Finding The $y$-Intercept Of $y = (2x + 14) / (x - 7)$ A Step-by-Step Guide

Hey everyone! Today, we're diving deep into the world of algebraic functions to tackle a common yet crucial concept: finding the $y$-intercept. Specifically, we're going to break down how to determine the $y$-intercept for the function $y = \frac{2x + 14}{x - 7}$. This might seem a bit daunting at first, but trust me, once we understand the underlying principles, it's a piece of cake. So, grab your thinking caps, and let's get started!

Understanding the $y$-Intercept

Before we jump into the specifics of our function, let's make sure we're all on the same page about what the $y$-intercept actually represents. In the simplest terms, the $y$-intercept is the point where a graph intersects the $y$-axis. Think of it as the spot where the line or curve "crosses" the vertical axis on our coordinate plane. Graphically, this point always has an $x$-coordinate of 0. That's because any point on the $y$-axis is, by definition, a certain distance above or below the origin (0, 0) but has no horizontal displacement.

Now, why is finding the $y$-intercept so important? Well, it gives us a valuable piece of information about the function's behavior. It tells us the value of $y$ when $x$ is zero, which can be significant in various real-world applications. For instance, in a supply-demand curve, the $y$-intercept might represent the price at which there's no demand for a product. In a linear equation representing the cost of a service, the $y$-intercept could be the fixed initial fee.

To find the $y$-intercept algebraically, we use this core concept: since the $x$-coordinate of the $y$-intercept is always 0, we simply substitute $x = 0$ into our function and solve for $y$. This gives us the $y$-coordinate of the point where the graph intersects the $y$-axis. This method works for any function, whether it's a linear equation, a quadratic equation, or, as in our case, a rational function.

The Significance of the $y$-Intercept

The $y$-intercept isn't just a random point on a graph; it's a key indicator of the function's behavior and characteristics. It acts as an anchor point, providing a starting value or baseline from which the function's behavior can be further analyzed. Understanding its significance allows us to interpret the function within a specific context. For example, consider a function that models the growth of a plant over time. The $y$-intercept would represent the initial height of the plant at the beginning of the observation period. If the function represents the balance of a bank account, the $y$-intercept would be the initial deposit amount.

Furthermore, the $y$-intercept plays a vital role in sketching the graph of a function. Along with other key points like the $x$-intercepts (where the graph crosses the $x$-axis) and any asymptotes (lines that the graph approaches but doesn't cross), the $y$-intercept helps define the shape and position of the graph. This can be incredibly useful for visualizing the function's behavior and making predictions based on the model.

In more complex functions, like the rational function we're dealing with today, the $y$-intercept can also provide clues about the function's overall structure and potential singularities. By examining the value of $y$ when $x$ is zero, we can gain insight into how the function behaves near the $y$-axis and identify any potential vertical asymptotes or discontinuities. So, as you can see, mastering the art of finding and interpreting the $y$-intercept is a fundamental skill in algebra and calculus, opening doors to a deeper understanding of mathematical functions and their applications.

Applying the Concept to $y = \frac{2x + 14}{x - 7}$

Now that we've solidified our understanding of the $y$-intercept, let's put our knowledge into action and find it for the given function: $y = \frac{2x + 14}{x - 7}$. Remember our golden rule: to find the $y$-intercept, we substitute $x = 0$ into the equation and solve for $y$. It's a straightforward process, but let's break it down step by step to ensure clarity.

First, we replace every instance of $x$ in the equation with 0. This gives us:

$y = \frac{2(0) + 14}{0 - 7}$

Next, we simplify the expression. The numerator becomes $2(0) + 14 = 0 + 14 = 14$, and the denominator becomes $0 - 7 = -7$. So, our equation now looks like this:

$y = \frac{14}{-7}$

Finally, we perform the division: $14$ divided by $-7$ is $-2$. Therefore, we have:

$y = -2$

And there you have it! The $y$-coordinate of the $y$-intercept is $-2$. This means that the graph of the function $y = \frac{2x + 14}{x - 7}$ intersects the $y$-axis at the point $(0, -2)$.

Visualizing the Result

To solidify our understanding, let's take a moment to visualize what this result means graphically. Imagine a coordinate plane with the $x$-axis running horizontally and the $y$-axis running vertically. The point $(0, -2)$ is located on the $y$-axis, two units below the origin (0, 0). This is the exact point where the graph of our function will cross the $y$-axis.

Now, while we've only found one point on the graph, the $y$-intercept, it provides valuable information about the function's behavior. We know that the graph passes through this specific point, which helps us to sketch a more accurate representation of the function. If we were to plot additional points or use graphing software, we would see that the function curves and approaches certain asymptotes, but it definitively passes through $(0, -2)$.

Furthermore, the fact that the $y$-intercept is negative tells us that the graph crosses the $y$-axis below the $x$-axis. This can be particularly useful when analyzing real-world scenarios modeled by the function. For instance, if this function represented the profit of a business, the $y$-intercept would indicate the initial loss before any sales were made. So, not only have we found the $y$-intercept algebraically, but we've also begun to interpret its meaning within the context of the function's graph and potential applications. This is the true power of understanding mathematical concepts – applying them to real-world situations and gaining meaningful insights.

Expressing the $y$-Intercept as a Coordinate Point

We've successfully calculated the $y$-coordinate of the $y$-intercept for the function $y = \frac{2x + 14}{x - 7}$, which is $-2$. However, it's crucial to express the $y$-intercept as a complete coordinate point to fully represent its location on the coordinate plane. Remember, a coordinate point is an ordered pair of the form $(x, y)$, where $x$ represents the horizontal position and $y$ represents the vertical position.

In the case of the $y$-intercept, we know that the $x$-coordinate is always 0. This is because the $y$-intercept lies on the $y$-axis, where all points have an $x$-coordinate of 0. We've already determined that the $y$-coordinate for our function is $-2$. Therefore, we can express the $y$-intercept as the coordinate point $(0, -2)$.

This notation provides a clear and concise way to communicate the location of the $y$-intercept. It tells us that the graph of the function intersects the $y$-axis at the point where $x$ is 0 and $y$ is $-2$. Writing the $y$-intercept as a coordinate point is essential for several reasons:

• Clarity: It avoids any ambiguity about the point's location. Simply stating "the $y$-intercept is $-2$" could be misinterpreted as a value on the $y$-axis rather than a specific point.
• Graphing: When plotting the graph of the function, we need the $y$-intercept as a coordinate point to accurately place it on the coordinate plane.
• Communication: In mathematical discussions and reports, using coordinate point notation ensures that everyone understands the precise location of the $y$-intercept.

The Importance of Proper Notation

Paying attention to proper mathematical notation, like expressing the $y$-intercept as a coordinate point, is a cornerstone of clear and effective communication in mathematics. It's not just about getting the right answer; it's about presenting the answer in a way that is easily understood and unambiguous. This is especially important when dealing with more complex concepts and functions.

Imagine trying to describe the intersection points of two graphs without using coordinate notation. It would be incredibly difficult and prone to misinterpretation. By using the standard $(x, y)$ notation, we can precisely identify each point and avoid any confusion. This level of precision is crucial in fields like engineering, physics, and computer science, where mathematical models are used to represent real-world phenomena.

Furthermore, consistent use of proper notation helps to build a solid foundation for more advanced mathematical studies. As you progress in your mathematical journey, you'll encounter increasingly complex concepts and notations. By mastering the fundamentals, like expressing the $y$-intercept as a coordinate point, you'll be well-equipped to tackle these challenges. So, remember, the details matter in mathematics, and proper notation is a key ingredient for success.

Conclusion: Mastering the $y$-Intercept

Alright, guys, we've reached the end of our journey to find the $y$-intercept of the function $y = \frac{2x + 14}{x - 7}$. We've covered a lot of ground, from understanding the basic concept of the $y$-intercept to applying it to a specific rational function and expressing the result as a coordinate point. By following the simple yet powerful rule of substituting $x = 0$ and solving for $y$, we were able to determine that the $y$-intercept is $(0, -2)$.

But our exploration didn't stop there. We delved into the significance of the $y$-intercept, recognizing its importance as a key indicator of a function's behavior and a valuable tool for graphing. We also emphasized the crucial role of proper notation in mathematics, highlighting the need to express the $y$-intercept as a coordinate point to ensure clarity and avoid ambiguity.

So, what have we learned? We've learned that finding the $y$-intercept is not just a mechanical process but a way to gain deeper insights into the characteristics of a function. It's a skill that builds upon fundamental algebraic principles and lays the groundwork for more advanced mathematical concepts. Whether you're dealing with linear equations, quadratic functions, or rational expressions, the ability to find the $y$-intercept will serve you well.

Keep Practicing and Exploring

The journey of mathematical understanding is a continuous one. While we've conquered the $y$-intercept today, there are countless other concepts and techniques to explore. The key is to keep practicing, keep asking questions, and keep challenging yourself. Try applying the method we've discussed to different functions, including those with more complex expressions. Explore the relationship between the $y$-intercept and other key features of a graph, such as the $x$-intercepts and asymptotes.

And remember, mathematics is not just about memorizing formulas and procedures; it's about developing a way of thinking, a logical and analytical approach to problem-solving. By mastering fundamental concepts like the $y$-intercept, you're not just adding to your mathematical toolkit; you're honing your critical thinking skills and expanding your capacity to understand the world around you. So, keep up the great work, and never stop exploring the fascinating world of mathematics!

If you have any questions or want to delve deeper into this topic, feel free to ask. Happy problem-solving!