First Step In Rewriting Quadratic Equations To Vertex Form

Transforming quadratic equations into different forms allows us to glean valuable information about their graphs and behavior. One particularly useful form is the vertex form, which provides direct insights into the vertex of the parabola, the axis of symmetry, and whether the parabola opens upwards or downwards. In this comprehensive guide, we will delve into the process of converting a quadratic equation from its standard form to vertex form, with a special focus on identifying the crucial first step. Our specific example will be the quadratic equation $y = -4x^2 + 2x - 7$, and we aim to rewrite it in the vertex form $y = a(x - h)^2 + k$. Understanding the initial step is paramount to successfully navigating the entire transformation process. Let's embark on this journey of mathematical exploration and unlock the secrets of vertex form!

Understanding the Vertex Form

The vertex form of a quadratic equation is expressed as $y = a(x - h)^2 + k$, where:

• 'a' determines the direction and steepness of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The larger the absolute value of 'a', the steeper the parabola.
• (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction, and it is either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards).
• The line x = h is the axis of symmetry of the parabola. This is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

The vertex form provides a clear and concise representation of these key features, making it a powerful tool for analyzing and graphing quadratic functions. By transforming a quadratic equation into vertex form, we can quickly identify the vertex, axis of symmetry, and direction of opening, which are essential for understanding the parabola's behavior.

The Standard Form and the Need for Transformation

The standard form of a quadratic equation is given by $y = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants. While the standard form is useful for certain purposes, such as easily identifying the y-intercept (which is 'c'), it does not readily reveal the vertex or axis of symmetry. This is where the transformation to vertex form becomes crucial.

Our given equation, $y = -4x^2 + 2x - 7$, is in standard form. To extract information about the vertex and axis of symmetry, we need to rewrite it in the form $y = a(x - h)^2 + k$. This transformation involves a process called completing the square, which is a technique used to rewrite a quadratic expression as a perfect square trinomial plus a constant.

The Crucial First Step Factoring out 'a'

The first and most critical step in rewriting a quadratic equation from standard form to vertex form is factoring out the coefficient of the $x^2$ term, which is 'a', from the terms containing $x^2$ and $x$. In our example, $y = -4x^2 + 2x - 7$, the coefficient of the $x^2$ term is -4. Therefore, the first step is to factor out -4 from the terms $-4x^2$ and $2x$.

This step is essential because it allows us to create a perfect square trinomial within the parentheses. A perfect square trinomial is a trinomial that can be factored as $(x + m)^2$ or $(x - m)^2$ for some constant 'm'. Completing the square relies on this structure, and factoring out 'a' sets the stage for this process.

Let's illustrate this step with our equation:

$y = -4x^2 + 2x - 7$

Factoring out -4 from the first two terms, we get:

y = -4(x^2 - rac{1}{2}x) - 7

Notice that we have divided both $-4x^2$ and $2x$ by -4. The constant term, -7, remains outside the parentheses as it is not involved in the factoring process at this stage.

Why This Step is Essential

The importance of factoring out 'a' lies in its role in creating the perfect square trinomial. When we have an expression of the form $x^2 + bx$ (where the coefficient of $x^2$ is 1), we can complete the square by adding and subtracting (rac{b}{2})^2. However, if the coefficient of $x^2$ is not 1, we cannot directly apply this method. Factoring out 'a' ensures that the expression inside the parentheses has a leading coefficient of 1, making it suitable for completing the square.

In our example, after factoring out -4, we have the expression x^2 - rac{1}{2}x inside the parentheses. Now, the coefficient of $x^2$ is 1, and we can proceed with completing the square by adding and subtracting (rac{-1/2}{2})^2 = (rac{-1}{4})^2 = rac{1}{16} inside the parentheses.

The Next Steps: Completing the Square

After factoring out -4, the next step is to complete the square inside the parentheses. This involves adding and subtracting the square of half the coefficient of the x term. In our case, the coefficient of the x term inside the parentheses is -rac{1}{2}. Half of this is -rac{1}{4}, and squaring it gives us rac{1}{16}.

We add and subtract rac{1}{16} inside the parentheses:

y = -4(x^2 - rac{1}{2}x + rac{1}{16} - rac{1}{16}) - 7

Now, the first three terms inside the parentheses form a perfect square trinomial:

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

We can rewrite the equation as:

y = -4((x - rac{1}{4})^2 - rac{1}{16}) - 7

Next, we distribute the -4 to both terms inside the parentheses:

y = -4(x - rac{1}{4})^2 + rac{1}{4} - 7

Finally, we combine the constant terms:

y = -4(x - rac{1}{4})^2 - rac{27}{4}

Now, the equation is in vertex form, $y = a(x - h)^2 + k$, where $a = -4$, h = rac{1}{4}, and k = -rac{27}{4}.

Identifying the Vertex and Axis of Symmetry

From the vertex form, y = -4(x - rac{1}{4})^2 - rac{27}{4}, we can easily identify the vertex and axis of symmetry:

• Vertex: The vertex is at the point (h, k), which is (rac{1}{4}, -rac{27}{4}).
• Axis of Symmetry: The axis of symmetry is the vertical line x = h, which is x = rac{1}{4}.

Additionally, since 'a' is -4 (negative), the parabola opens downwards.

Common Mistakes to Avoid

When transforming quadratic equations to vertex form, there are a few common mistakes to watch out for:

1. Forgetting to factor out 'a': This is the most critical first step, and skipping it will lead to an incorrect vertex form.
2. Incorrectly completing the square: Make sure to add and subtract (rac{b}{2})^2 inside the parentheses, where 'b' is the coefficient of the x term after factoring out 'a'.
3. Distributing 'a' incorrectly: When distributing 'a' back into the parentheses, remember to multiply it by both terms, including the constant term that was added and subtracted to complete the square.
4. Misidentifying the vertex: The vertex is (h, k), not (-h, k) or (h, -k). Pay close attention to the signs in the vertex form equation.

By understanding these common pitfalls and carefully following the steps, you can confidently transform quadratic equations into vertex form.

Conclusion

In conclusion, the first and foremost step in rewriting the quadratic equation $y = -4x^2 + 2x - 7$ in the vertex form $y = a(x - h)^2 + k$ is to factor out -4 from the terms $-4x^2 + 2x$. This crucial step sets the stage for completing the square and ultimately reveals the vertex and axis of symmetry of the parabola. By mastering this technique, you gain a deeper understanding of quadratic functions and their graphical representations. Remember to always prioritize factoring out the coefficient of the $x^2$ term as the initial step, and the rest of the transformation will follow smoothly. With practice and a clear understanding of the process, you can confidently navigate the world of quadratic equations and their various forms.

By correctly transforming the equation, we can readily identify the vertex, axis of symmetry, and the direction in which the parabola opens. This information is invaluable for graphing the quadratic function and solving related problems. Always remember to double-check your work and be mindful of common mistakes to ensure accurate results.

Therefore, the correct answer is A. -4 must be factored from $-4x^2 + 2x$. This foundational step unlocks the path to vertex form and a clearer understanding of quadratic equations.

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First Step Rewriting Quadratic Equations to Vertex Form