Finding The Standard Form Of A Quadratic Function An Example

Introduction

In the realm of mathematics, quadratic functions hold a significant place, especially in algebra and calculus. These functions, characterized by their parabolic graphs, have the general form of f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. Understanding the different forms of quadratic functions is crucial for solving various mathematical problems, from finding roots to determining the vertex of a parabola. In this article, we will delve into the standard form of a quadratic function and how to identify it. We'll use a specific example, the function f(x) = -x² - 8x + 10, to illustrate the process. This exploration will not only clarify the concept but also enhance your problem-solving skills in mathematics.

The standard form of a quadratic function is expressed as f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. This form is particularly useful because it directly reveals the coefficients that determine the shape and position of the parabola. The coefficient 'a' dictates the direction and steepness of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The coefficients 'b' and 'c' influence the parabola's position on the coordinate plane. The term 'c' specifically represents the y-intercept of the parabola. The standard form allows for easy identification of these key features, making it a fundamental concept in the study of quadratic functions. Recognizing and manipulating quadratic functions in standard form is a critical skill in various mathematical contexts, including solving quadratic equations, graphing parabolas, and understanding real-world applications like projectile motion and optimization problems.

What is the Standard Form of a Quadratic Function?

The standard form of a quadratic function is mathematically represented as f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. This form is fundamental in the study of quadratic functions because it readily provides key information about the parabola's characteristics. The coefficient a determines the parabola's direction: if a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The magnitude of a also affects the steepness of the parabola; a larger absolute value of a results in a narrower parabola, while a smaller absolute value results in a wider one. The coefficient b influences the parabola's position along the x-axis, and c represents the y-intercept, which is the point where the parabola intersects the y-axis. The standard form is essential for various mathematical operations, such as solving quadratic equations using the quadratic formula, completing the square, and graphing parabolas. It allows mathematicians and students to quickly identify the critical parameters of the function and understand its behavior. For instance, the standard form makes it straightforward to find the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves, and the vertex, which is the highest or lowest point on the parabola. Therefore, mastering the standard form is crucial for anyone working with quadratic functions, as it provides a clear and concise representation that facilitates both analysis and manipulation of these functions.

Analyzing the Given Function

Now, let's consider the function f(x) = -x² - 8x + 10. This function is already presented in the standard form f(x) = ax² + bx + c, which makes it easier to analyze its characteristics. By comparing the given function with the standard form, we can directly identify the coefficients: a = -1, b = -8, and c = 10. The coefficient a = -1 indicates that the parabola opens downwards, since a is negative. This also means that the parabola has a maximum point, or vertex. The coefficient b = -8 affects the position of the parabola along the x-axis, influencing the location of the axis of symmetry and the vertex. Specifically, the x-coordinate of the vertex can be found using the formula -b/(2a), which in this case is -(-8)/(2*(-1)) = -4. The coefficient c = 10 tells us that the y-intercept of the parabola is at the point (0, 10). This is the point where the parabola intersects the y-axis. Understanding these coefficients allows us to sketch a rough graph of the function and predict its behavior. For example, we know that the parabola opens downwards, its vertex is to the left of the y-axis (since the x-coordinate of the vertex is negative), and it intersects the y-axis at (0, 10). This preliminary analysis is invaluable for solving problems related to this quadratic function, such as finding its roots, determining its range, and understanding its transformations.

Identifying the Correct Standard Form

To ensure we have correctly identified the standard form of the given function, we need to compare it against the options provided. The function is f(x) = -x² - 8x + 10. The standard form, as we discussed, is f(x) = ax² + bx + c. By direct comparison, it's clear that the function is already in the standard form. Now, let’s evaluate the given options to confirm which one matches our function:

A. f(x) = -x² - 8x + 10: This option exactly matches the given function. The coefficients are a = -1, b = -8, and c = 10, which aligns perfectly with our analysis.

B. f(x) = x² - 8x + 10: This option has a positive leading coefficient (a = 1), which means the parabola would open upwards, not downwards as in our function. Thus, this is not the correct standard form.

C. f(x) = x² - 4x + 10: This option also has a positive leading coefficient (a = 1) and a different middle term (-4x instead of -8x), indicating that this is not the correct representation of the function.

D. N(x) = -x² - 6x + 10: This option has the correct leading coefficient (a = -1*) but differs in the middle term (-6x instead of -8x), making it an incorrect match.

Therefore, after a thorough comparison, it is evident that option A, f(x) = -x² - 8x + 10, is the correct standard form of the given quadratic function. This exercise underscores the importance of carefully comparing coefficients and understanding the implications of each term in the standard form of a quadratic function.

Conclusion

In conclusion, identifying the standard form of a quadratic function is a fundamental skill in mathematics. The standard form, f(x) = ax² + bx + c, provides a clear and concise representation that allows for easy analysis of the function's characteristics. By understanding the role of the coefficients a, b, and c, we can determine the parabola's direction, steepness, and position on the coordinate plane. In the case of the function f(x) = -x² - 8x + 10, we successfully identified that it is already in standard form and that option A, f(x) = -x² - 8x + 10, correctly represents it. This process involved comparing the given function with the general standard form and evaluating the provided options to ensure an exact match. Mastering this skill is crucial for solving various mathematical problems related to quadratic functions, including finding roots, determining vertices, and graphing parabolas. The ability to recognize and manipulate quadratic functions in standard form is not only essential for academic success in mathematics but also for practical applications in fields such as physics, engineering, and economics, where quadratic models are frequently used to describe real-world phenomena. Therefore, a solid understanding of the standard form lays a strong foundation for further studies and applications in mathematics and related disciplines.