Determining The Range Of F(x) = -(x+3)^2 + 7 A Comprehensive Guide

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In the vast landscape of mathematics, quadratic functions hold a prominent position. These functions, defined by the general form f(x) = ax^2 + bx + c, where a, b, and c are constants, exhibit a characteristic parabolic curve when graphed. The range of a function, a fundamental concept in mathematics, refers to the set of all possible output values (y-values) that the function can produce. Determining the range of a quadratic function is crucial for understanding its behavior and applications. This article will delve into the process of finding the range of the specific quadratic function f(x) = -(x+3)^2 + 7, providing a comprehensive explanation and step-by-step guidance.

Understanding the Vertex Form

The given function, f(x) = -(x+3)^2 + 7, is presented in vertex form. The vertex form of a quadratic function is expressed as f(x) = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. The vertex is a critical point on the parabola, as it indicates either the maximum or minimum value of the function. In our case, the function f(x) = -(x+3)^2 + 7 is already in vertex form, allowing us to readily identify the vertex. By comparing the given function with the vertex form, we can see that h = -3 and k = 7. Thus, the vertex of the parabola is (-3, 7). The coefficient 'a' in the vertex form determines the direction in which the parabola opens. If 'a' is positive, the parabola opens upwards, and the vertex represents the minimum point of the function. Conversely, if 'a' is negative, the parabola opens downwards, and the vertex represents the maximum point of the function. In our function, f(x) = -(x+3)^2 + 7, the coefficient 'a' is -1, which is negative. This indicates that the parabola opens downwards, and the vertex (-3, 7) represents the maximum point of the function. This understanding is pivotal in determining the range of the function.

The Significance of the Vertex

The vertex of the parabola is the cornerstone in determining the range of a quadratic function. As we established, the vertex of the function f(x) = -(x+3)^2 + 7 is (-3, 7), and the parabola opens downwards. This implies that the function reaches its maximum value at the vertex, which is y = 7. Since the parabola opens downwards, all other y-values of the function will be less than or equal to the y-coordinate of the vertex. Therefore, the range of the function consists of all real numbers less than or equal to 7. Visualizing the graph of the function reinforces this concept. The parabola extends downwards from the vertex, encompassing all y-values below 7. The vertex serves as the upper boundary of the range, dictating the maximum attainable value of the function. This connection between the vertex and the range is a fundamental characteristic of quadratic functions.

Deconstructing the Function

To meticulously determine the range, let's analyze the function f(x) = -(x+3)^2 + 7 step by step. We start with the term (x+3)^2. Since squaring any real number always yields a non-negative result, (x+3)^2 will always be greater than or equal to zero. This fundamental property of squares is crucial in understanding the function's behavior. Next, we consider the negative sign in front of the squared term. Multiplying a non-negative quantity by -1 results in a non-positive quantity (i.e., less than or equal to zero). Therefore, -(x+3)^2 will always be less than or equal to zero. This transformation flips the parabola, making it open downwards. Finally, we add 7 to the expression. Adding 7 to a non-positive quantity results in a quantity that is less than or equal to 7. Hence, f(x) = -(x+3)^2 + 7 will always be less than or equal to 7. This step-by-step analysis solidifies our understanding of how the function's components interact to determine its maximum value.

Visualizing the Parabola

The graphical representation of f(x) = -(x+3)^2 + 7 as a parabola provides a visual confirmation of our findings. The parabola opens downwards, with its vertex at (-3, 7). The vertex, as the highest point on the graph, represents the maximum value of the function. The parabola extends downwards indefinitely, encompassing all y-values less than 7. The graph visually demonstrates that the function's output values are bounded above by 7, reinforcing the concept that the range consists of all real numbers less than or equal to 7. The symmetry of the parabola around the vertical line passing through the vertex further illustrates the function's behavior and reinforces the connection between the vertex and the range.

Expressing the Range

Based on our analysis, the range of the function f(x) = -(x+3)^2 + 7 is the set of all real numbers less than or equal to 7. We can express this range using various notations:

  • Inequality notation: y ≤ 7
  • Interval notation: (-∞, 7]

Both notations accurately represent the same set of values, encompassing all real numbers from negative infinity up to and including 7. The use of interval notation provides a concise way to express the range, while inequality notation explicitly states the condition that the y-values must satisfy. The choice of notation often depends on the context and the preference of the individual.

In conclusion, by understanding the vertex form of a quadratic function, analyzing the function's components, and visualizing its graph, we have successfully determined that the range of f(x) = -(x+3)^2 + 7 is all real numbers less than or equal to 7. This comprehensive exploration highlights the importance of the vertex in determining the range of a quadratic function and provides a framework for analyzing similar functions. The ability to determine the range of a function is a fundamental skill in mathematics, with applications extending to various fields, including physics, engineering, and economics. Understanding the behavior of functions, including their range, is crucial for modeling real-world phenomena and solving complex problems.

By dissecting the function, we unveiled the mechanism behind its range, cementing the understanding of quadratic functions and their behavior. This knowledge empowers us to tackle similar problems and apply these principles to more complex mathematical scenarios. The range, a fundamental property of functions, dictates the set of possible output values, and its determination is pivotal in understanding the function's behavior and its applicability in various contexts.