First Step Converting F(x) = 3x^2 + 6x - 8 To Vertex Form

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Introduction

When dealing with quadratic functions, transforming them into different forms can reveal various properties and make them easier to analyze. One such form is the vertex form, which provides immediate insights into the vertex (the maximum or minimum point) of the parabola represented by the quadratic function. The given quadratic function is f(x) = 3x^2 + 6x - 8, and the question at hand is: What is the first step in rewriting this function in vertex form? The options provided are:

A. Factor out 3 from each term. B. Write the trinomial as a binomial squared. C. Form a perfect square trinomial by keeping the value of the function equivalent. D. Factor out 3.

This article will delve into the process of converting a quadratic function from its standard form to vertex form, with a particular focus on identifying the crucial first step. We will explore each option in detail to clarify the correct approach and provide a comprehensive understanding of the underlying mathematical principles. Whether you're a student learning about quadratic functions or someone looking to refresh your knowledge, this guide aims to provide a clear and concise explanation.

Understanding Vertex Form

Before we dive into the steps, it's essential to understand what the vertex form of a quadratic function actually looks like. The vertex form is generally expressed as:

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

Where:

  • a determines the direction and stretch of the parabola.
  • (h, k) represents the vertex of the parabola. The vertex is a crucial point because it indicates the maximum or minimum value of the function. If a > 0, the parabola opens upwards, and the vertex is the minimum point. If a < 0, the parabola opens downwards, and the vertex is the maximum point.

Knowing the vertex form allows us to quickly identify the vertex of the parabola and understand its orientation. Converting a quadratic function to vertex form often involves a process called completing the square, which we will explore in the subsequent sections. This method ensures that we rewrite the quadratic expression in a way that highlights the squared term, making it easy to identify h and k.

Analyzing the Options

Now, let's dissect each option provided in the question to determine the correct first step in converting f(x) = 3x^2 + 6x - 8 into vertex form.

A. Factor out 3 from each term.

This option suggests factoring out the coefficient of the x^2 term, which is 3, from the quadratic expression. This is indeed a crucial step in the process of completing the square. Factoring out 3 from 3x^2 + 6x - 8 gives us:

f(x) = 3(x^2 + 2x) - 8

Notice that we've only factored 3 from the terms containing x. The constant term, -8, remains outside the parentheses. This is because we want to isolate the x^2 and x terms to complete the square within the parentheses. Factoring out the leading coefficient is essential because it simplifies the process of finding the value needed to complete the square. When the coefficient of x^2 is 1, it becomes much easier to determine the constant term that makes the expression inside the parentheses a perfect square trinomial. This step sets the stage for creating the binomial squared term in the vertex form.

B. Write the trinomial as a binomial squared.

This option is premature. We cannot directly write the given trinomial 3x^2 + 6x - 8 as a binomial squared without further manipulation. A binomial squared has the form (x + a)^2 or (x - a)^2, which expands to x^2 + 2ax + a^2 or x^2 - 2ax + a^2, respectively. The given trinomial does not readily fit this form. Moreover, we have a coefficient of 3 for the x^2 term, which complicates matters further. Therefore, this option is not the correct first step. We need to complete the square, and this involves more steps than simply writing the trinomial as a binomial squared right away. The process of completing the square requires us to manipulate the quadratic expression to create a perfect square trinomial, which then can be written as a binomial squared.

C. Form a perfect square trinomial by keeping the value of the function equivalent.

This option describes the goal of completing the square, but it's not the immediate first step. Forming a perfect square trinomial is a process that involves several steps, and it's something we aim to do after factoring out the leading coefficient. To form a perfect square trinomial, we need to add and subtract a specific value inside the parentheses to maintain the equivalence of the function. This value is determined by taking half of the coefficient of the x term and squaring it. However, we can't perform this step until we've factored out the leading coefficient. Therefore, while this option correctly identifies a necessary part of the process, it is not the first step.

D. Factor out 3

This option is similar to option A but lacks the precision of specifying that the factoring should be done from the terms containing x. While factoring out 3 is a necessary component, the crucial detail is factoring it out from the x^2 and x terms, not the constant term. This distinction is important because it sets up the expression for completing the square. Factoring out the leading coefficient only from the x^2 and x terms allows us to focus on the quadratic and linear terms within the parentheses, making the process of completing the square more manageable. Thus, while the general idea is correct, option A provides a more accurate description of the first step.

The Correct First Step: Factoring out 3 from the x^2 and x terms

Based on the analysis above, the correct first step in writing f(x) = 3x^2 + 6x - 8 in vertex form is to factor out 3 from the x^2 and x terms. This step simplifies the quadratic expression and prepares it for the process of completing the square. By factoring out the leading coefficient, we make the coefficient of x^2 inside the parentheses equal to 1, which is essential for easily determining the constant term needed to complete the square.

Detailed Explanation of the First Step

Let's reiterate the importance of this first step with a more detailed explanation. Starting with f(x) = 3x^2 + 6x - 8, we factor out 3 from the first two terms:

f(x) = 3(x^2 + 2x) - 8

Here’s why this is crucial:

  1. Simplifies Completing the Square: When the coefficient of x^2 is 1, completing the square becomes straightforward. We only need to focus on the coefficient of the x term.
  2. Prepares for Perfect Square Trinomial: Factoring out 3 allows us to create a perfect square trinomial inside the parentheses. This trinomial can then be expressed as a binomial squared.
  3. Maintains Function Equivalence: By only factoring from the x^2 and x terms and leaving the constant term outside, we ensure that the value of the function remains unchanged.

This first step is the foundation for the subsequent steps in converting to vertex form. It's a strategic move that simplifies the overall process and sets the stage for a successful transformation.

Subsequent Steps in Completing the Square

Now that we’ve identified the first step, let's briefly outline the subsequent steps involved in completing the square to convert f(x) = 3x^2 + 6x - 8 to vertex form.

  1. Complete the Square Inside the Parentheses:
    • Take half of the coefficient of the x term (which is 2), square it (1), and add and subtract it inside the parentheses: f(x) = 3(x^2 + 2x + 1 - 1) - 8
  2. Rewrite as a Binomial Squared:
    • Rewrite the perfect square trinomial as a binomial squared: f(x) = 3((x + 1)^2 - 1) - 8
  3. Distribute and Simplify:
    • Distribute the 3 and combine the constant terms: f(x) = 3(x + 1)^2 - 3 - 8 f(x) = 3(x + 1)^2 - 11

Now, the function is in vertex form, f(x) = 3(x + 1)^2 - 11. The vertex of the parabola is at (-1, -11).

Conclusion

In conclusion, the first and foremost step in writing the quadratic function f(x) = 3x^2 + 6x - 8 in vertex form is to factor out 3 from the x^2 and x terms. This crucial step simplifies the expression and sets the stage for completing the square. By understanding this initial step and the subsequent steps involved, you can confidently convert any quadratic function from standard form to vertex form, gaining valuable insights into the parabola's properties, particularly its vertex. Mastering this process is essential for anyone studying quadratic functions and their applications in various fields of mathematics and beyond. Remember to always focus on creating a perfect square trinomial within the parentheses after factoring out the leading coefficient, and the rest will follow seamlessly. This approach not only simplifies the algebraic manipulations but also enhances your understanding of the underlying mathematical principles. Whether you are tackling homework problems or working on more complex mathematical models, the ability to convert quadratic functions into vertex form is a powerful tool in your mathematical toolkit.