Parabola Equation: Vertex At (-2, 0) Explained!

Hey guys! Let's dive into the fascinating world of parabolas and figure out which equation perfectly captures a parabola with its vertex gracefully sitting at the point (-2, 0). We'll break down each option, making sure we understand exactly why one choice shines above the rest. So, grab your mental graph paper, and let's get started!

Understanding Parabolas and Vertex Form

Before we jump into the options, let's quickly refresh our understanding of parabolas and their equations. A parabola is a U-shaped curve, and its most basic form is represented by the equation y = ax², where 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0) and how wide or narrow it is. The vertex is the turning point of the parabola—the minimum point if it opens upwards, or the maximum point if it opens downwards.

But here’s where it gets super useful: we can express the equation of a parabola in vertex form, which directly reveals the vertex coordinates. The vertex form looks like this: y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is a game-changer because it allows us to immediately identify the vertex just by looking at the equation. Remember, the h value in the equation is the x-coordinate of the vertex, and the k value is the y-coordinate.

Understanding this vertex form is crucial for solving our problem. We're looking for an equation where the vertex is at (-2, 0). This means we need to find an equation that, when written in vertex form, has h = -2 and k = 0. Keep this in mind as we explore the options. The key to success in mathematics often lies in understanding the fundamental forms and how they relate to the properties of the shapes or functions they represent. By grasping the vertex form, we transform a potentially complex problem into a straightforward pattern-matching exercise. This approach not only helps in solving this particular question but also builds a solid foundation for tackling more advanced problems involving parabolas and other conic sections. So, let's use this knowledge to dissect each option and pinpoint the one that fits our criteria perfectly.

Dissecting the Options

Now, let's put our knowledge of vertex form to the test by examining each option and determining which one gives us a parabola with a vertex at (-2, 0).

Option A: y = -2x²

This equation looks simple enough, doesn't it? It's in the basic form of a parabola, y = ax², where a = -2. This tells us the parabola opens downwards because a is negative. However, let's think about the vertex. In this form, we can consider h and k to be 0, so the vertex is actually at (0, 0). This is because the equation can be rewritten as y = -2(x - 0)² + 0. So, while it's a parabola, its vertex isn't at (-2, 0), meaning this option is not the correct one. It's a classic example of a parabola centered at the origin, and understanding why it doesn't fit our criteria is just as important as finding the correct answer. Recognizing these subtle differences is what sharpens our mathematical intuition and problem-solving skills. The negative coefficient also tells us that the parabola is reflected across the x-axis, adding another layer to our understanding of its graphical representation. So, while option A might be tempting at first glance, a closer look reveals that its vertex is not where we need it to be.

Option B: y = (x + 2)²

Ah, this one looks more promising! This equation is already in, or very close to, vertex form. We can rewrite it as y = 1(x + 2)² + 0. Notice anything familiar? Let's break it down. We have h hiding inside the parenthesis as +2, but remember, the vertex form has (x - h). So, to find the actual value of h, we need to think about what number, when subtracted from x, gives us (x + 2). That number is -2! So, h = -2. And k, the vertical shift, is clearly 0. This means the vertex is at (-2, 0). Bingo! This option perfectly matches our criteria. The coefficient a is 1, which means the parabola opens upwards, but the critical point is that the vertex is indeed at (-2, 0). This highlights the power of the vertex form in instantly revealing key characteristics of the parabola. We've successfully identified an equation that meets our requirements, but let's not stop here. It's always a good practice to examine the remaining options to solidify our understanding and ensure we haven't missed anything.

Option C: y = (x - 2)²

Okay, let's take a look at option C: y = (x - 2)². This equation is also in a form that's very close to the vertex form, which is great for us! We can rewrite it as y = 1(x - 2)² + 0. Now, let's pinpoint the vertex. Here, we see (x - 2), which directly tells us that h = 2 (remember, the vertex form is y = a(x - h)² + k). And just like in option B, k = 0. So, the vertex for this parabola is at (2, 0). Notice the subtle but crucial difference between this and option B! The x-coordinate of the vertex is positive 2 here, not -2. This means that while this equation represents a parabola, it's not the one we're looking for. It's shifted to the right compared to the parabola we need, which is centered at x = -2. This comparison highlights the importance of paying close attention to the signs in the equation, as they directly affect the position of the parabola in the coordinate plane. By carefully analyzing the h value, we can quickly determine the horizontal shift of the parabola and whether it matches our desired vertex location. So, while option C is a valid parabola, it doesn't fit the specific criteria of having a vertex at (-2, 0).

Option D: y = x² - 2

Lastly, let's consider option D: y = x² - 2. This equation is also a parabola, but it's slightly different in form compared to options B and C. We can rewrite it in vertex form as y = 1(x - 0)² - 2. Now, let's identify the vertex. Here, h = 0 (since we have (x - 0)²), and k = -2. This means the vertex of this parabola is at (0, -2). Notice how the -2 is outside the parenthesis, indicating a vertical shift downwards. This parabola is shifted vertically downwards compared to the basic parabola y = x². So, while it's still a parabola, its vertex is not at (-2, 0). It's on the y-axis, specifically at y = -2. This option serves as a good reminder that the h and k values in the vertex form control the horizontal and vertical position of the parabola, respectively. By understanding how these values shift the graph, we can quickly eliminate options that don't meet our vertex requirements. Therefore, option D, although a parabola, does not have its vertex at the desired location of (-2, 0).

The Verdict: Option B is the Champion!

After carefully examining all the options, we've definitively found our answer. Option B, y = (x + 2)², is the only equation that represents a parabola with a vertex at (-2, 0). We saw how rewriting it in vertex form, y = 1(x + 2)² + 0, clearly shows h = -2 and k = 0, confirming our vertex location. This entire process has been a fantastic exercise in understanding the vertex form of a parabola and how it directly relates to the graph's characteristics. We not only found the correct answer but also reinforced our knowledge of parabolas, vertex form, and how to analyze equations to extract key information. This understanding will undoubtedly be valuable as we continue our mathematical journey.

Final Answer

Therefore, the equation that has a graph that is a parabola with a vertex at (-2, 0) is:

B. y = (x + 2)²