Rewriting $y=2x^2-8x+9$ In Vertex Form A Step-by-Step Guide

The question at hand asks us to rewrite the quadratic equation $y = 2x^2 - 8x + 9$ in vertex form. This is a fundamental concept in algebra, particularly when dealing with parabolas. The vertex form provides valuable insights into the parabola's key features, such as its vertex (the point where the parabola changes direction), axis of symmetry, and whether it opens upwards or downwards. Understanding how to convert a quadratic equation from standard form to vertex form is crucial for solving various mathematical problems and real-world applications. This article will delve deep into the process, explaining each step in detail and providing a clear understanding of the underlying principles.

Understanding the Vertex Form of a Quadratic Equation

Before we dive into the conversion process, let's first understand what the vertex form of a quadratic equation actually is. The vertex form is expressed as:

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

Where:

• y represents the dependent variable (usually plotted on the vertical axis).
• x represents the independent variable (usually plotted on the horizontal axis).
• a determines the direction and steepness 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 – a larger absolute value means a steeper parabola.
• (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction. It is the minimum point if the parabola opens upwards and the maximum point if the parabola opens downwards.
• h represents the x-coordinate of the vertex.
• k represents the y-coordinate of the vertex.

The vertex form is incredibly useful because it directly reveals the vertex of the parabola, which is a critical point for understanding the parabola's behavior and properties. Knowing the vertex allows us to easily determine the axis of symmetry, which is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation of the axis of symmetry is x = h.

Methods for Converting to Vertex Form

There are two primary methods for converting a quadratic equation from standard form ($y = ax^2 + bx + c$) to vertex form: completing the square and using the vertex formula. Both methods will yield the same result, but they approach the problem from different angles. We will explore both methods in detail, starting with completing the square.

1. Completing the Square

Completing the square is an algebraic technique used to rewrite a quadratic expression into a perfect square trinomial, which can then be factored into the square of a binomial. This method is particularly useful for converting quadratic equations to vertex form because it directly manipulates the equation to reveal the vertex coordinates. Let's break down the steps involved in completing the square for the given equation, $y = 2x^2 - 8x + 9$.

Step 1: Factor out the coefficient of the $x^2$ term from the first two terms.

In our equation, the coefficient of the $x^2$ term is 2. We factor this out from the first two terms: $y = 2(x^2 - 4x) + 9$. This step is crucial because it ensures that the expression inside the parentheses has a leading coefficient of 1, which is necessary for completing the square.

Step 2: Complete the square inside the parentheses.

To complete the square, we take half of the coefficient of the x term (which is -4), square it, and add it inside the parentheses. Half of -4 is -2, and (-2)^2 is 4. So, we add 4 inside the parentheses. However, because we've added 4 inside the parentheses, which are being multiplied by 2, we've effectively added 2 * 4 = 8 to the right side of the equation. To maintain the equation's balance, we must subtract 8 outside the parentheses. This gives us: $y = 2(x^2 - 4x + 4) + 9 - 8$

Step 3: Rewrite the expression inside the parentheses as a squared binomial.

The expression $x^2 - 4x + 4$ is a perfect square trinomial, which can be factored as $(x - 2)^2$. So, we rewrite the equation as: $y = 2(x - 2)^2 + 9 - 8$

Step 4: Simplify the equation.

Finally, we simplify the constant terms outside the parentheses: $y = 2(x - 2)^2 + 1$. This is the equation in vertex form.

By completing the square, we have successfully transformed the given quadratic equation into vertex form. From this form, we can easily identify the vertex of the parabola as (2, 1) and the value of a as 2, which indicates that the parabola opens upwards and has a vertical stretch.

2. Using the Vertex Formula

Another method for converting a quadratic equation to vertex form is by using the vertex formula. This method involves directly calculating the coordinates of the vertex using the coefficients of the quadratic equation in standard form. The vertex formula is derived from the process of completing the square, but it provides a more direct approach for finding the vertex.

Step 1: Identify the coefficients a, b, and c in the standard form equation.

Our equation is $y = 2x^2 - 8x + 9$. Comparing this to the standard form equation $y = ax^2 + bx + c$, we can identify the coefficients as: a = 2, b = -8, and c = 9.

Step 2: Calculate the x-coordinate of the vertex (h).

The x-coordinate of the vertex, h, can be calculated using the formula: $h = -b / (2a)$. Substituting the values of a and b, we get: $h = -(-8) / (2 * 2) = 8 / 4 = 2$.

Step 3: Calculate the y-coordinate of the vertex (k).

The y-coordinate of the vertex, k, can be calculated by substituting the value of h back into the original equation: $k = y(h) = 2(2)^2 - 8(2) + 9 = 2(4) - 16 + 9 = 8 - 16 + 9 = 1$.

Step 4: Substitute the values of a, h, and k into the vertex form equation.

Now that we have the values of a (which we identified in Step 1), h, and k, we can substitute them into the vertex form equation $y = a(x - h)^2 + k$: $y = 2(x - 2)^2 + 1$.

As we can see, using the vertex formula yields the same vertex form equation as completing the square. This method provides a more direct approach for finding the vertex coordinates, especially when the quadratic equation is already in standard form.

Solution and Verification

Both methods, completing the square and using the vertex formula, have led us to the same vertex form equation: $y = 2(x - 2)^2 + 1$. This matches option C in the given choices. Therefore, the correct answer is:

C. $y = 2(x - 2)^2 + 1$

To further verify our solution, we can expand the vertex form equation and see if it simplifies back to the original standard form equation:

$y = 2(x - 2)^2 + 1$ $y = 2(x^2 - 4x + 4) + 1$ $y = 2x^2 - 8x + 8 + 1$ $y = 2x^2 - 8x + 9$

This confirms that our vertex form equation is indeed equivalent to the original standard form equation.

Importance of Vertex Form

The vertex form of a quadratic equation is not just a different way of writing the same equation; it provides valuable information about the parabola's characteristics and behavior. As we've discussed, the vertex form directly reveals the vertex of the parabola, which is crucial for understanding the parabola's minimum or maximum value and its axis of symmetry. This information is essential in various applications, such as:

• Optimization problems: Many real-world problems involve finding the maximum or minimum value of a quantity that can be modeled by a quadratic equation. For example, determining the maximum height of a projectile, the maximum profit a business can make, or the minimum cost of production.
• Graphing parabolas: The vertex form makes it easy to graph a parabola by plotting the vertex and using the value of a to determine the direction and steepness of the curve.
• Solving quadratic equations: The vertex form can be used to solve quadratic equations by setting y to zero and solving for x. This is particularly useful when the equation cannot be easily factored.
• Understanding transformations: The vertex form highlights the transformations that have been applied to the basic parabola $y = x^2$, such as horizontal and vertical shifts and stretches or compressions.

Conclusion

In this article, we have thoroughly explored the process of rewriting the quadratic equation $y = 2x^2 - 8x + 9$ in vertex form. We discussed the two primary methods for doing so: completing the square and using the vertex formula. Both methods yielded the same result, $y = 2(x - 2)^2 + 1$, which corresponds to option C. We also verified our solution by expanding the vertex form equation and confirming that it simplifies back to the original standard form equation.

Furthermore, we emphasized the importance of the vertex form in understanding the characteristics and behavior of parabolas, highlighting its applications in optimization problems, graphing, solving equations, and understanding transformations. Mastering the conversion to vertex form is a fundamental skill in algebra and provides a powerful tool for analyzing and solving a wide range of mathematical problems and real-world scenarios. By understanding the underlying principles and practicing the techniques, you can confidently tackle any quadratic equation and unlock its hidden insights.

This comprehensive guide should provide a solid understanding of how to rewrite quadratic equations in vertex form and the significance of doing so. Remember to practice these techniques regularly to solidify your understanding and develop your problem-solving skills.