. Another common mistake is incorrectly manipulating the equation while solving for $x$. This can include errors in arithmetic, algebra, or the order of operations. It's crucial to pay close attention to detail and to double-check each step to ensure accuracy. Additionally, some students may struggle with the initial step of setting $y = 0$. It's important to remember that the $x$-intercept is the point where the graph crosses the $x$-axis, and on the $x$-axis, the $y$-coordinate is always zero. By understanding this fundamental concept, students can avoid this common mistake. By being aware of these potential pitfalls, students can approach the process of finding $x$-intercepts with greater confidence and accuracy.

Conclusion

In conclusion, finding the $x$-intercept of a function, such as $y = \frac{4x + 24}{x - 6}$, is a fundamental skill in mathematics. By setting $y = 0$ and solving for $x$, we can determine the point where the graph of the function intersects the $x$-axis. This point provides valuable information about the function's behavior and its relationship to the coordinate plane. Throughout this guide, we've emphasized the importance of each step, from setting up the equation to verifying the solution. We've also highlighted common mistakes to avoid, such as forgetting to check for extraneous solutions. Mastering this skill is not only essential for success in mathematics courses but also for applying mathematical concepts to real-world problems. The ability to find $x$-intercepts is a valuable tool in various fields, including physics, engineering, economics, and computer science. Therefore, a thorough understanding of this concept is crucial for anyone pursuing studies or careers in STEM fields. By following the steps outlined in this guide and practicing regularly, you can confidently find the $x$-intercept of any function and enhance your overall mathematical proficiency.