Rewriting Y=-3x²-12x-2 In Vertex Form A Comprehensive Guide

Understanding quadratic equations is crucial in mathematics, especially when dealing with parabolas. The vertex form of a quadratic equation provides valuable insights into the properties of the parabola, such as its vertex and axis of symmetry. This article delves into the process of converting a quadratic equation from its standard form to vertex form, focusing on the equation y = -3x² - 12x - 2. We will explore the steps involved in completing the square, a fundamental technique used to rewrite quadratic equations in vertex form. By understanding this transformation, you can easily identify the vertex of the parabola and gain a better understanding of its graphical representation.

The standard form of a quadratic equation is given by y = ax² + bx + c, where a, b, and c are constants. In our case, the equation y = -3x² - 12x - 2 is in standard form, with a = -3, b = -12, and c = -2. The vertex form, on the other hand, is expressed as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is a critical point on the parabola, representing either the maximum or minimum value of the quadratic function. Converting from standard form to vertex form involves a process called completing the square, which we will explore in detail.

To begin the transformation, we first factor out the coefficient of the x² term, which is a = -3, from the first two terms of the equation. This gives us y = -3(x² + 4x) - 2. The next step is to complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (which is 4), square it (which gives us 4), and add it inside the parentheses. However, since we're adding it inside the parentheses that are being multiplied by -3, we must also subtract the same amount multiplied by -3 outside the parentheses to maintain the equation's balance. This process ensures that we are not changing the overall value of the equation, only its form. The completed square allows us to rewrite the quadratic expression as a perfect square, making it easier to identify the vertex.

H2: Step-by-Step Conversion of y=-3x²-12x-2 to Vertex Form

Converting quadratic equations requires a systematic approach. To convert the quadratic equation y = -3x² - 12x - 2 into vertex form, we follow a step-by-step process known as completing the square. This method allows us to rewrite the equation in the form y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Understanding this process is not just about finding the right answer; it's about grasping the underlying mathematical principles that govern quadratic equations and their graphical representations. Let's break down the conversion into manageable steps:

1. Factor out the coefficient of the x² term: The first step involves factoring out the coefficient of the x² term, which in this case is -3, from the first two terms of the equation. This gives us y = -3(x² + 4x) - 2. Factoring out the leading coefficient is crucial because it isolates the quadratic and linear terms, making it easier to complete the square. This step sets the stage for creating a perfect square trinomial within the parentheses, which is the key to converting to vertex form. Remember, the goal is to rewrite the quadratic expression in a way that highlights the vertex of the parabola.

2. Complete the square: Inside the parentheses, we need to complete the square. This involves taking half of the coefficient of the x term (which is 4), squaring it (which gives us 4), and adding it inside the parentheses. However, because we're adding this value inside the parentheses that are being multiplied by -3, we must also subtract the same amount multiplied by -3 outside the parentheses to keep the equation balanced. This means we add (-3) * 4 = -12 inside the parentheses and subtract -12 outside, which is equivalent to adding 12. The equation now looks like this: y = -3(x² + 4x + 4) - 2 + 12. Completing the square is a technique that transforms a quadratic expression into a perfect square trinomial, which can then be factored into a squared binomial.

3. Rewrite as a perfect square: Now we can rewrite the expression inside the parentheses as a perfect square. The expression x² + 4x + 4 is equivalent to (x + 2)². So the equation becomes y = -3(x + 2)² - 2 + 12. Rewriting the expression as a perfect square is a critical step in identifying the vertex. The squared binomial directly corresponds to the h value in the vertex form equation, which represents the x-coordinate of the vertex.

4. Simplify the equation: Finally, we simplify the equation by combining the constant terms outside the parentheses. In this case, -2 + 12 = 10. So the equation in vertex form is y = -3(x + 2)² + 10. Simplifying the equation to its final vertex form makes it easy to read off the vertex coordinates and understand the parabola's characteristics, such as its direction and vertical stretch.

By following these steps, we have successfully converted the quadratic equation y = -3x² - 12x - 2 into vertex form, which is y = -3(x + 2)² + 10. This form allows us to easily identify the vertex of the parabola as (-2, 10). Understanding the process of completing the square is essential for working with quadratic equations and their applications in various mathematical and real-world scenarios. This technique provides a powerful tool for analyzing and interpreting quadratic functions.

H2: Identifying the Vertex from the Vertex Form

The vertex form of a quadratic equation, y = a(x - h)² + k, is particularly useful because it directly reveals the vertex of the parabola represented by the equation. The vertex is the point where the parabola changes direction, and it represents either the maximum or minimum value of the quadratic function. In the vertex form, the vertex coordinates are given by (h, k). Understanding how to extract the vertex from the equation is crucial for graphing parabolas and analyzing their properties.

In our example, the equation in vertex form is y = -3(x + 2)² + 10. Comparing this to the general vertex form y = a(x - h)² + k, we can identify the values of h and k. Notice that the equation has (x + 2), which can be rewritten as (x - (-2)). Therefore, h = -2 and k = 10. This means that the vertex of the parabola is at the point (-2, 10). The vertex is a critical point on the parabola, and its coordinates provide valuable information about the function's behavior.

The value of a in the vertex form, which is -3 in our example, also provides important information about the parabola. Since a is negative, the parabola opens downward, indicating that the vertex is the maximum point of the function. If a were positive, the parabola would open upward, and the vertex would be the minimum point. The absolute value of a also determines the vertical stretch of the parabola. A larger absolute value of a means the parabola is stretched vertically, making it narrower, while a smaller absolute value makes it wider.

Identifying the vertex allows us to easily sketch the graph of the parabola. We know the vertex is at (-2, 10), and since the parabola opens downward, we can visualize the shape of the curve. We can also find additional points on the parabola by plugging in different values of x into the equation. For example, we can find the y-intercept by setting x = 0 in the original equation y = -3x² - 12x - 2, which gives us y = -2. This provides another point (0, -2) on the parabola. By plotting the vertex and a few additional points, we can create an accurate graph of the quadratic function.

In summary, the vertex form y = a(x - h)² + k provides a direct way to identify the vertex of a parabola, which is the point (h, k). The sign of a determines whether the parabola opens upward or downward, and the absolute value of a affects the vertical stretch. Understanding these properties allows us to quickly analyze and graph quadratic functions. The ability to convert between standard form and vertex form is a fundamental skill in algebra and calculus, enabling us to solve a wide range of problems involving quadratic relationships.

H2: Comparing Vertex Form Options for y=-3x²-12x-2

When converting a quadratic equation to vertex form, it's essential to verify that the resulting equation is equivalent to the original. In this case, we've converted y = -3x² - 12x - 2 to y = -3(x + 2)² + 10. Now, let's consider the given options and compare them to our result to determine the correct answer. This comparison involves understanding how small differences in the equation can significantly impact the graph of the parabola and its vertex.

Option A is y = -3(x + 2)² + 10. This matches the vertex form we derived through completing the square. The vertex of this parabola is (-2, 10), and the parabola opens downward since the coefficient of the squared term is negative. This option appears to be the correct answer, but we should still compare it to the other options to ensure there are no discrepancies.

Option B is y = -3(x - 2)² + 10. This equation has a different vertex compared to Option A. The vertex for this parabola would be (2, 10). The change in the sign inside the parentheses shifts the parabola horizontally. Expanding this equation, we get y = -3(x² - 4x + 4) + 10 = -3x² + 12x - 12 + 10 = -3x² + 12x - 2. This is not the same as the original equation y = -3x² - 12x - 2, so Option B is incorrect.

Option C is y = -3(x + 2)² - 14. This equation also has a different vertex compared to Option A. The vertex for this parabola would be (-2, -14). The difference in the constant term shifts the parabola vertically. Expanding this equation, we get y = -3(x² + 4x + 4) - 14 = -3x² - 12x - 12 - 14 = -3x² - 12x - 26. This is not the same as the original equation y = -3x² - 12x - 2, so Option C is incorrect.

Option D is y = -3(x - 2)² - 2. This equation has a vertex of (2, -2). Expanding this equation, we get y = -3(x² - 4x + 4) - 2 = -3x² + 12x - 12 - 2 = -3x² + 12x - 14. This is also not the same as the original equation y = -3x² - 12x - 2, so Option D is incorrect.

By comparing the vertex form options and expanding them to verify their equivalence to the original equation, we can confidently conclude that Option A, y = -3(x + 2)² + 10, is the correct answer. This exercise highlights the importance of careful comparison and verification when working with quadratic equations and their transformations. Understanding how changes in the vertex form affect the graph of the parabola is crucial for accurately analyzing and interpreting quadratic functions.

H2: Conclusion

In conclusion, converting a quadratic equation to vertex form is a fundamental skill in algebra that provides valuable insights into the properties of the parabola represented by the equation. The vertex form, y = a(x - h)² + k, directly reveals the vertex of the parabola as (h, k), which is essential for graphing and analyzing quadratic functions. By completing the square, we can transform a quadratic equation from its standard form to vertex form, allowing us to easily identify the vertex and understand the parabola's characteristics.

In the specific example of y = -3x² - 12x - 2, we demonstrated the step-by-step process of completing the square to obtain the vertex form y = -3(x + 2)² + 10. This form clearly shows that the vertex of the parabola is at (-2, 10). The negative coefficient of the squared term indicates that the parabola opens downward, making the vertex the maximum point of the function. This comprehensive guide has walked through the process, ensuring a clear understanding of each step involved.

We also compared the vertex form options provided and verified their equivalence to the original equation. By expanding each option and comparing it to the original equation, we confirmed that Option A, y = -3(x + 2)² + 10, is the correct answer. This exercise emphasized the importance of careful verification and comparison when working with quadratic equations and their transformations.

Understanding the vertex form and the process of completing the square is crucial for solving a wide range of mathematical problems, including optimization problems, projectile motion analysis, and curve fitting. The ability to convert between standard form and vertex form empowers us to analyze and interpret quadratic functions effectively. This skill is not only essential for academic success in mathematics but also for practical applications in various fields of science and engineering. The knowledge and techniques discussed in this article provide a solid foundation for further exploration of quadratic functions and their applications.