Quadratic Function In Standard Form With A=-3.5, B=2.7, And C=-8.2

In the realm of mathematics, particularly within the study of quadratic functions, understanding the standard form is crucial. The standard form provides a clear structure for analyzing and interpreting the properties of these functions. A quadratic function, a fundamental concept in algebra, is typically expressed in the standard form as f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. This article delves into the process of identifying the correct quadratic function in standard form given specific values for the coefficients a, b, and c. We will explore how these coefficients influence the shape and position of the parabola, which is the graphical representation of a quadratic function. Understanding the role of each coefficient is essential for solving quadratic equations, graphing parabolas, and applying quadratic functions in various real-world scenarios.

This exploration is vital not only for students learning algebra but also for anyone dealing with mathematical modeling in fields such as physics, engineering, and economics. Quadratic functions are used to model a wide array of phenomena, from the trajectory of a projectile to the optimization of business processes. By mastering the standard form of a quadratic function, we gain a powerful tool for analyzing and predicting these phenomena. In this article, we will walk through a specific example, demonstrating step-by-step how to construct the correct quadratic function when given the values of a, b, and c. This will reinforce the importance of understanding the structure of the standard form and how the coefficients fit into it. Let's embark on this mathematical journey to solidify our understanding of quadratic functions and their applications.

To effectively tackle the problem at hand, it's imperative to have a solid grasp of the standard form of a quadratic function. As previously mentioned, the standard form is expressed as f(x) = ax² + bx + c, where a, b, and c are constants. The coefficient a determines the direction and width of the parabola; if a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The magnitude of a affects the parabola's width – a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider one. The coefficient b influences the position of the parabola's axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. The axis of symmetry is given by the equation x = -b / 2a. Thus, b plays a crucial role in determining the horizontal placement of the parabola.

The constant c represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. In other words, when x = 0, the value of the function is f(0) = c. This provides a direct visual reference on the graph of the function. Understanding these individual roles of a, b, and c allows us to quickly analyze and interpret quadratic functions. When given a quadratic function in standard form, we can immediately identify the key features of its graph, such as its direction, width, axis of symmetry, and y-intercept. Conversely, if we have information about these features, we can construct the corresponding quadratic function in standard form.

The standard form is not only useful for graphing but also for solving quadratic equations. By setting f(x) = 0, we can find the x-intercepts (or roots) of the parabola, which are the points where the parabola intersects the x-axis. These roots can be found using various methods, such as factoring, completing the square, or using the quadratic formula. The standard form provides a convenient framework for applying these methods. In the context of our problem, we are given the values of a, b, and c, and our task is to correctly substitute these values into the standard form. This highlights the importance of understanding the structure of the standard form and how the coefficients fit into it. Let's now proceed to solve the problem by applying our knowledge of the standard form.

In this particular problem, we are presented with the values a = -3.5, b = 2.7, and c = -8.2. Our objective is to identify which of the given options correctly represents the quadratic function in standard form, which, as we've established, is f(x) = ax² + bx + c. This task requires a direct application of our understanding of the standard form. We need to substitute the given values of a, b, and c into the general form and then compare the result with the provided options. This seemingly simple process is a fundamental skill in algebra and serves as a building block for more complex problem-solving. It reinforces the importance of attention to detail and accurate substitution.

Before we delve into the options, let's first perform the substitution ourselves. Substituting a = -3.5, b = 2.7, and c = -8.2 into f(x) = ax² + bx + c, we get: f(x) = -3.5x² + 2.7x - 8.2. This is the quadratic function in standard form with the given coefficients. Now, we can compare this result with the options provided in the problem. This step is crucial as it tests our ability to not only substitute correctly but also to recognize the correct form among distractors. The options may differ in the order of terms, the signs of the coefficients, or other subtle ways. Therefore, a careful comparison is essential to avoid errors.

The process of identifying the correct quadratic function highlights the importance of precision in mathematics. A small mistake in substitution or a misinterpretation of the standard form can lead to an incorrect answer. This problem serves as a reminder to always double-check our work and to have a clear understanding of the underlying concepts. By mastering the standard form of a quadratic function, we can confidently tackle problems like this and build a strong foundation for further mathematical studies. Now, let's examine the given options and determine which one matches our derived function.

Now that we have determined the quadratic function with the given coefficients to be f(x) = -3.5x² + 2.7x - 8.2, we must compare this result to the provided options. This step is crucial for ensuring we select the correct answer. Let's examine each option:

• Option A: f(x) = -3.5x² - 8.2x + 2.7. This option has the correct coefficient for the x² term (-3.5) but incorrectly assigns the coefficients for the x term and the constant term. The coefficient of x should be 2.7, and the constant term should be -8.2. Thus, this option is incorrect.

• Option B: f(x) = 2.7x² - 8.2x - 3.5. This option has the coefficients in completely the wrong order. The coefficient of x² should be -3.5, the coefficient of x should be 2.7, and the constant term should be -8.2. This option is clearly incorrect.

• Option C: f(x) = -3.5x² + 2.7x - 8.2. This option perfectly matches the quadratic function we derived by substituting the given values. The coefficient of x² is -3.5, the coefficient of x is 2.7, and the constant term is -8.2. Therefore, this option is the correct answer.

• Option D: f(x) = 2.7x² - 3.5x - 8.2. Like option B, this option also has the coefficients in the wrong order. The coefficient of x² is incorrect, as is the coefficient of x. This option is incorrect.

By carefully analyzing each option, we can confidently conclude that Option C is the correct answer. This process highlights the importance of methodical comparison and attention to detail when working with mathematical expressions. It also reinforces our understanding of the standard form of a quadratic function and how the coefficients are arranged.

In conclusion, the quadratic function in standard form with a = -3.5, b = 2.7, and c = -8.2 is f(x) = -3.5x² + 2.7x - 8.2. This result was obtained by directly substituting the given values into the standard form f(x) = ax² + bx + c and then comparing the resulting function with the provided options. The process underscored the significance of understanding the standard form of a quadratic function, which is a cornerstone concept in algebra. The standard form allows us to readily identify key characteristics of the parabola represented by the function, such as its direction, width, axis of symmetry, and y-intercept.

The problem-solving approach we employed involved several critical steps. First, we reiterated the definition of the standard form of a quadratic function and highlighted the role of each coefficient. Then, we performed the substitution of the given values into the standard form, carefully ensuring accuracy. Finally, we meticulously compared the derived function with the provided options, eliminating incorrect choices based on coefficient mismatches. This step-by-step method not only led us to the correct answer but also reinforced our understanding of the underlying mathematical principles.

This exercise demonstrates the importance of precision and attention to detail in mathematics. Even a minor error in substitution or a misinterpretation of the standard form can lead to an incorrect solution. Therefore, it is crucial to double-check our work and to have a solid grasp of the fundamental concepts. Quadratic functions are not only important in algebra but also have numerous applications in various fields, including physics, engineering, and economics. Mastering the standard form and the associated problem-solving techniques is essential for success in these areas. By working through problems like this, we strengthen our mathematical skills and build a solid foundation for future learning.