Finding The Equation Of A Parabola With Vertex At (-3, 2)
In the realm of quadratic functions, the vertex form of a parabola's equation offers a straightforward method to identify key features, such as the vertex itself. Understanding how to convert between different forms of quadratic equations is crucial for solving a variety of mathematical problems. This article delves into the process of determining the equation of a parabola given its vertex, and we'll illustrate this with a specific example.
The core concept revolves around the vertex form of a quadratic equation, which is expressed as y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. The coefficient 'a' dictates the parabola's direction (upward if positive, downward if negative) and its vertical stretch or compression. By substituting the given vertex coordinates into this form and expanding the equation, we can identify the standard form of the quadratic equation. The standard form, y = ax^2 + bx + c, is another common way to represent quadratic functions, and it allows us to determine the coefficients that define the parabola's shape and position.
This article will meticulously walk through the steps of converting from vertex form to standard form, providing a clear understanding of the algebraic manipulations involved. We'll start with the given vertex (-3, 2) and explore how to substitute these values into the vertex form equation. From there, we'll expand the squared term, distribute the coefficient 'a', and combine like terms to arrive at the standard form. By comparing the resulting equation with the provided options, we'll pinpoint the correct equation that represents the parabola with the specified vertex. This process not only reinforces the understanding of quadratic equations but also hones the skills necessary for solving similar problems in mathematics and related fields.
To effectively tackle the problem of finding the equation of a parabola with a specific vertex, it's essential to first grasp the significance of the vertex form of a quadratic equation. The vertex form, expressed as y = a(x - h)^2 + k, provides a direct representation of the parabola's vertex, making it a powerful tool for analysis and problem-solving. In this form, (h, k) precisely denotes the coordinates of the vertex, which is the point where the parabola changes direction—either the minimum or maximum point on the curve. The coefficient 'a' plays a pivotal role in determining the parabola's shape and orientation; if 'a' is positive, the parabola opens upwards, indicating a minimum value at the vertex, while a negative 'a' signifies a downward-opening parabola with a maximum value at the vertex. Furthermore, the magnitude of 'a' influences the parabola's vertical stretch or compression, with larger absolute values resulting in a narrower parabola and smaller values leading to a wider one.
Understanding the vertex form goes beyond simply recognizing its structure; it involves comprehending how each component contributes to the overall characteristics of the parabola. The values of 'h' and 'k' dictate the horizontal and vertical shifts of the parabola from the origin, respectively. For instance, if 'h' is positive, the parabola is shifted to the right, and if 'h' is negative, it's shifted to the left. Similarly, a positive 'k' indicates an upward shift, and a negative 'k' represents a downward shift. The coefficient 'a', as mentioned earlier, not only determines the direction of opening but also affects the parabola's curvature. A larger 'a' value implies a steeper curve, while a smaller 'a' value results in a gentler curve.
The versatility of the vertex form lies in its ability to readily reveal the vertex coordinates and the direction of the parabola, making it invaluable for graphing quadratic functions and solving optimization problems. By simply examining the equation in vertex form, one can quickly identify the parabola's turning point and whether it represents a minimum or maximum value. This information is crucial in various applications, such as determining the maximum height of a projectile or the minimum cost in a production scenario. Moreover, the vertex form serves as a bridge to other forms of quadratic equations, such as the standard form, allowing for seamless conversion and analysis. The ability to convert between vertex form and standard form is a fundamental skill in algebra and calculus, enabling a deeper understanding of quadratic functions and their applications.
The process of converting a quadratic equation from vertex form to standard form is a fundamental skill in algebra, enabling a more comprehensive understanding of the parabola's properties and behavior. The vertex form, y = a(x - h)^2 + k, explicitly reveals the vertex (h, k), while the standard form, y = ax^2 + bx + c, facilitates the identification of the y-intercept and provides a different perspective on the parabola's curvature. The conversion involves algebraic manipulation, primarily focusing on expanding the squared term and simplifying the equation to match the standard form.
The initial step in converting from vertex form to standard form is to expand the squared term, (x - h)^2. This expansion utilizes the binomial square formula, which states that (x - h)^2 = x^2 - 2hx + h^2. By applying this formula, we transform the equation into y = a(x^2 - 2hx + h^2) + k. This step is crucial as it removes the squared term, paving the way for distributing the coefficient 'a' and rearranging the terms into the standard form.
Next, we distribute the coefficient 'a' across the expanded trinomial, resulting in y = ax^2 - 2ahx + ah^2 + k. This step involves multiplying 'a' with each term inside the parentheses, ensuring that the distributive property is correctly applied. The equation now contains individual terms that can be rearranged to match the standard form's structure. The key here is to maintain the correct signs and coefficients during the multiplication process, as any error in this step can lead to an incorrect final equation.
The final step involves combining the constant terms and rearranging the equation to match the standard form y = ax^2 + bx + c. We identify the coefficients of the x^2, x, and constant terms. The coefficient of x^2 remains 'a', the coefficient of x is '-2ah', and the constant term 'c' is the sum of 'ah^2' and 'k'. By rearranging the terms in this order, we obtain the standard form of the quadratic equation. This form allows us to easily identify the y-intercept, which is the value of 'c', and provides a different perspective on the parabola's vertical stretch and position. The standard form is particularly useful for finding the roots of the quadratic equation using the quadratic formula or for analyzing the parabola's symmetry and other properties. The ability to seamlessly convert between vertex form and standard form enhances one's problem-solving capabilities in algebra and calculus, enabling a more thorough understanding of quadratic functions and their applications.
To determine which equation represents a graph with a vertex at (-3, 2), we'll employ the vertex form of a quadratic equation and the process of converting it to standard form. The vertex form, as we know, is given by y = a(x - h)^2 + k, where (h, k) is the vertex. In this case, the vertex is (-3, 2), so we have h = -3 and k = 2. Substituting these values into the vertex form, we get y = a(x - (-3))^2 + 2, which simplifies to y = a(x + 3)^2 + 2.
Now, we need to expand and simplify this equation to match the standard form, y = ax^2 + bx + c. First, we expand the squared term: (x + 3)^2 = x^2 + 6x + 9. Substituting this back into our equation, we have y = a(x^2 + 6x + 9) + 2. Next, we distribute the coefficient 'a': y = ax^2 + 6ax + 9a + 2. This equation is now in a form that closely resembles the standard form, where the coefficient of x^2 is 'a', the coefficient of x is '6a', and the constant term is '9a + 2'.
To find the correct equation among the given options, we need to determine the value of 'a'. The options provided are in standard form, and we can compare their coefficients with our derived equation. Let's consider the options:
A. y = 4x^2 + 24x + 38 B. y = 4x^2 - 24x + 38 C. y = 4x^2 + 12x + 2 D. y = 4x^2 + 16x + 13
By comparing the coefficients of x^2, we can see that in options A, B, C, and D, the value of 'a' is 4. So, let's substitute a = 4 into our derived equation: y = 4x^2 + 6(4)x + 9(4) + 2, which simplifies to y = 4x^2 + 24x + 36 + 2, and further simplifies to y = 4x^2 + 24x + 38.
Comparing this final equation with the given options, we find that it matches option A. Therefore, the equation that represents a graph with a vertex at (-3, 2) is y = 4x^2 + 24x + 38. This process demonstrates how the vertex form and its conversion to standard form can be effectively used to solve problems involving quadratic equations and their graphs.
In conclusion, the journey of finding the equation of a parabola with a given vertex at (-3, 2) underscores the significance of understanding and applying the vertex form of a quadratic equation. By leveraging the vertex form, y = a(x - h)^2 + k, and mastering the technique of converting it to standard form, y = ax^2 + bx + c, we can effectively navigate and solve a wide array of mathematical problems related to parabolas. This exercise not only reinforces the fundamental concepts of quadratic functions but also hones the algebraic skills necessary for more advanced mathematical pursuits.
The process began with recognizing the importance of the vertex form in directly representing the vertex coordinates (h, k). Substituting the given vertex (-3, 2) into the vertex form yielded y = a(x + 3)^2 + 2, a crucial intermediate step. The subsequent conversion to standard form involved expanding the squared term, distributing the coefficient 'a', and combining like terms. This algebraic manipulation transformed the equation into y = ax^2 + 6ax + 9a + 2, a form that allowed for direct comparison with the provided options.
By carefully analyzing the coefficients in the standard form and comparing them with the options, we deduced that the value of 'a' was 4. Substituting a = 4 into our derived equation led to the final equation, y = 4x^2 + 24x + 38, which precisely matched option A. This methodical approach highlights the power of algebraic manipulation and the importance of understanding the relationship between different forms of quadratic equations. The ability to seamlessly convert between vertex form and standard form is a valuable asset in solving quadratic equations and analyzing their graphical representations.
Ultimately, this exercise demonstrates that a solid grasp of quadratic functions, coupled with proficiency in algebraic techniques, empowers us to tackle complex mathematical problems with confidence and precision. The skills acquired through this process extend beyond the specific problem at hand, providing a foundation for understanding and solving a broader range of mathematical challenges in various fields of study and application. The journey from vertex form to standard form not only reveals the equation of a parabola but also illuminates the beauty and interconnectedness of mathematical concepts.
