Finding The Equation Of A Parabola Given Vertex (-3, -2)

In the realm of quadratic functions, parabolas hold a prominent position as U-shaped curves defined by a second-degree polynomial equation. The vertex of a parabola, representing its minimum or maximum point, plays a crucial role in determining its equation and overall characteristics. Understanding the vertex form of a parabola empowers us to decipher its equation when the vertex coordinates are known. This article delves into the intricacies of identifying the equation of a parabola given its vertex, focusing on a specific scenario where the vertex is located at (-3, -2). We will explore the vertex form equation, analyze how the vertex coordinates are incorporated, and meticulously examine the provided options to pinpoint the correct equation. The journey involves understanding the relationship between the vertex, the equation's coefficients, and the parabola's graphical representation.

When confronted with the task of determining a parabola's equation, the vertex form serves as an indispensable tool. This particular form, expressed as y = a(x - h)^2 + k, directly incorporates the vertex coordinates (h, k) into the equation. The parameter a governs the parabola's direction (upward if a > 0, downward if a < 0) and its vertical stretch or compression. In our case, the vertex is situated at (-3, -2), implying that h = -3 and k = -2. Substituting these values into the vertex form, we obtain y = a(x - (-3))^2 + (-2), which simplifies to y = a(x + 3)^2 - 2. The remaining task involves identifying the appropriate value for a from the provided options. The value of a dictates the parabola's concavity and width. A negative a value indicates a downward-opening parabola, while a positive a value corresponds to an upward-opening parabola. The magnitude of a influences the parabola's width; a larger magnitude results in a narrower parabola, and a smaller magnitude leads to a wider parabola. Considering the given options, we can narrow down the possibilities based on the vertex coordinates and the general shape of the parabola. The vertex form provides a direct link between the vertex coordinates and the equation, making it a powerful tool for analysis. Further, understanding the role of the coefficient 'a' is crucial in determining the parabola's orientation and stretch.

Now, let's meticulously examine the provided options, keeping in mind the vertex form equation y = a(x + 3)^2 - 2 and the vertex coordinates (-3, -2):

• Option A: y = -2(x - 3)^2 - 2

  This equation, however, presents a discrepancy in the x-coordinate within the parentheses. Instead of (x + 3), it features (x - 3), indicating a vertex at (3, -2), which contradicts the given vertex of (-3, -2). This discrepancy immediately disqualifies Option A as a potential solution. The sign difference within the parenthesis is crucial, as it directly affects the x-coordinate of the vertex. A negative sign implies a shift to the right, while a positive sign indicates a shift to the left. Thus, even though the y-coordinate matches, the incorrect x-coordinate makes this option invalid. Furthermore, the coefficient -2 suggests a downward-opening parabola, which is consistent with the other options but doesn't compensate for the incorrect vertex.

• Option B: y = -2(x + 3)^2 - 2

  This equation precisely aligns with the vertex form y = a(x + 3)^2 - 2, incorporating the correct x-coordinate of -3 within the parentheses. The y-coordinate of -2 is also accurately represented. Furthermore, the coefficient a = -2 signifies a downward-opening parabola, a characteristic that doesn't contradict any given information. Therefore, Option B emerges as a strong contender and warrants further scrutiny to confirm its validity. The coefficient -2 also indicates a vertical stretch, making the parabola narrower compared to the standard parabola y = x^2. This option perfectly captures the essence of the vertex form with the correct vertex coordinates and a suitable leading coefficient. Consequently, it stands out as the most likely candidate for the parabola's equation.

• Option C: y = -2(x - 3)^2 + 2

  Similar to Option A, this equation falters in representing the x-coordinate within the parentheses. The (x - 3) term suggests a vertex x-coordinate of 3, deviating from the specified -3. Additionally, the y-coordinate in this equation is +2, contradicting the given vertex y-coordinate of -2. These dual inconsistencies unequivocally eliminate Option C from consideration. The positive sign for the y-coordinate shifts the vertex upward, which is inconsistent with the problem statement. The combination of the incorrect x and y coordinates makes this option clearly wrong. Therefore, option C can be safely discarded as it does not represent the parabola with the given vertex.

• Option D: y = -2(x + 3)^2 + 2

  While this equation accurately captures the x-coordinate of the vertex as -3, it errs in the y-coordinate representation. The equation displays a y-coordinate of +2, which contradicts the given vertex y-coordinate of -2. This single discrepancy renders Option D incorrect. The incorrect y-coordinate represents a vertical shift of the parabola, moving it upwards instead of downwards as required by the given vertex. Even though the x-coordinate is correctly represented, the incorrect y-coordinate disqualifies this option. Therefore, option D cannot be the equation of the parabola with the specified vertex.

After a meticulous analysis of all the options, Option B: y = -2(x + 3)^2 - 2 emerges as the sole equation that accurately represents a parabola with a vertex at (-3, -2). The equation adheres to the vertex form, incorporates the correct vertex coordinates, and possesses a coefficient that indicates a downward-opening parabola. This comprehensive alignment solidifies Option B as the definitive solution.

The vertex of a parabola serves as a cornerstone in understanding its properties and behavior. It represents the point where the parabola changes direction, transitioning from decreasing to increasing or vice versa. For a downward-opening parabola (where a < 0), the vertex signifies the maximum point, while for an upward-opening parabola (where a > 0), it represents the minimum point. The vertex also plays a pivotal role in determining the axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation of the axis of symmetry is given by x = h, where (h, k) are the vertex coordinates. In our case, the axis of symmetry is x = -3. Understanding the vertex not only helps in identifying the equation but also provides insights into the parabola's overall shape and position in the coordinate plane. The vertex, along with the coefficient 'a', completely defines the parabola in vertex form, highlighting its importance in quadratic function analysis.

The vertex also plays a significant role in real-world applications of parabolas. For instance, the trajectory of a projectile, such as a ball thrown into the air, often follows a parabolic path. The vertex in this scenario represents the highest point the projectile reaches. Similarly, the design of satellite dishes and parabolic reflectors relies on the properties of parabolas and their vertices to focus incoming signals or light rays. The vertex form of the equation provides a convenient way to model these situations and make predictions. By knowing the vertex and one other point on the parabola, one can uniquely determine the equation and analyze the behavior of the system. Therefore, a strong understanding of the vertex and its significance is crucial for both theoretical and practical applications of parabolas.

In conclusion, determining the equation of a parabola given its vertex involves a thorough understanding of the vertex form equation and a meticulous analysis of the provided options. The vertex form, y = a(x - h)^2 + k, directly incorporates the vertex coordinates (h, k), simplifying the equation identification process. By substituting the given vertex coordinates into the vertex form and carefully examining the remaining parameters, one can effectively pinpoint the correct equation. In the specific scenario discussed, where the vertex is located at (-3, -2), Option B: y = -2(x + 3)^2 - 2 emerged as the definitive solution after a rigorous evaluation of all possibilities. This exercise underscores the importance of mastering the vertex form and its application in solving parabola-related problems. The ability to correctly identify and interpret the vertex form is a fundamental skill in algebra and precalculus, paving the way for understanding more complex mathematical concepts. Furthermore, the understanding gained in this process extends to practical applications in physics, engineering, and other fields where parabolic relationships are prevalent.

Ultimately, the key to successfully working with parabolas lies in a deep understanding of their properties and the ability to translate these properties into mathematical equations. The vertex form provides a powerful framework for this translation, making it an indispensable tool in the study of quadratic functions and their applications. By consistently applying the principles discussed in this article, one can confidently tackle a wide range of problems involving parabolas and their equations.