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

Jul 15, 2025 by ADMIN 56 views

Understanding the Equation of a Parabola with Vertex at (-3, -2)

The question at hand asks us to identify the equation of a parabola given its vertex. The vertex of a parabola is a crucial point, representing either the minimum or maximum value of the quadratic function. In this case, we are told that the vertex is located at the point (-3, -2). To determine the correct equation, we need to understand the vertex form of a parabola's equation and how the vertex coordinates relate to that form.

Vertex Form of a Parabola

The vertex form of a parabola's equation is given by:

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

where:

(h, k) represents the coordinates of the vertex.
a determines the direction the parabola opens (upward if a > 0, downward if a < 0) and the width of the parabola. A larger absolute value of 'a' means a narrower parabola, while a smaller absolute value means a wider parabola.

Understanding this form is fundamental to solving this problem. It allows us to directly plug in the vertex coordinates and narrow down the possibilities. By recognizing the structure of this equation, we can quickly assess which of the given options could represent a parabola with the specified vertex.

Applying the Vertex Information

We are given that the vertex of the parabola is at (-3, -2). This means that h = -3 and k = -2. Let's substitute these values into the vertex form equation:

y = a(x - (-3))^2 + (-2)

Simplifying, we get:

y = a(x + 3)^2 - 2

Now, we need to compare this general form with the options provided in the question to see which one matches. The crucial part here is the (x + 3)^2 term and the -2 constant term. These directly reflect the x and y coordinates of the vertex, respectively. The coefficient 'a' will determine the parabola's direction and width, but the vertex coordinates are fixed by the (x + 3) and -2 terms.

Analyzing the Options

Let's examine the provided options in light of our derived equation:

Option A: y = -2(x - 3)^2 + 2 This option has (x - 3)^2, which indicates a vertex x-coordinate of 3, not -3. Also, the +2 indicates a vertex y-coordinate of 2, not -2. Therefore, this option is incorrect.
Option B: y = -2(x + 3)^2 + 2 This option correctly has (x + 3)^2, indicating the x-coordinate of the vertex is -3. However, the +2 indicates a y-coordinate of 2, not -2. Thus, this option is also incorrect.
Option C: y = -2(x + 3)^2 - 2 This option has (x + 3)^2, correctly indicating the x-coordinate of the vertex is -3, and -2, correctly indicating the y-coordinate of the vertex is -2. The coefficient -2 simply tells us the parabola opens downward and is a bit narrower than the standard parabola. Therefore, this option is the most likely to be correct.

By carefully analyzing each option and comparing it with the vertex form of the equation, we can definitively identify the correct answer. This methodical approach, based on a solid understanding of the underlying concepts, is key to success in mathematics.

Importance of the Coefficient 'a'

While we focused primarily on the (h, k) values to determine the vertex, it's important to briefly discuss the role of the coefficient 'a'. In the correct option (Option C), 'a' is -2. This negative value tells us that the parabola opens downward. If 'a' were positive, the parabola would open upward. The absolute value of 'a' (in this case, |-2| = 2) affects the 'width' of the parabola. A larger absolute value means the parabola is narrower, while a smaller absolute value means it is wider. In the context of this problem, the value of 'a' helps us understand the overall shape of the parabola, but the crucial element for determining the correct equation is still the vertex (h, k).

Understanding the interplay between 'a' and the vertex coordinates provides a complete picture of how the equation defines the parabola's shape and position in the coordinate plane. This comprehensive understanding is invaluable for tackling more complex problems involving parabolas and quadratic functions.

Conclusion

The vertex form of a parabola's equation is a powerful tool for understanding and manipulating quadratic functions. By recognizing the relationship between the vertex coordinates (h, k) and the equation y = a(x - h)^2 + k, we can quickly identify the equation of a parabola given its vertex. In this case, option C, y = -2(x + 3)^2 - 2, is the correct answer because it accurately reflects the vertex at (-3, -2). This problem highlights the importance of understanding the fundamental forms of equations in mathematics and how they relate to graphical representations. Mastering these concepts will pave the way for success in more advanced mathematical topics.

Determining the equation of a parabola when provided with its vertex is a common problem in algebra and precalculus. The vertex, which represents the point where the parabola changes direction, holds significant information about the quadratic function. Understanding the vertex form of a parabola's equation is crucial for solving such problems effectively. This discussion will delve into the process of identifying the correct equation given the vertex, highlighting key concepts and problem-solving strategies.

Understanding the Vertex and Its Significance

The vertex of a parabola is the point where the curve changes direction. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, the vertex is the maximum point. The coordinates of the vertex, often denoted as (h, k), are directly related to the equation of the parabola. The x-coordinate (h) represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The y-coordinate (k) represents the minimum or maximum value of the quadratic function.

The vertex is a critical feature of a parabola, and its location provides substantial information about the quadratic function it represents. Knowing the vertex allows us to write the equation of the parabola in a specific form that simplifies analysis and problem-solving. This form, known as the vertex form, is the key to solving the problem at hand.

The Vertex Form of a Quadratic Equation

The vertex form of a parabola's equation is given by:

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

where:

(h, k) are the coordinates of the vertex.
a is a constant that determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards. 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 results in a wider parabola.

This form is extremely useful because it directly incorporates the vertex coordinates. Once we know the vertex (h, k), we can substitute these values into the equation. The remaining unknown is 'a', which can be determined if we have an additional point on the parabola or other information about its shape. However, in this particular problem, we are only given the vertex, so we will focus on matching the (h, k) values in the options provided.

Applying the Vertex Form to the Problem

In the problem, we are given that the vertex of the parabola is at (-3, -2). This means h = -3 and k = -2. Substituting these values into the vertex form equation, we get:

y = a(x - (-3))^2 + (-2)

Simplifying this equation, we have:

y = a(x + 3)^2 - 2

Now, we need to compare this equation with the given options and identify the one that matches this form. The crucial elements to look for are the (x + 3)^2 term, which indicates h = -3, and the -2 constant term, which indicates k = -2. The coefficient 'a' can vary, but the vertex coordinates must match.

Analyzing the Multiple-Choice Options

Let's analyze the given options in the context of the vertex form equation we derived:

Option A: y = -2(x - 3)^2 + 2 This option has (x - 3)^2, which corresponds to a vertex x-coordinate of h = 3, not -3. The constant term is +2, indicating a vertex y-coordinate of k = 2, not -2. Therefore, this option is incorrect.
Option B: y = -2(x + 3)^2 + 2 This option correctly has (x + 3)^2, indicating a vertex x-coordinate of h = -3. However, the constant term is +2, indicating a vertex y-coordinate of k = 2, not -2. Thus, this option is also incorrect.
Option C: y = -2(x + 3)^2 - 2 This option has (x + 3)^2, correctly indicating a vertex x-coordinate of h = -3, and a constant term of -2, correctly indicating a vertex y-coordinate of k = -2. The coefficient a = -2 indicates that the parabola opens downwards. This option aligns perfectly with the vertex form equation we derived and the given vertex coordinates.

By systematically comparing each option with the vertex form equation, we can confidently identify the correct answer. This approach emphasizes the importance of understanding the fundamental forms of equations and how they relate to graphical features.

The Role of the Coefficient 'a' in the Equation

While the vertex coordinates (h, k) are the primary focus when determining the equation given the vertex, the coefficient 'a' plays an important role in defining the parabola's shape and direction. In the correct option (Option C), a = -2. The negative sign indicates that the parabola opens downwards, and the magnitude of 2 suggests that the parabola is narrower than the standard parabola (y = x^2).

Understanding the influence of 'a' on the parabola's shape provides a deeper understanding of quadratic functions. It allows us to visualize the parabola's orientation and relative width, which can be helpful in various applications, such as modeling projectile motion or optimizing certain processes.

Conclusion: Mastering the Vertex Form

The vertex form of a parabola's equation is a powerful tool for solving problems involving quadratic functions. By understanding the relationship between the vertex coordinates (h, k) and the equation y = a(x - h)^2 + k, we can efficiently determine the equation of a parabola given its vertex. In this case, Option C, y = -2(x + 3)^2 - 2, is the correct answer because it accurately represents a parabola with a vertex at (-3, -2). This problem illustrates the significance of mastering fundamental equation forms in mathematics and their connection to graphical representations. Proficiency in these concepts will facilitate success in more advanced mathematical studies.

Finding the equation of a parabola when its vertex is known is a fundamental skill in algebra. The vertex, being the turning point of the parabola, provides key information about the quadratic function. This discussion aims to provide a comprehensive understanding of how to identify the correct equation of a parabola given its vertex, emphasizing the importance of the vertex form of a quadratic equation.

The Significance of the Vertex in Parabola Equations

The vertex of a parabola is the point where the curve changes direction. It represents either the minimum value (for parabolas opening upwards) or the maximum value (for parabolas opening downwards) of the quadratic function. The vertex coordinates are typically represented as (h, k), where 'h' is the x-coordinate and 'k' is the y-coordinate.

The vertex is a crucial element in defining a parabola because it directly corresponds to the shift and vertical position of the parabola from the basic form y = x^2. Knowing the vertex coordinates allows us to express the parabola's equation in a form that explicitly shows these shifts, making it easier to analyze and manipulate the function.

Understanding the Vertex Form of a Parabola

The vertex form of a parabola's equation is expressed as:

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

where:

(h, k) represents the coordinates of the vertex.
'a' is a coefficient that determines the direction and the width of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards. The absolute value of 'a' indicates the 'steepness' or 'flatness' of the parabola; a larger |a| means a narrower parabola, and a smaller |a| means a wider parabola.

This form is incredibly useful because it directly incorporates the vertex coordinates (h, k). By substituting the known values of h and k into the equation, we can simplify the problem and focus on identifying the correct form among the given options. The coefficient 'a' will further refine the shape and direction of the parabola, but the vertex coordinates are the primary identifiers in this case.

Applying Vertex Information to Find the Equation

In the given problem, the vertex of the parabola is at (-3, -2). This means that h = -3 and k = -2. We substitute these values into the vertex form equation:

y = a(x - (-3))^2 + (-2)

Simplifying the equation, we get:

y = a(x + 3)^2 - 2

Now, we compare this general form with the multiple-choice options to find the one that matches. The critical parts to match are the (x + 3)^2 term, indicating the x-coordinate of the vertex is -3, and the -2 term, indicating the y-coordinate of the vertex is -2. The coefficient 'a' can vary among the options, but these vertex terms must be correct.

Step-by-Step Analysis of the Options

Let's examine each option and compare it to the equation we derived:

Option A: y = -2(x - 3)^2 + 2 This option has (x - 3)^2, which indicates a vertex x-coordinate of 3 (not -3), and +2, which indicates a vertex y-coordinate of 2 (not -2). Thus, this option is incorrect.
Option B: y = -2(x + 3)^2 + 2 This option has (x + 3)^2, which correctly indicates a vertex x-coordinate of -3. However, it has +2, which indicates a vertex y-coordinate of 2 (not -2). So, this option is also incorrect.
Option C: y = -2(x + 3)^2 - 2 This option has (x + 3)^2, correctly indicating a vertex x-coordinate of -3, and -2, correctly indicating a vertex y-coordinate of -2. The coefficient -2 indicates that the parabola opens downward. This option perfectly matches the derived equation and the given vertex coordinates, making it the correct choice.

By methodically comparing each option with the vertex form equation, we can confidently determine the correct answer. This process underscores the value of understanding and applying fundamental equation forms in problem-solving.

The Impact of the Leading Coefficient 'a'

While the vertex coordinates (h, k) are essential for identifying the equation, the coefficient 'a' provides additional information about the parabola's shape and direction. In the correct option (Option C), a = -2. The negative sign signifies that the parabola opens downwards, and the magnitude of 2 implies that it is narrower than the standard parabola y = x^2.

Grasping the influence of 'a' on the parabola's graphical representation offers a more complete understanding of quadratic functions. It allows us to visualize the parabola's orientation and its relative 'steepness', which is valuable in numerous applications, such as modeling trajectories or optimizing geometric shapes.

Final Thoughts: Mastering Parabola Equations

The vertex form of a parabola's equation is a powerful tool for tackling problems involving quadratic functions. By understanding the direct relationship between the vertex coordinates (h, k) and the equation y = a(x - h)^2 + k, we can efficiently identify the equation of a parabola when its vertex is given. In this scenario, Option C, y = -2(x + 3)^2 - 2, is the correct equation for a parabola with a vertex at (-3, -2). This example reinforces the significance of mastering key equation forms in mathematics and their connection to graphical representations. Developing proficiency in these concepts will enable success in more advanced mathematical explorations.