Finding The Vertex Form Of F(x) = 2x² + 28x - 5 A Step-by-Step Guide
Introduction: Understanding Quadratic Functions and Vertex Form
In the realm of mathematics, quadratic functions hold a prominent position, appearing in diverse applications ranging from physics to engineering and economics. These functions, characterized by their parabolic curves, are typically expressed in two primary forms: the standard form and the vertex form. While the standard form provides a straightforward representation of the coefficients, the vertex form offers valuable insights into the function's key features, most notably its vertex, which represents the maximum or minimum point of the parabola. This article delves into the intricacies of converting a quadratic function from standard form to vertex form, using the example of f(x) = 2x² + 28x - 5. We will explore the underlying concepts, step-by-step procedures, and the significance of the vertex form in analyzing quadratic functions.
Decoding Quadratic Functions: Standard Form vs. Vertex Form
A quadratic function is defined as a polynomial function of degree two, generally expressed in the standard form as f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. This form readily reveals the coefficients of the quadratic, linear, and constant terms, but it does not explicitly display the vertex of the parabola. On the other hand, the vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex form provides immediate information about the parabola's vertex, axis of symmetry (x = h), and the direction of opening (determined by the sign of 'a'). When 'a' is positive, the parabola opens upwards, indicating a minimum value at the vertex, while a negative 'a' signifies a downward-opening parabola with a maximum value at the vertex. The vertex form is particularly useful for graphing quadratic functions and solving optimization problems.
The Significance of Vertex Form in Quadratic Function Analysis
The vertex form of a quadratic function, f(x) = a(x - h)² + k, offers a powerful lens through which to analyze the function's behavior and characteristics. The vertex (h, k) is the cornerstone of this form, providing the coordinates of the parabola's turning point. This point represents either the minimum value (if a > 0) or the maximum value (if a < 0) of the function, making it crucial for optimization problems. The x-coordinate of the vertex, h, also defines the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The value of 'a' not only determines the direction of the parabola's opening but also influences its width; a larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value leads to a wider parabola. By simply examining the vertex form, we can readily extract essential information about the quadratic function's graph, behavior, and key features, making it an indispensable tool in mathematical analysis and problem-solving. Understanding vertex form significance simplifies complex quadratic equation interpretations.
Transforming to Vertex Form: A Step-by-Step Guide
To convert the given quadratic function, f(x) = 2x² + 28x - 5, from standard form to vertex form, we employ the technique of completing the square. This method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a squared binomial. Let's embark on this transformation step by step:
Step 1: Factor out the Leading Coefficient
The first step involves factoring out the coefficient of the x² term (which is 2 in this case) from the first two terms of the expression. This yields:
f(x) = 2(x² + 14x) - 5
Step 2: Completing the Square
To complete the square within the parentheses, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is 14, so half of it is 7, and its square is 49. We add and subtract 49 inside the parentheses:
f(x) = 2(x² + 14x + 49 - 49) - 5
Step 3: Forming the Perfect Square Trinomial
The first three terms inside the parentheses now form a perfect square trinomial, which can be factored as (x + 7)²:
f(x) = 2((x + 7)² - 49) - 5
Step 4: Distribute and Simplify
Distribute the 2 and simplify the expression:
f(x) = 2(x + 7)² - 98 - 5
f(x) = 2(x + 7)² - 103
Thus, the vertex form transformation reveals the quadratic function in its vertex form.
Identifying the Vertex: Unveiling the Parabola's Turning Point
Now that we have successfully converted the quadratic function f(x) = 2x² + 28x - 5 into vertex form, f(x) = 2(x + 7)² - 103, we can readily identify the vertex of the parabola. Recall that the vertex form is expressed as f(x) = a(x - h)² + k, where (h, k) represents the vertex. By comparing our transformed equation with the general vertex form, we can extract the values of h and k.
Decoding the Vertex Coordinates
In our case, we have f(x) = 2(x + 7)² - 103, which can be rewritten as f(x) = 2(x - (-7))² + (-103). This directly reveals that h = -7 and k = -103. Therefore, the vertex of the parabola is located at the point (-7, -103). The vertex identification process is crucial for understanding parabola properties.
Interpreting the Vertex: Minimum Value and Axis of Symmetry
The vertex (-7, -103) provides valuable insights into the behavior of the quadratic function. Since the coefficient 'a' (which is 2) is positive, the parabola opens upwards, indicating that the vertex represents the minimum point of the function. Thus, the minimum value of f(x) is -103, which occurs when x = -7. Furthermore, the x-coordinate of the vertex, -7, defines the axis of symmetry, which is the vertical line x = -7. This line divides the parabola into two symmetrical halves, highlighting the symmetrical nature of quadratic functions. The vertex and axis of symmetry are fundamental characteristics of a parabola, guiding its shape and position in the coordinate plane.
Comparing with Answer Choices: Selecting the Correct Vertex Form
Having derived the vertex form of the quadratic function f(x) = 2x² + 28x - 5 as f(x) = 2(x + 7)² - 103, we now turn our attention to the provided answer choices to identify the correct option. This step involves a careful comparison of our derived vertex form with the given alternatives, ensuring that the coefficients and constants align perfectly.
Evaluating the Options: A Precise Match
The answer choices presented are:
- f(x) = 2(x + 14)² - 14
- f(x) = 2(x + 7)² - 103
- f(x) = 2(x + 7)² - 19
- f(x) = 2(x + 14)² - 98
By meticulously comparing these options with our derived vertex form, f(x) = 2(x + 7)² - 103, we can clearly see that the second option, f(x) = 2(x + 7)² - 103, is an exact match. This confirms that our transformation and calculations are accurate, leading us to the correct vertex form representation of the given quadratic function. Comparing vertex forms ensures solution accuracy.
Eliminating Incorrect Choices: Identifying Discrepancies
To further solidify our confidence in the correct answer, it is beneficial to examine the incorrect choices and understand why they are not valid representations of the vertex form. The first and fourth options, f(x) = 2(x + 14)² - 14 and f(x) = 2(x + 14)² - 98, have a different value inside the parenthesis, (x+14) instead of (x+7), indicating an incorrect horizontal shift of the parabola. The third option, f(x) = 2(x + 7)² - 19, shares the correct horizontal shift but has a different constant term (-19 instead of -103), implying an incorrect vertical shift. By identifying these discrepancies, we reinforce our understanding of the vertex form and its relationship to the quadratic function's parameters.
Conclusion: Mastering Vertex Form Transformations
In this comprehensive exploration, we have successfully navigated the process of converting a quadratic function from standard form to vertex form, using the example of f(x) = 2x² + 28x - 5. By employing the technique of completing the square, we transformed the function into its vertex form, f(x) = 2(x + 7)² - 103. This transformation allowed us to readily identify the vertex of the parabola as (-7, -103), which represents the minimum point of the function. Furthermore, we recognized the axis of symmetry as the vertical line x = -7, highlighting the symmetrical nature of the parabola.
The Power of Vertex Form: A Summary of Key Insights
The vertex form of a quadratic function provides a powerful tool for analyzing its behavior and characteristics. It directly reveals the vertex, axis of symmetry, and the direction of opening, enabling us to quickly sketch the graph and solve optimization problems. The ability to convert between standard form and vertex form is a fundamental skill in algebra and calculus, empowering us to gain deeper insights into the world of quadratic functions. Mastering vertex form unlocks deeper quadratic function understanding.
Further Exploration: Expanding Your Quadratic Function Toolkit
This article serves as a stepping stone in your journey to mastering quadratic functions. We encourage you to further explore related topics, such as finding the roots of quadratic equations, analyzing the discriminant, and applying quadratic functions to real-world scenarios. By expanding your knowledge and skills, you will be well-equipped to tackle a wide range of mathematical challenges. The exploration of quadratic function skills is crucial for mathematical proficiency.
