Rewriting Quadratic Functions Understanding H(x) = 31x² + 77x + 41

Jul 11, 2025 by ADMIN 67 views

Understanding Quadratic Functions: Rewriting h(x) = 31x² + 77x + 41

In the realm of mathematics, understanding functions is crucial, especially quadratic functions. This article delves into the intricacies of rewriting a given quadratic function, h(x) = 31x² + 77x + 41, exploring different equivalent forms and their implications. We will dissect the function, analyze its components, and determine which of the provided options accurately represents the same function. This exploration will not only solidify your understanding of quadratic functions but also enhance your ability to manipulate and interpret them in various contexts.

Dissecting the Quadratic Function

The given function, h(x) = 31x² + 77x + 41, is a classic example of a quadratic function. Quadratic functions are characterized by their highest degree term being x², which gives them a distinctive parabolic shape when graphed. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. In our case, a = 31, b = 77, and c = 41. Understanding these coefficients is key to understanding the function's behavior.

The coefficient a (31 in this case) determines the parabola's concavity. Since a is positive, the parabola opens upwards. The larger the absolute value of a, the narrower the parabola. The coefficient b (77 in this case) influences the parabola's position along the x-axis, and the constant c (41 in this case) represents the y-intercept of the parabola, the point where the graph intersects the y-axis. Analyzing these coefficients provides valuable insights into the function's graph and overall characteristics.

Furthermore, the function notation h(x) simply means that the output of the function, often represented by y, depends on the input value x. Therefore, h(x) and y are interchangeable in this context. This understanding is crucial when examining the provided options and determining which one correctly represents the given function.

Analyzing the Options

Let's examine each option provided and determine its validity in representing the function h(x) = 31x² + 77x + 41:

A. y + 41 = 31x² + 77x This option attempts to represent the function by isolating the constant term. However, it fails to accurately represent the original function. To see this, we can subtract 41 from both sides of the original equation, h(x) = 31x² + 77x + 41, which yields h(x) - 41 = 31x² + 77x. This is different from option A, which has y + 41 on the left side. Therefore, option A is incorrect.
B. h(x) + 41 = 31x² + 77x This option adds 41 to the function h(x), which fundamentally changes the function's output. This equation suggests that adding 41 to the original function results in 31x² + 77x. However, this is not the case. As we saw in the analysis of option A, subtracting 41 from h(x) gives us 31x² + 77x, not adding 41. Therefore, option B is incorrect.
C. y = 31x² + 77x + 41 This option is a direct restatement of the original function, simply replacing h(x) with y. As we discussed earlier, h(x) and y are often used interchangeably to represent the output of a function. This option maintains the original relationship between x and the function's output, making it a valid representation of h(x) = 31x² + 77x + 41. Therefore, option C is the correct answer.
D. y = 31x² + 77x - 41 This option alters the constant term of the function, changing it from +41 to -41. This significantly impacts the function's graph, shifting it vertically. The y-intercept would now be -41 instead of 41. This change makes option D an incorrect representation of the original function. Therefore, option D is incorrect.

The Correct Representation

After analyzing each option, it becomes clear that option C, y = 31x² + 77x + 41, is the only accurate representation of the function h(x) = 31x² + 77x + 41. This option simply replaces the function notation h(x) with the more common variable y for the output, while maintaining the original mathematical relationship. This reinforces the understanding that h(x) and y are interchangeable in the context of function representation.

Key Takeaways

This exercise highlights the importance of understanding function notation and the components of a quadratic function. By carefully analyzing the coefficients and the structure of the equation, we can accurately identify equivalent representations of the same function. This skill is essential for solving quadratic equations, graphing parabolas, and applying quadratic functions in various real-world scenarios. Remember, the key to success in mathematics lies in a solid understanding of fundamental concepts and the ability to apply them in different contexts. Always double-check your work and ensure that any transformations you make preserve the original mathematical relationship. In conclusion, option C, y = 31x² + 77x + 41, is the correct answer because it is a direct and accurate restatement of the original function, using y as an alternative representation of h(x).

Delving deeper into quadratic functions, it's essential to grasp the significance of each component within the equation. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants that dictate the shape and position of the parabola. In the function h(x) = 31x² + 77x + 41, we can clearly identify these components: a = 31, b = 77, and c = 41. Let's explore the role each of these constants plays in defining the function's characteristics.

The coefficient a (31 in our case) is the leading coefficient and exerts a significant influence on the parabola's shape. Its sign determines the direction of the parabola's opening. A positive a value, as we have here, indicates that the parabola opens upwards, forming a U-shape. Conversely, a negative a value would result in a parabola opening downwards, resembling an inverted U-shape. The magnitude of a also 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 parabola. Understanding the impact of a is crucial for visualizing the parabola's basic form.

The coefficient b (77 in our case) plays a role in determining the horizontal position of the parabola's vertex, the point where the parabola changes direction. The vertex's x-coordinate can be calculated using the formula x = -b / 2a. In our example, this would be x = -77 / (2 * 31) ≈ -1.24. This value represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The b coefficient, therefore, influences the parabola's location along the x-axis. It is essential to consider both a and b to fully understand the vertex's position.

The constant term c (41 in our case) represents the y-intercept of the parabola. This is the point where the graph intersects the y-axis, occurring when x = 0. In our function, when x = 0, h(x) = 31(0)² + 77(0) + 41 = 41. Thus, the parabola intersects the y-axis at the point (0, 41). The y-intercept provides a crucial reference point for graphing the parabola and understanding its position relative to the coordinate axes.

By carefully analyzing the values of a, b, and c, we can gain a comprehensive understanding of a quadratic function's behavior and graphical representation. This knowledge is invaluable for solving quadratic equations, optimizing functions, and applying quadratic models in various fields.

In mathematics, a single function can be represented in multiple equivalent forms. Understanding these alternative representations is crucial for problem-solving and gaining a deeper insight into the function's properties. In the case of h(x) = 31x² + 77x + 41, we explored the common representation using function notation h(x) and its equivalent form using the variable y. However, there are other ways to manipulate and express this function while preserving its underlying mathematical relationship.

The most straightforward alternative representation is simply replacing h(x) with y, resulting in y = 31x² + 77x + 41. This substitution is valid because h(x) is defined as the output of the function for a given input x, and y is the conventional variable used to represent the output. This representation is often preferred when graphing the function or performing algebraic manipulations.

Another way to represent a quadratic function is through its vertex form: f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. Converting the given function to vertex form involves completing the square. This process involves manipulating the equation to create a perfect square trinomial. While the vertex form provides valuable information about the parabola's vertex and axis of symmetry, it is not one of the options presented in the original question. However, understanding this form enhances our understanding of alternative representations.

Furthermore, a quadratic function can also be represented in its factored form: f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots or x-intercepts of the function. Finding the factored form involves finding the roots of the quadratic equation, which can be done through factoring, completing the square, or using the quadratic formula. The factored form is particularly useful for determining the x-intercepts of the parabola and solving quadratic equations.

The ability to move between different representations of a function is a powerful tool in mathematics. Each form provides unique insights and facilitates different types of problem-solving. By understanding the connections between these representations, we can develop a more comprehensive understanding of functions and their applications.

In the context of mathematics, accuracy in representation is paramount. A slight alteration in an equation can lead to significant changes in the function's behavior and its graphical representation. This is evident in the given problem, where only one option accurately represents the original function h(x) = 31x² + 77x + 41. Let's delve into the implications of inaccurate representations and the importance of maintaining mathematical integrity.

Consider option A, y + 41 = 31x² + 77x. This equation, while seemingly similar to the original, represents a different function. Subtracting 41 from both sides, we get y = 31x² + 77x - 41. This new function has a y-intercept of -41, whereas the original function has a y-intercept of 41. This seemingly small change dramatically shifts the parabola's position on the coordinate plane.

Similarly, option B, h(x) + 41 = 31x² + 77x, represents a function distinct from the original. Subtracting 41 from both sides yields h(x) = 31x² + 77x - 41, again resulting in a different y-intercept and a vertically shifted parabola. These subtle differences highlight the sensitivity of mathematical expressions to alterations.

Option D, y = 31x² + 77x - 41, further underscores the significance of accurate representation. This equation, as we've seen, has a different y-intercept compared to the original function. The y-intercept, a crucial point for graphing and understanding the function, is altered by changing the constant term. This illustrates that even minor changes in the equation can lead to substantial deviations in the function's behavior.

The correct option, C, y = 31x² + 77x + 41, is the only representation that maintains the original function's characteristics. It simply replaces the function notation h(x) with the variable y, a valid substitution that preserves the mathematical relationship. This accuracy is crucial for solving problems correctly, interpreting graphs accurately, and applying mathematical models effectively.

The importance of accurate representation extends beyond simple equations. In complex mathematical models and scientific applications, even small errors in representation can have significant consequences. Therefore, a meticulous approach to mathematical expressions and a deep understanding of fundamental concepts are essential for success in mathematics and related fields. Always double-check your work, verify your representations, and ensure that your manipulations preserve the original mathematical relationships.

In conclusion, this exploration of rewriting the quadratic function h(x) = 31x² + 77x + 41 highlights the crucial aspects of function representation and manipulation in mathematics. We've dissected the function, analyzed its components, and examined various alternative representations. By carefully evaluating the provided options, we identified option C, y = 31x² + 77x + 41, as the only accurate representation of the original function. This exercise underscores the importance of understanding function notation, the significance of each coefficient in a quadratic function, and the need for precision in mathematical expressions. The ability to accurately represent and manipulate functions is a cornerstone of mathematical proficiency and essential for success in various fields. Remember to always double-check your work and ensure that your representations maintain the original mathematical relationships. By mastering these fundamental concepts, you can confidently tackle more complex mathematical challenges.