Parabola Equation With Vertex At (-2, 0) A Step-by-Step Guide

When delving into the world of quadratic equations, parabolas stand out as a fundamental concept. Parabolas are U-shaped curves that play a crucial role in various fields, from physics to engineering. The vertex, being the extreme point of the parabola (either the maximum or minimum), is a key feature that helps define its position and orientation in the coordinate plane. In this article, we will explore how to identify the equation of a parabola given its vertex, focusing specifically on parabolas with a vertex at (-2, 0). We will break down the standard forms of quadratic equations, explain how the vertex form reveals the vertex coordinates directly, and meticulously analyze the given options to determine the correct equation. By the end of this discussion, you will have a solid understanding of how to connect the equation of a parabola to its graphical representation, particularly its vertex.

Standard Forms of Quadratic Equations

To effectively identify the equation of a parabola, it's essential to first grasp the standard forms of quadratic equations. The two most commonly used forms are the standard form and the vertex form. Each form provides unique insights into the properties of the parabola.

1. Standard Form: The standard form of a quadratic equation is expressed as:

   $y = ax^2 + bx + c$

   where a, b, and c are constants, and a ≠ 0. While the standard form is useful for various algebraic manipulations, it does not directly reveal the vertex of the parabola. The coefficients a, b, and c do, however, influence the shape and position of the parabola. Specifically, a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and its magnitude affects the parabola's width. The b and c coefficients, together with a, determine the parabola's position in the coordinate plane.

2. Vertex Form: The vertex form of a quadratic equation is given by:

   $y = a(x - h)^2 + k$

   Here, (h, k) represents the vertex of the parabola, and a is the same coefficient as in the standard form, determining the parabola's direction and width. The vertex form is particularly advantageous because it explicitly provides the coordinates of the vertex, making it straightforward to identify the highest or lowest point on the curve. The value of h represents the x-coordinate of the vertex, and k represents the y-coordinate. The sign in front of h is crucial; for example, if the equation is $y = (x - 2)^2$, the x-coordinate of the vertex is 2, and if the equation is $y = (x + 2)^2$, the x-coordinate is -2. Understanding the vertex form is paramount when the goal is to find or verify the vertex of a parabola.

The vertex form directly reveals the vertex of the parabola, which is a key advantage when dealing with problems that specify the vertex coordinates. By recognizing and utilizing the vertex form, one can quickly determine the equation of a parabola that satisfies the given conditions.

Analyzing the Vertex Form: $y = a(x - h)^2 + k$

The vertex form of a quadratic equation, $y = a(x - h)^2 + k$, is instrumental in identifying the vertex of a parabola. The vertex is the point where the parabola changes direction, representing either the minimum or maximum value of the quadratic function. In this form, (h, k) directly corresponds to the vertex coordinates. Let's delve deeper into how each component of the vertex form contributes to the parabola's characteristics.

• (h, k) - The Vertex Coordinates: The values h and k pinpoint the exact location of the vertex on the coordinate plane. The h value shifts the parabola horizontally, while the k value shifts it vertically. It is essential to note the sign convention in the equation: a positive h shifts the parabola to the left, and a negative h shifts it to the right. The k value, on the other hand, behaves more intuitively; a positive k shifts the parabola upwards, and a negative k shifts it downwards. For instance, in the equation $y = (x - 3)^2 + 4$, the vertex is at (3, 4), whereas in the equation $y = (x + 3)^2 - 4$, the vertex is at (-3, -4). Understanding this relationship is crucial for accurately interpreting and constructing quadratic equations.

• a - The Direction and Width of the Parabola: The coefficient a plays a dual role in determining the parabola's shape. First, it dictates whether the parabola opens upwards or downwards. If a > 0, the parabola opens upwards, indicating that the vertex is the minimum point. Conversely, if a < 0, the parabola opens downwards, making the vertex the maximum point. Second, the magnitude of a affects the parabola's width. A larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider parabola. For example, $y = 2x^2$ is narrower than $y = 0.5x^2$. This characteristic is vital for sketching parabolas and understanding their behavior graphically.

The vertex form not only simplifies the identification of the vertex but also provides a clear understanding of how the parabola is positioned and shaped in the coordinate plane. This form is particularly useful when solving problems that involve finding the maximum or minimum value of a quadratic function or determining the equation of a parabola given its vertex and another point.

Step-by-Step Solution for the Given Problem

In this section, we will methodically determine which equation represents a parabola with a vertex at (-2, 0). Our approach involves utilizing the vertex form of a quadratic equation and comparing it with the given options.

1. Recall the Vertex Form: The vertex form of a quadratic equation is $y = a(x - h)^2 + k$, where (h, k) is the vertex of the parabola.

2. Substitute the Given Vertex: We are given that the vertex is at (-2, 0). Therefore, h = -2 and k = 0. Substituting these values into the vertex form, we get:

   $y = a(x - (-2))^2 + 0$

   Simplifying, we have:

   $y = a(x + 2)^2$

   This equation represents all parabolas with a vertex at (-2, 0). The value of a will determine the direction and width of the parabola, but the vertex remains at (-2, 0) regardless of a.

3. Evaluate the Options:

   Now, let's examine each of the given options to see which one matches our derived equation:

   A. $y = -2x^2$: This equation is in standard form. The vertex of this parabola is at (0, 0), not (-2, 0). Thus, this option is incorrect.

   B. $y = (x + 2)^2$: This equation is in vertex form with a = 1, h = -2, and k = 0. Therefore, the vertex is indeed at (-2, 0). This option matches our derived equation and is a potential answer.

   C. $y = (x - 2)^2$: This equation is in vertex form with a = 1, h = 2, and k = 0. The vertex of this parabola is at (2, 0), not (-2, 0). Thus, this option is incorrect.

   D. $y = x^2 - 2$: This equation can be seen as vertex form with a = 1, h = 0, and k = -2. The vertex of this parabola is at (0, -2), not (-2, 0). Thus, this option is incorrect.

4. Select the Correct Option: Based on our analysis, only option B, $y = (x + 2)^2$, has a vertex at (-2, 0). Therefore, option B is the correct answer.

By following this step-by-step approach, we have successfully identified the equation of the parabola with the specified vertex. This method highlights the importance of understanding the vertex form and its direct relationship to the parabola's vertex coordinates.

Detailed Analysis of the Options

To reinforce our understanding, let's delve into a more detailed analysis of each option, highlighting why some are incorrect and affirming why option B is the correct choice. This detailed examination will solidify your ability to identify parabolic equations based on their vertex.

Option A: $y = -2x^2$

This equation is presented in the standard form, $y = ax^2 + bx + c$, where a = -2, b = 0, and c = 0. To find the vertex of a parabola in standard form, we can use the formula for the x-coordinate of the vertex, which is $h = -b / (2a)$. In this case, $h = -0 / (2 * -2) = 0$. Substituting x = 0 into the equation, we find the y-coordinate of the vertex: $y = -2(0)^2 = 0$. Therefore, the vertex of the parabola represented by $y = -2x^2$ is at (0, 0). This clearly does not match the required vertex of (-2, 0), making option A incorrect. Furthermore, the negative coefficient a = -2 indicates that the parabola opens downwards.

Option B: $y = (x + 2)^2$

This equation is in the vertex form, $y = a(x - h)^2 + k$, where a = 1, h = -2, and k = 0. Directly from the vertex form, we can identify the vertex as (h, k) = (-2, 0). This perfectly matches the given vertex. The positive coefficient a = 1 indicates that the parabola opens upwards. Expanding this equation, we get $y = x^2 + 4x + 4$, which is also a valid representation of the same parabola in standard form. The consistency between the vertex form and the given vertex confirms that option B is the correct answer.

Option C: $y = (x - 2)^2$

This equation is also in vertex form, $y = a(x - h)^2 + k$, but here, a = 1, h = 2, and k = 0. Thus, the vertex is at (h, k) = (2, 0). This vertex is different from the required vertex of (-2, 0), making option C incorrect. The parabola opens upwards since a = 1 is positive. Expanding this equation, we obtain $y = x^2 - 4x + 4$, further illustrating that it represents a different parabola than the one we are looking for.

Option D: $y = x^2 - 2$

This equation can be interpreted as being in vertex form with a = 1, h = 0, and k = -2. Therefore, the vertex is at (h, k) = (0, -2). This does not match the required vertex of (-2, 0), making option D incorrect. The positive coefficient a = 1 indicates that the parabola opens upwards. The equation can also be seen as a vertical shift of the basic parabola $y = x^2$ downwards by 2 units.

By thoroughly analyzing each option, we have not only identified the correct answer but also reinforced our understanding of how the equation of a parabola dictates its vertex and orientation. This comprehensive approach ensures that you can confidently tackle similar problems in the future.

Conclusion

In conclusion, identifying the equation of a parabola with a specific vertex requires a solid understanding of the vertex form of quadratic equations. By recognizing that the vertex form, $y = a(x - h)^2 + k$, directly provides the vertex coordinates (h, k), we can efficiently determine the correct equation. In the given problem, the step-by-step analysis and detailed examination of each option clearly demonstrate that option B, $y = (x + 2)^2$, is the only equation representing a parabola with a vertex at (-2, 0).

This process underscores the importance of being familiar with different forms of quadratic equations and their properties. The standard form, vertex form, and factored form each offer unique insights into the parabola's characteristics, such as its vertex, axis of symmetry, and intercepts. Mastering these forms and knowing when to apply them is crucial for success in algebra and beyond.

By practicing and applying these concepts, you can confidently tackle a wide range of problems involving parabolas and quadratic equations. Understanding parabolas is not just about solving mathematical problems; it also provides a foundation for understanding various real-world phenomena, from the trajectory of projectiles to the design of satellite dishes. Therefore, a thorough grasp of these concepts is invaluable.

This exploration has hopefully deepened your understanding of parabolas and equipped you with the tools necessary to identify and analyze their equations effectively. Keep practicing, and you'll become proficient in navigating the world of quadratic functions and their graphical representations.