Finding Equivalent Quadratic Equations: A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a significant place, and mastering their various forms is crucial for problem-solving. This article delves into the process of identifying equivalent forms of a given quadratic equation, specifically focusing on the function f(x) = 2x² - 12x + 11. We will explore how to manipulate this equation to match one of the provided options, highlighting the underlying principles and techniques involved. Understanding these manipulations not only helps in solving this particular problem but also strengthens the understanding of quadratic equations in general.
The Significance of Equivalent Forms
When we talk about equivalent forms of a quadratic equation, we're essentially referring to different ways of expressing the same relationship between x and f(x). These different forms can reveal different aspects of the quadratic function, such as its vertex, axis of symmetry, and roots. The standard form of a quadratic equation is ax² + bx + c, where a, b, and c are constants. The given equation, f(x) = 2x² - 12x + 11, is already in this standard form. However, there's another form that's particularly useful: the vertex form, which is a(x - h)² + k, where (h, k) represents the vertex of the parabola. Transforming the standard form into vertex form is a common technique in algebra, and it's precisely what we need to do to solve this problem.
Completing the Square: The Key Technique
The method of completing the square is a powerful algebraic technique used to rewrite a quadratic expression in vertex form. This method involves manipulating the expression to create a perfect square trinomial, which can then be factored into the square of a binomial. Let's apply this technique to our equation, f(x) = 2x² - 12x + 11. The first step is to factor out the coefficient of the x² term (which is 2 in this case) from the first two terms:
f(x) = 2(x² - 6x) + 11
Now, we focus on the expression inside the parentheses, x² - 6x. To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of x is -6, half of which is -3, and the square of -3 is 9. So, we add and subtract 9 inside the parentheses:
f(x) = 2(x² - 6x + 9 - 9) + 11
Notice that we've added and subtracted the same value, so we haven't changed the overall equation. However, the first three terms inside the parentheses now form a perfect square trinomial, which can be factored as (x - 3)². Our equation becomes:
f(x) = 2((x - 3)² - 9) + 11
Next, we distribute the 2 back into the parentheses:
f(x) = 2(x - 3)² - 18 + 11
Finally, we combine the constant terms:
f(x) = 2(x - 3)² - 7
Comparing with the Options
Now that we have the equation in vertex form, f(x) = 2(x - 3)² - 7, we can directly compare it with the given options:
- A. y = 2(x + 3)² - 7
- B. y = 2(x + 6)² + 2
- C. y = 2(x - 3)² - 7
- D. y = 2(x - 6)² + 5
By observation, we can see that option C, y = 2(x - 3)² - 7, exactly matches the vertex form we derived. Therefore, option C is the correct answer.
Why Other Options Are Incorrect
It's also insightful to understand why the other options are incorrect. Option A, y = 2(x + 3)² - 7, has a (x + 3)² term, which would correspond to a vertex with an x-coordinate of -3, while our original equation's vertex has an x-coordinate of 3. Options B and D have different constant terms and different values inside the squared term, indicating that they represent different parabolas altogether.
Conclusion: Mastering Quadratic Transformations
In conclusion, the correct equation that models the same quadratic relationship as f(x) = 2x² - 12x + 11 is C. y = 2(x - 3)² - 7. This was determined by completing the square and transforming the equation into vertex form. This exercise highlights the importance of understanding the different forms of quadratic equations and the techniques used to convert between them. Mastering these concepts is essential for success in algebra and beyond. The ability to manipulate equations and recognize equivalent forms is a cornerstone of mathematical problem-solving, and this example serves as a valuable illustration of that principle.
By understanding how to complete the square and convert between standard and vertex forms, students can gain a deeper appreciation for the properties of quadratic functions and their graphs. This knowledge is not only useful for solving specific problems but also for developing a more intuitive understanding of mathematical relationships in general.
Which of the following equations represents the same quadratic relationship as the function f(x) = 2x² - 12x + 11?
- A. y = 2(x + 3)² - 7
- B. y = 2(x + 6)² + 2
- C. y = 2(x - 3)² - 7
- D. y = 2(x - 6)² + 5
Finding Equivalent Quadratic Equations A Step-by-Step Guide
