Finding The Vertex Position Of The Parabola Function 3x + 4y = 3x + 4

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Hey guys! Today, we're diving into the fascinating world of parabolas and how to pinpoint their vertex. You know, that crucial point where the parabola changes direction? We'll tackle a specific problem, breaking it down step-by-step so you can conquer these questions with confidence. Let's get started!

Understanding the Problem: Unveiling the Parabola

The question at hand asks us to determine the vertex of the parabola represented by the function 3x + 4y = 3x + 4. But hold on a second! This equation doesn't quite look like the standard form of a quadratic function we're used to, which is ax² + bx + c, where a ≠ 0. So, our first mission is to manipulate the equation into a recognizable quadratic form. This involves isolating 'y' on one side of the equation to express it as a function of 'x'. Once we have the equation in the standard quadratic form, we can easily identify the coefficients 'a', 'b', and 'c', which are essential for finding the vertex. Remember, the vertex is the turning point of the parabola, and it’s either the minimum or maximum point depending on the parabola’s orientation (whether it opens upwards or downwards). The 'a' coefficient plays a crucial role here; if 'a' is positive, the parabola opens upwards, indicating a minimum point, and if 'a' is negative, it opens downwards, indicating a maximum point. Understanding these foundational concepts is vital because they guide our approach to solving the problem, ensuring we’re not just plugging numbers into formulas but actually grasping the underlying principles. Once we've massaged the equation into the familiar quadratic form, we'll be ready to roll up our sleeves and calculate the vertex coordinates. It's like preparing the canvas before painting a masterpiece – you need the right foundation to create something truly beautiful. So, let's sharpen our pencils and dive into the algebraic maneuvering that will reveal the hidden parabola within the given equation.

Transforming the Equation: From Implicit to Explicit

So, the key here is to rewrite the given equation, 3x + 4y = 3x + 4, into the standard quadratic form, ax² + bx + c. To do this, we need to isolate 'y' on one side. Let's start by subtracting 3x from both sides of the equation. This gives us 4y = (3x - 3x) + 4, which simplifies to 4y = 4. Now, to get 'y' by itself, we divide both sides by 4. This results in y = 1. Whoa! This looks simpler than we initially thought. But wait a minute... where's the x² term? Where's the 'x' term? This is actually a special case. The equation y = 1 represents a horizontal line, not a parabola. A parabola, by definition, is a U-shaped curve described by a quadratic equation, which includes an x² term. The absence of the x² term means that the curve flattens out into a straight line. This might seem like a curveball (pun intended!), but it's a crucial observation. It highlights the importance of carefully examining the equation's form before blindly applying formulas. We've essentially encountered a degenerate case of a parabola, where the quadratic term vanishes. Think of it like a parabola that has been stretched out infinitely wide, turning into a flat line. This is a great reminder that math often throws unexpected twists our way, and it's our job to be flexible and adapt our approach accordingly. So, while we were expecting a curved parabola, we've discovered a straight line hiding in plain sight. Now, how does this straight line relate to our quest for the vertex? Well, that’s the next piece of the puzzle we need to unravel.

Finding the Vertex: A Straight Line's Peculiarity

Now, here's a tricky part. The equation y = 1 represents a horizontal line. Horizontal lines don't have a single vertex in the same way a parabola does. A parabola has a distinct turning point, either a minimum or maximum, which we call the vertex. However, a horizontal line extends infinitely in both directions without changing its direction. So, how do we reconcile this with the question asking for the vertex? Well, we need to think about what the vertex represents. It's the point where the parabola changes direction. For a horizontal line, there's no change in direction; it's a constant, flat line. In a sense, we could say that every point on the line is a vertex because there's no single point that stands out as the turning point. It's a bit like asking for the highest point on a flat plain – there isn't one! They are all equally high. This might seem like a paradoxical situation, but it highlights the nuances of mathematical concepts. The definition of a vertex is intrinsically linked to the curvature of a parabola. A straight line, lacking this curvature, doesn't fit neatly into the vertex definition. So, instead of trying to force-fit the concept of a vertex onto a horizontal line, we need to acknowledge that the question might be a bit misleading. A more accurate way to phrase it might be to ask for a characteristic point on the line. However, given the multiple-choice options, we need to choose the answer that best reflects the nature of the line y = 1. This means we'll be looking for a point that lies on this line, a point where the y-coordinate is equal to 1. It's like finding a landmark on our flat plain – any point will do, as long as it's on the level ground.

Evaluating the Options: Choosing the Right Point

Okay, let's put on our detective hats and examine the options provided. We have: A) v = (3, 7), B) v = (0, 4), C) v = (1, 1), and D) v = (2, -5). Remember, we're looking for a point that lies on the line y = 1. This means the y-coordinate of the point must be 1. So, let's go through each option:

  • A) v = (3, 7): The y-coordinate is 7, which is not equal to 1. So, this option is incorrect.
  • B) v = (0, 4): The y-coordinate is 4, which is also not equal to 1. This option is out too.
  • C) v = (1, 1): Aha! The y-coordinate is 1. This point lies on the line y = 1. This looks promising.
  • D) v = (2, -5): The y-coordinate is -5, which is definitely not equal to 1. So, this option is incorrect.

It seems like option C, v = (1, 1), is the only point that satisfies the condition of lying on the line y = 1. It's like finding the single landmark that matches our description of being on the flat plain. The other options are like landmarks located on hills or in valleys – they just don't fit the criteria. So, based on our analysis, option C appears to be the most logical answer. It's the only point among the choices that aligns with the equation we derived. This process of elimination is a powerful tool in problem-solving. By systematically ruling out incorrect answers, we can narrow down the possibilities and arrive at the correct solution with greater confidence. Now, let's solidify our understanding by confirming why this is the most suitable answer in the context of the question.

The Verdict: Option C is the Answer

Therefore, the answer is C) v = (1, 1). Even though the original equation simplifies to a horizontal line rather than a parabola, we've successfully navigated the question by understanding the underlying concepts and applying logical reasoning. The key takeaway here is that not every equation will neatly fit the expected form. Sometimes, we encounter special cases that require us to think outside the box. In this instance, the absence of the x² term transformed the problem from finding a parabola's vertex to identifying a point on a horizontal line. This highlights the importance of careful observation and a flexible approach to problem-solving. We didn't just blindly apply a formula; we analyzed the equation, understood its geometric representation, and then selected the answer that best fit the situation. Math is not just about memorizing formulas; it's about developing critical thinking skills and the ability to adapt to different scenarios. And that's what makes it so fascinating! So, next time you encounter a math problem, remember to take a step back, analyze the situation, and don't be afraid to think creatively. You might just surprise yourself with what you can achieve. And hey, if you ever get stuck, just remember this little adventure we had with the