Vertex Of Y=(x+7)^2-9: Finding The Correct Graph

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Hey everyone! Today, we're diving into a super important concept in algebra: finding the vertex of a parabola. Specifically, we're going to figure out the vertex of the graph represented by the equation y = (x + 7)^2 - 9. This is a classic problem, and understanding how to solve it will help you tackle all sorts of quadratic equation questions. So, let's jump right in and make sure we nail this down!

Understanding the Vertex Form of a Quadratic Equation

To find the vertex, we first need to understand the vertex form of a quadratic equation. This form is super helpful because it directly tells us the coordinates of the vertex. The vertex form looks like this:

  • y = a(x - h)^2 + k

Where:

  • (h, k) is the vertex of the parabola.
  • 'a' determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards. The larger the absolute value of 'a', the narrower the parabola.

Now, why is this form so useful? Well, it transforms the standard form of a quadratic equation (ax^2 + bx + c = 0) into a format where the vertex is immediately visible. The vertex is the point where the parabola changes direction – it’s either the lowest point (minimum) if the parabola opens upwards or the highest point (maximum) if it opens downwards. Identifying the vertex is crucial for graphing the parabola accurately and for solving related problems.

Think of it like this: the vertex is the peak of a mountain or the bottom of a valley. Once you know where that point is, you can easily sketch the rest of the parabola. The 'h' value shifts the parabola horizontally, and the 'k' value shifts it vertically. The 'a' value then stretches or compresses the parabola and flips it upside down if needed. So, by understanding the vertex form, we unlock a powerful tool for analyzing and graphing quadratic functions. This knowledge is not just for textbook problems; it's essential in various real-world applications, from physics to engineering, where parabolic paths and shapes are common.

Identifying the Vertex in Our Equation: y = (x + 7)^2 - 9

Alright, now let's apply this knowledge to our specific equation: y = (x + 7)^2 - 9. Our main goal here is to match this equation to the vertex form y = a(x - h)^2 + k so we can easily read off the vertex coordinates. This is where a little bit of algebraic sleuthing comes in handy!

First, let’s rewrite our equation slightly to make the correspondence even clearer. We can rewrite (x + 7) as (x - (-7)). This might seem like a small change, but it’s crucial for correctly identifying the h value. So now our equation looks like:

  • y = 1(x - (-7))^2 + (-9)

Notice how we've also explicitly written the '1' in front of the parentheses. This helps us see that a = 1. This means our parabola opens upwards, which is good to know! The coefficient 'a' being positive tells us we have a smile-shaped parabola, not a frown. The magnitude of 'a' (which is 1 in this case) also tells us about the width of the parabola, but for finding the vertex, we mainly care about the sign.

Now, let's match the parts to the vertex form y = a(x - h)^2 + k. We can see that:

  • h = -7
  • k = -9

Remember, the vertex is the point (h, k). So, in our case, the vertex is (-7, -9). It's super important to pay attention to the signs here. The form has (x - h), so when we have (x + 7), that actually means h is negative. This is a common spot for mistakes, so always double-check your signs!

Understanding this process of matching equations to the vertex form isn't just about getting the right answer for this problem. It's about building a solid foundation for understanding quadratic functions in general. Once you're comfortable with this, you can quickly identify key features of a parabola, like its minimum or maximum point, and use that information to solve a variety of problems. So, keep practicing, and you’ll become a vertex-finding pro in no time!

Why the Other Options Are Incorrect

Now that we've confidently identified the correct vertex as (-7, -9), let's take a moment to understand why the other options are incorrect. This is a great way to reinforce your understanding and avoid similar mistakes in the future. Often, incorrect options are based on common errors or misunderstandings, so breaking them down can be very insightful.

  • Option A: Vertex at (7, 9)
    • This option gets the signs completely flipped. It seems like someone might have simply taken the numbers 7 and 9 from the equation without considering the form (x - h) and the addition of k. Remember, the equation has (x + 7), which means h is actually negative 7, not positive. And the k value is -9, not 9. So, this option is incorrect because it misunderstands the vertex form equation's structure.
  • Option C: Vertex at (7, -9)
    • This option correctly identifies the k value as -9 but makes the same sign error with the h value. It takes the 7 from (x + 7) directly but forgets that the vertex form has (x - h). So, the x-coordinate of the vertex should be the opposite sign of the number inside the parentheses.
  • Option D: Vertex at (-7, 9)
    • This option gets the h value correct as -7 but incorrectly identifies the k value as 9. The equation clearly shows -9 outside the parentheses, which directly corresponds to the k value in the vertex form. So, this option misunderstands the role of the constant term in determining the vertical position of the vertex.

By understanding why these options are wrong, you're not just memorizing the correct answer; you're actually learning the underlying concepts. This is super important for building a strong understanding of quadratic functions. When you can identify and explain common errors, you're much less likely to make them yourself! So, always take the time to analyze incorrect options – it’s a valuable learning exercise.

Conclusion: Vertex at (-7, -9)

So, to wrap things up, the correct answer is B. Vertex at (-7, -9). We found this by recognizing the vertex form of a quadratic equation, y = a(x - h)^2 + k, and carefully matching the parts of our equation, y = (x + 7)^2 - 9, to this form. Remember, the vertex is the point (h, k), and by correctly identifying h and k, we pinpointed the vertex as (-7, -9).

We also took a look at why the other options were incorrect, which helped us reinforce the key concepts. We saw how sign errors and misunderstandings of the vertex form equation can lead to wrong answers. By analyzing these mistakes, we've strengthened our understanding and can avoid similar pitfalls in the future.

Finding the vertex is a fundamental skill in algebra, and it's essential for graphing parabolas and solving related problems. Mastering this skill opens the door to more advanced topics in mathematics and has practical applications in various fields. So, keep practicing, keep analyzing, and you'll become a quadratic equation whiz in no time!