Vertex Form: Rewriting Y=9x^2+9x-1 Explained

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Hey guys! Today, we're going to dive deep into the fascinating world of quadratic equations and, more specifically, how to convert a quadratic equation from its standard form to vertex form. This is a super useful skill in algebra, as the vertex form gives us a ton of information about the parabola the equation represents. Let's tackle the equation y=9x2+9x−1y = 9x^2 + 9x - 1. We'll break down the process step-by-step, making it easy to follow along, even if you're just starting your algebra journey. Stick with me, and you'll be a vertex form pro in no time! Understanding vertex form is not just about manipulating equations; it's about gaining a deeper insight into the nature of quadratic functions and their graphical representations.

What is Vertex Form?

Before we jump into the solution, let's make sure we're all on the same page about what vertex form actually is. The vertex form of a quadratic equation is given by:

y=a(x−h)2+ky = a(x - h)^2 + k

Where:

  • (h, k) represents the vertex of the parabola.
  • a is the same leading coefficient as in the standard form (ax2+bx+cax^2 + bx + c).

The vertex, as you might already know, is the point where the parabola changes direction – it's either the minimum or maximum point on the graph. Converting to vertex form makes it incredibly easy to identify this crucial point. Why is this important? Well, the vertex gives us a wealth of information about the quadratic function, such as its maximum or minimum value and the axis of symmetry. This knowledge is invaluable in various applications, from physics problems involving projectile motion to optimization problems in economics and engineering. Furthermore, the vertex form allows us to quickly sketch the graph of the parabola, providing a visual representation of the function's behavior. By understanding the vertex form, we can gain a deeper appreciation for the relationship between algebraic equations and their corresponding graphical representations.

Steps to Convert to Vertex Form

Now, let's get to the fun part – actually converting our equation, y=9x2+9x−1y = 9x^2 + 9x - 1, into vertex form. We'll use a technique called completing the square. Don't worry; it sounds more intimidating than it is! Just follow these steps:

1. Factor out the leading coefficient from the x2x^2 and xx terms

Our equation is y=9x2+9x−1y = 9x^2 + 9x - 1. The leading coefficient is 9, so we factor it out from the first two terms:

y=9(x2+x)−1y = 9(x^2 + x) - 1

Why do we do this? Factoring out the leading coefficient allows us to focus on completing the square within the parentheses, making the process more manageable. This step sets the stage for the next steps, where we'll manipulate the expression inside the parentheses to create a perfect square trinomial. By isolating the x2x^2 and xx terms, we can work with a simpler quadratic expression that is easier to complete the square for. This technique is crucial for successfully converting the equation to vertex form.

2. Complete the square inside the parentheses

To complete the square, we need to add and subtract a specific value inside the parentheses. This value is equal to (b/2)2(b/2)^2, where b is the coefficient of the x term inside the parentheses. In our case, the coefficient of the x term is 1. So, we calculate:

(1/2)2=1/4(1/2)^2 = 1/4

Now, we add and subtract this value inside the parentheses:

y=9(x2+x+1/4−1/4)−1y = 9(x^2 + x + 1/4 - 1/4) - 1

Why do we add and subtract the same value? This might seem like a strange move, but it's the key to completing the square without changing the value of the equation. By adding and subtracting the same quantity, we're essentially adding zero, which preserves the equation's integrity. This allows us to rewrite the expression inside the parentheses as a perfect square trinomial, which is crucial for achieving vertex form. The added term, 1/4, creates the perfect square, while the subtracted term, -1/4, compensates for the addition, ensuring the equation remains balanced. This clever maneuver is the essence of the completing the square technique.

3. Rewrite the perfect square trinomial

The first three terms inside the parentheses (x2+x+1/4x^2 + x + 1/4) now form a perfect square trinomial. We can rewrite it as:

(x+1/2)2(x + 1/2)^2

So our equation becomes:

y=9((x+1/2)2−1/4)−1y = 9((x + 1/2)^2 - 1/4) - 1

Recognizing and rewriting the perfect square trinomial is a critical step in the conversion process. It allows us to condense the expression into a squared term, which is the hallmark of vertex form. The perfect square trinomial arises from the specific value we added and subtracted in the previous step. By rewriting it as a squared term, we're essentially reversing the process of expanding a binomial square, simplifying the equation and bringing us closer to the desired vertex form. This step highlights the algebraic manipulation skills necessary for working with quadratic equations.

4. Distribute and simplify

Now, we distribute the 9 back into the parentheses:

y=9(x+1/2)2−9/4−1y = 9(x + 1/2)^2 - 9/4 - 1

And simplify the constant terms:

y=9(x+1/2)2−9/4−4/4y = 9(x + 1/2)^2 - 9/4 - 4/4

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

Distributing and simplifying are essential steps in isolating the squared term and the constant term, which are the key components of vertex form. Distributing the leading coefficient ensures that the equation remains balanced and that all terms are properly accounted for. Simplifying the constant terms allows us to combine them into a single value, representing the vertical shift of the parabola. This process of distribution and simplification is a standard algebraic technique that is widely used in various mathematical contexts. By carefully executing these steps, we ensure that the equation is transformed accurately into its final vertex form.

The Answer

There you have it! The equation y=9x2+9x−1y = 9x^2 + 9x - 1 rewritten in vertex form is:

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

So, the correct answer is B. But more importantly, you now understand how to get to the answer! Understanding the process is far more valuable than memorizing a single solution. By mastering the technique of completing the square, you can confidently convert any quadratic equation into vertex form. This skill will not only help you solve problems in algebra but also provide you with a deeper understanding of quadratic functions and their graphical representations. Remember, the goal is not just to arrive at the correct answer but to develop a strong foundation in mathematical concepts.

What Does This Tell Us?

Now that we have the equation in vertex form, let's see what information we can glean from it:

  • Vertex: The vertex is at the point (-1/2, -13/4). Remember, the vertex form is y=a(x−h)2+ky = a(x - h)^2 + k, so the x-coordinate of the vertex is the opposite of the value inside the parentheses.
  • Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. Its equation is x=−1/2x = -1/2.
  • Direction of Opening: Since the leading coefficient (9) is positive, the parabola opens upwards.

This is the beauty of vertex form! It provides us with immediate insights into the parabola's key features. The vertex tells us the minimum or maximum point of the function, the axis of symmetry divides the parabola into two symmetrical halves, and the direction of opening reveals whether the parabola opens upwards or downwards. This information is invaluable for sketching the graph of the quadratic function and understanding its behavior. By analyzing the vertex form, we can quickly identify these characteristics without having to resort to other methods, such as plotting points or using the quadratic formula. This makes vertex form a powerful tool for analyzing and understanding quadratic functions.

Practice Makes Perfect

The best way to solidify your understanding of vertex form is to practice! Try converting other quadratic equations into vertex form. You can even create your own equations and challenge yourself. The more you practice, the more comfortable you'll become with the process. Remember, mathematics is a skill that is developed through consistent effort and practice. Don't be discouraged if you encounter difficulties along the way. Instead, view them as opportunities to learn and grow. Seek out additional resources, such as textbooks, online tutorials, or your teacher, if you need further assistance. With dedication and perseverance, you can master the art of converting quadratic equations into vertex form and unlock a deeper understanding of quadratic functions.

Conclusion

Converting to vertex form might seem tricky at first, but with practice, it becomes second nature. You've got this! We've walked through the steps together, and now you have a powerful tool in your algebra arsenal. Keep practicing, and you'll be amazed at how much you can achieve. Remember, understanding the underlying concepts is crucial for success in mathematics. Don't just memorize formulas; strive to comprehend the logic and reasoning behind them. This will not only help you solve problems more effectively but also foster a deeper appreciation for the beauty and elegance of mathematics. So, keep exploring, keep questioning, and keep learning! The world of mathematics is vast and fascinating, and there's always something new to discover. And remember, we're all in this together, so don't hesitate to reach out for help or share your knowledge with others. Happy solving, guys!