Converting $y=9x^2+9x-1$ To Vertex Form A Step-by-Step Guide

Jul 11, 2025 by Admin 61 views

Which Equation is $y=9x^2+9x-1$ Rewritten in Vertex Form?

Understanding Vertex Form

When we delve into quadratic equations, understanding the various forms in which they can be expressed is crucial for analysis and application. One such form is the vertex form, which provides a clear insight into the quadratic's key features, namely its vertex. The vertex form of a quadratic equation is generally given by y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction, marking either the minimum or maximum value of the quadratic function. The coefficient 'a' determines the direction and width of the parabola; if 'a' is positive, the parabola opens upwards, indicating a minimum value, while a negative 'a' means it opens downwards, indicating a maximum value. The absolute value of 'a' also affects the parabola's width; a larger absolute value makes the parabola narrower, and a smaller one makes it wider.

In the context of our problem, we have the quadratic equation y = 9x² + 9x - 1, which is in standard form (y = ax² + bx + c). Our goal is to transform this equation into vertex form. The process involves completing the square, a technique that allows us to rewrite the quadratic expression as a squared term plus a constant. This method is fundamental in algebra and is widely used in various mathematical contexts, including calculus and complex analysis. By completing the square, we can identify the vertex (h, k) and rewrite the equation in the desired vertex form, making it easier to analyze and graph the quadratic function.

To effectively convert the standard form to vertex form, we must follow a systematic approach. First, we factor out the coefficient of the x² term from the first two terms of the equation. This step is crucial because it sets the stage for creating a perfect square trinomial. Next, we complete the square within the parentheses by adding and subtracting a value that makes the expression a perfect square. This value is calculated by taking half of the coefficient of the x term, squaring it, and then adding and subtracting it within the parentheses. The addition ensures we create a perfect square trinomial, while the subtraction compensates for the added value, maintaining the equation's balance. Finally, we simplify the equation by combining like terms and rewriting the perfect square trinomial in its factored form, which gives us the vertex form of the quadratic equation. This form immediately reveals the vertex coordinates, providing valuable information about the parabola's position and orientation.

Completing the Square

To convert the given equation, y = 9x² + 9x - 1, into vertex form, we employ the technique of completing the square. This method allows us to rewrite the quadratic expression as a squared term plus a constant, making the vertex of the parabola readily apparent. The first step in this process is to factor out the coefficient of the x² term, which is 9 in this case, from the first two terms of the equation. This gives us y = 9(x² + x) - 1. Factoring out the leading coefficient is crucial as it simplifies the process of creating a perfect square trinomial inside the parentheses. It sets the stage for adding and subtracting the appropriate constant to complete the square effectively.

Next, we focus on the expression inside the parentheses, x² + x. To complete the square, we need to add and subtract a value that will turn this expression into a perfect square trinomial. This value is determined by taking half of the coefficient of the x term, which is 1, squaring it, and we get (1/2)². Thus, we add and subtract (1/2)² = 1/4 inside the parentheses. This results in y = 9(x² + x + 1/4 - 1/4) - 1. Adding and subtracting the same value ensures that we are not changing the equation's overall value, but merely rearranging its terms. The addition of 1/4 helps us create a perfect square trinomial, while the subtraction of 1/4 compensates for the added term, maintaining the equation's balance.

Now, we can rewrite the perfect square trinomial x² + x + 1/4 as (x + 1/2)². This simplifies our equation to y = 9[(x + 1/2)² - 1/4] - 1. The next step is to distribute the 9 back into the brackets, which gives us y = 9(x + 1/2)² - 9/4 - 1. Finally, we combine the constant terms, -9/4 and -1, to get y = 9(x + 1/2)² - 13/4. This is the vertex form of the given quadratic equation. The vertex form allows us to easily identify the vertex of the parabola, which is (-1/2, -13/4). This point represents the minimum value of the quadratic function since the coefficient of the squared term is positive.

Identifying the Correct Vertex Form

After completing the square, we have successfully transformed the quadratic equation y = 9x² + 9x - 1 into its vertex form, which is y = 9(x + 1/2)² - 13/4. This form is particularly useful because it directly reveals the vertex of the parabola, which is the point where the parabola changes direction. In vertex form, the vertex is given by the coordinates (h, k), where h is the value inside the parentheses (with the opposite sign) and k is the constant term outside the parentheses. Therefore, in our equation, the vertex is at (-1/2, -13/4).

Now, let's compare our result with the given options to identify the correct vertex form. The options provided are:

y = 9(x + 1/2)² - 13/4
y = 9(x + 1/2)² - 1
y = 9(x + 1/2)² + 5/4
y = 9(x + 1/2)² - 5/4

By comparing these options with our derived vertex form, y = 9(x + 1/2)² - 13/4, it is clear that the first option matches our result exactly. This confirms that y = 9(x + 1/2)² - 13/4 is indeed the correct vertex form of the given quadratic equation. The other options have different constant terms, indicating different vertical shifts of the parabola, and thus, they do not represent the same quadratic function.

The correct identification of the vertex form is crucial for various applications. For instance, it allows us to easily determine the minimum or maximum value of the quadratic function, which occurs at the vertex. In this case, since the coefficient of the squared term (9) is positive, the parabola opens upwards, and the vertex represents the minimum point of the function. The y-coordinate of the vertex, -13/4, gives us the minimum value of the function. Additionally, the vertex form simplifies graphing the parabola, as we know the vertex and the axis of symmetry (which is the vertical line passing through the vertex). This information helps in quickly sketching the graph and understanding the behavior of the quadratic function. Therefore, accurately converting to and recognizing the vertex form is a valuable skill in quadratic function analysis.

Conclusion

In conclusion, we successfully converted the quadratic equation y = 9x² + 9x - 1 into its vertex form by employing the method of completing the square. This process involved factoring out the leading coefficient, adding and subtracting a constant to create a perfect square trinomial, and simplifying the equation to the form y = a(x - h)² + k. Through these steps, we arrived at the vertex form y = 9(x + 1/2)² - 13/4. This transformation is a fundamental technique in algebra, providing a clear and concise representation of the quadratic function's key features.

By comparing our result with the given options, we identified the correct vertex form as y = 9(x + 1/2)² - 13/4. This form not only matches our derived equation but also allows us to easily identify the vertex of the parabola, which is at the point (-1/2, -13/4). The vertex represents the minimum value of the function, since the coefficient of the squared term is positive, indicating that the parabola opens upwards. This information is invaluable for understanding the behavior and characteristics of the quadratic function.

Understanding the vertex form of a quadratic equation is essential for various mathematical applications. It simplifies the process of graphing parabolas, determining the minimum or maximum values of quadratic functions, and solving optimization problems. The ability to convert between standard form and vertex form equips students and practitioners with a powerful tool for analyzing and manipulating quadratic equations. The method of completing the square, as demonstrated in this problem, is a versatile technique that extends beyond quadratic equations, finding applications in calculus, complex analysis, and other areas of mathematics. Therefore, mastering this technique and understanding the significance of the vertex form is crucial for a comprehensive understanding of quadratic functions and their applications.