Rewriting Quadratic Equations The Vertex Form Of Y = -6x^2 + 3x + 2

Jul 10, 2025 by ADMIN 68 views

Unveiling the Vertex Form of a Quadratic Equation: A Step-by-Step Guide

In the realm of quadratic equations, the vertex form holds a special significance. It unveils the vertex, the highest or lowest point on the parabola, and provides a clear understanding of the parabola's symmetry and overall shape. In this comprehensive guide, we will embark on a journey to transform the given quadratic equation, $y = -6x^2 + 3x + 2$ , into its vertex form, while unraveling the underlying concepts and techniques. Let's delve into the world of quadratic equations and master the art of vertex form conversion.

The vertex form of a quadratic equation is expressed as:

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

where:

(h, k) represents the vertex of the parabola
'a' determines the direction and steepness of the parabola

Our mission is to rewrite the given equation, $y = -6x^2 + 3x + 2$ , in this coveted form. To achieve this, we will employ the powerful technique of completing the square. This method allows us to manipulate the equation algebraically, ultimately revealing the vertex form.

Step 1: Factor out the coefficient of $x^2$

Our initial equation is:

y = -6x^2 + 3x + 2

We begin by factoring out the coefficient of the $x^2$ term, which is -6, from the first two terms:

y = -6(x^2 - rac{1}{2}x) + 2

This step sets the stage for completing the square within the parentheses.

Step 2: Complete the square

Now, we focus on the expression inside the parentheses: $x^2 - rac{1}{2}x$ . To complete the square, we need to add and subtract a specific value. This value is determined by taking half of the coefficient of the x term (-1/2), squaring it, and adding and subtracting it within the parentheses. The coefficient of our x term is -1/2. Half of -1/2 is -1/4, and squaring -1/4 gives us 1/16. So, we add and subtract 1/16 inside the parentheses:

y = -6(x^2 - rac{1}{2}x + rac{1}{16} - rac{1}{16}) + 2

Notice that we've added and subtracted the same value, ensuring that the equation remains balanced.

The first three terms inside the parentheses, $x^2 - rac{1}{2}x + rac{1}{16}$ , now form a perfect square trinomial. We can rewrite this as:

y = -6${(x - rac{1}{4})^2 - rac{1}{16}}$ + 2

Step 3: Distribute and simplify

Next, we distribute the -6 to both terms inside the brackets:

y = -6(x - rac{1}{4})^2 + -6(- rac{1}{16}) + 2

Simplifying, we get:

y = -6(x - rac{1}{4})^2 + rac{3}{8} + 2

To combine the constant terms, we need a common denominator. We rewrite 2 as 16/8:

y = -6(x - rac{1}{4})^2 + rac{3}{8} + rac{16}{8}

Finally, we add the fractions:

y = -6(x - rac{1}{4})^2 + rac{19}{8}

And there we have it! The equation is now in vertex form.

Step 4: Identify the vertex

By comparing our transformed equation, $y = -6(x - rac{1}{4})^2 + rac{19}{8}$ , with the general vertex form, $y = a(x - h)^2 + k$ , we can readily identify the vertex:

h = 1/4
k = 19/8

Therefore, the vertex of the parabola is (1/4, 19/8).

Step 5: Select the correct answer

Now, let's revisit the original question and the given options:

Which equation is $y = -6x^2 + 3x + 2$ rewritten in vertex form?

A. $y = -8(x - 1)^2 + 8$ B. $y = -6(x + rac{1}{4})^2 + rac{13}{8}$ C. $y = -6(x - rac{1}{4})^2 + rac{19}{8}$ D. $y = -6(x - rac{1}{2})^2 + rac{7}{2}$

Comparing our derived vertex form, $y = -6(x - rac{1}{4})^2 + rac{19}{8}$ , with the options, we can clearly see that option C matches our result. Therefore, the correct answer is:

C. $y = -6(x - rac{1}{4})^2 + rac{19}{8}$

Understanding the Significance of Vertex Form

The vertex form of a quadratic equation is not merely a different way of writing the equation; it provides valuable insights into the parabola's characteristics. The vertex, (h, k), reveals the maximum or minimum point of the parabola, depending on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and the vertex represents the minimum point. Conversely, if 'a' is negative, the parabola opens downwards, and the vertex represents the maximum point.

Furthermore, the vertex form allows us to easily identify the axis of symmetry, which is a vertical line passing through the vertex. The equation of the axis of symmetry is x = h. The axis of symmetry divides the parabola into two symmetrical halves.

In our example, the vertex is (1/4, 19/8), and the coefficient 'a' is -6, which is negative. This tells us that the parabola opens downwards, and the vertex represents the maximum point. The axis of symmetry is the vertical line x = 1/4.

Real-World Applications of Quadratic Equations and Vertex Form

Quadratic equations and their vertex forms have numerous applications in various fields, including:

Physics: Projectile motion, such as the trajectory of a ball thrown in the air, can be modeled using quadratic equations. The vertex represents the highest point reached by the projectile.
Engineering: The design of parabolic arches, bridges, and satellite dishes relies on the principles of quadratic equations. The vertex plays a crucial role in determining the optimal shape and dimensions.
Economics: Quadratic functions can be used to model cost, revenue, and profit functions. The vertex helps in identifying the point of maximum profit or minimum cost.
Computer Graphics: Parabolas are used in computer graphics to create smooth curves and shapes. The vertex form allows for precise control over the curve's shape and position.

Mastering the Art of Completing the Square: Tips and Tricks

Completing the square is a fundamental technique in algebra, and mastering it is essential for solving quadratic equations and understanding their properties. Here are some tips and tricks to help you become proficient in completing the square:

Practice makes perfect: The more you practice completing the square, the more comfortable you will become with the process. Work through various examples and exercises to solidify your understanding.
Pay attention to signs: Be mindful of the signs of the coefficients and terms while completing the square. A small error in sign can lead to an incorrect result.
Fractional coefficients: Don't be intimidated by fractional coefficients. The process remains the same, just be careful while performing the arithmetic operations.
Check your work: After completing the square, you can always expand the vertex form back to the standard form to verify your answer.
Visualize the process: Try to visualize the process of completing the square geometrically. This can help you understand why the method works and make it easier to remember.

Conclusion: Embracing the Power of Vertex Form

In this comprehensive guide, we have successfully transformed the quadratic equation $y = -6x^2 + 3x + 2$ into its vertex form, $y = -6(x - rac{1}{4})^2 + rac{19}{8}$ . We explored the significance of the vertex form in revealing the vertex, axis of symmetry, and overall shape of the parabola. Furthermore, we discussed the real-world applications of quadratic equations and the importance of mastering the technique of completing the square.

The vertex form is more than just a mathematical representation; it's a key that unlocks the secrets of quadratic equations and their applications. By understanding the vertex form, we gain a deeper appreciation for the beauty and power of mathematics in describing the world around us.

So, embrace the vertex form, practice completing the square, and continue your mathematical journey with confidence and curiosity. The world of quadratic equations awaits your exploration!