Vertex Form Explained Convert G(x)=2x^2+12x+21 And Find Vertex

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In the realm of quadratic equations, the vertex form stands out as a particularly insightful representation. It provides a clear and direct way to identify the vertex of the parabola, which is a crucial point for understanding the behavior and characteristics of the quadratic function. In this article, we will embark on a journey to transform the given quadratic equation, g(x) = 2x² + 12x + 21, into its vertex form and, in doing so, pinpoint the coordinates of the vertex. Understanding vertex form not only simplifies graphing quadratic equations but also offers a profound understanding of their properties, such as the minimum or maximum value of the function. The journey to rewrite quadratics into vertex form involves a process called completing the square, a powerful algebraic technique that allows us to rewrite the quadratic expression in a way that reveals the vertex. Let's delve into the step-by-step process of converting the equation and unveil the vertex that lies at the heart of this parabola.

Transforming to Vertex Form: A Step-by-Step Guide

The key to converting a quadratic equation into vertex form lies in a technique called completing the square. This method allows us to rewrite the quadratic expression as a squared term plus a constant, which directly corresponds to the vertex form g(x) = a(x - h)² + k, where (h, k) represents the vertex. This transformation is not merely a mathematical exercise; it's a journey into understanding the very essence of the parabola, revealing its most critical point: the vertex. Let's embark on this step-by-step transformation, unlocking the secrets held within the equation g(x) = 2x² + 12x + 21. The journey begins with factoring out the leading coefficient from the terms containing x², followed by the crucial step of completing the square, and finally, simplifying the expression to reveal the vertex form. Each step is a careful maneuver, a mathematical dance that brings us closer to unveiling the parabola's vertex. This process is not just about manipulating numbers; it's about gaining a deeper appreciation for the structure and properties of quadratic equations.

  1. Factor out the leading coefficient: Begin by factoring out the coefficient of the x² term (which is 2 in this case) from the first two terms of the equation. This step sets the stage for completing the square.

    • g(x) = 2(x² + 6x) + 21

  2. Completing the square: Inside the parentheses, complete the square by taking half of the coefficient of the x term (which is 6), squaring it (3² = 9), and adding and subtracting it within the parentheses. This maintains the equation's balance while setting up the squared term. Completing the square is the heart of the transformation process. It's a technique that allows us to rewrite a quadratic expression as a perfect square trinomial, which can then be factored into a squared binomial. The addition and subtraction of the same value within the parentheses is a clever maneuver that ensures we don't change the value of the expression while manipulating its form.

    • g(x) = 2(x² + 6x + 9 - 9) + 21

  3. Rewrite as a squared term: Rewrite the trinomial (x² + 6x + 9) as a squared binomial (x + 3)². This is the culmination of the completing the square process, where the perfect square trinomial is elegantly expressed as a squared binomial. This step is not just a simplification; it's a revelation, as it brings us closer to the vertex form of the equation.

    • g(x) = 2((x + 3)² - 9) + 21

  4. Distribute and simplify: Distribute the 2 back into the parentheses and simplify the expression. This involves multiplying the 2 by both the squared term and the constant, and then combining the constant terms. This step is a matter of tidying up the expression, ensuring that all the terms are properly accounted for and combined. It's a crucial step in arriving at the final vertex form.

    • g(x) = 2(x + 3)² - 18 + 21
    • g(x) = 2(x + 3)² + 3

Unveiling the Vertex: Decoding the Coordinates

Now that we have successfully transformed the quadratic equation into vertex form, g(x) = 2(x + 3)² + 3, the vertex is readily apparent. The vertex form g(x) = a(x - h)² + k provides a direct mapping to the vertex coordinates, where (h, k) represents the vertex. In our transformed equation, we can identify the values of h and k by carefully comparing it to the general vertex form. The value of h is the x-coordinate of the vertex, and the value of k is the y-coordinate. This direct correspondence between the vertex form and the vertex coordinates is one of the key advantages of this representation. It allows us to quickly and easily identify the vertex without having to resort to more complex methods. The vertex is not just a point; it's a landmark, a critical point that defines the parabola's shape and position in the coordinate plane. It's the point where the parabola changes direction, and it represents either the minimum or maximum value of the quadratic function.

Comparing our equation g(x) = 2(x + 3)² + 3 with the general vertex form, we can deduce the following:

  • h = -3 (Note the sign change due to the (x - h) form)
  • k = 3

Therefore, the vertex of the parabola represented by the equation g(x) = 2x² + 12x + 21 is (-3, 3). This point is not just a coordinate; it's the heart of the parabola, the point around which the entire curve is symmetrically shaped. The vertex is the key to understanding the parabola's behavior, its direction, and its minimum or maximum value. It's a fundamental concept in the study of quadratic equations and their graphical representations.

Conclusion: The Power of Vertex Form

In conclusion, we have successfully navigated the transformation of the quadratic equation g(x) = 2x² + 12x + 21 into its vertex form, g(x) = 2(x + 3)² + 3. This transformation, achieved through the powerful technique of completing the square, has not only rewritten the equation but has also unveiled the vertex of the parabola, which we have identified as (-3, 3). The vertex form is more than just a different way of writing a quadratic equation; it's a key that unlocks a deeper understanding of the parabola's properties. It provides a direct and intuitive way to identify the vertex, which is the most critical point on the graph. The vertex is the turning point of the parabola, and its coordinates reveal the minimum or maximum value of the quadratic function. Understanding vertex form empowers us to quickly analyze and interpret quadratic equations, making it an invaluable tool in mathematics and various applications. The journey of converting to vertex form and identifying the vertex is a testament to the elegance and power of algebraic manipulation. It's a process that not only yields a specific answer but also deepens our understanding of the underlying mathematical concepts. The vertex form is a window into the heart of the parabola, revealing its secrets and making its behavior readily understandable.

Final Answer:

  • Writing in the vertex form: g(x) = 2(x + 3)² + 3
  • Vertex: (-3, 3)