Analyzing The Quadratic Function F(x) = 3(x+y)^2 - 2 Axis Of Symmetry, Domain, And Range

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This article delves into the characteristics of the quadratic function f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2, exploring its axis of symmetry, domain, and range. Understanding these key aspects provides a comprehensive view of the function's behavior and graphical representation. We will break down each component, offering clear explanations and insights to enhance your understanding of quadratic functions.

Axis of Symmetry

The axis of symmetry is a crucial feature of a parabola, the U-shaped curve that represents a quadratic function. It is a vertical line that divides the parabola into two symmetrical halves. To determine the axis of symmetry for the given function, f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2, we first need to recognize the standard form of a quadratic function, which is f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h, k) represents the vertex of the parabola. The axis of symmetry is then given by the vertical line x=hx = h.

In our function, f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2, we can rewrite it as f(x)=3(x−(−y))2−2f(x) = 3(x - (-y))^2 - 2. Comparing this to the standard form, we can identify a=3a = 3, h=−yh = -y, and k=−2k = -2. Therefore, the vertex of the parabola is (−y,−2)(-y, -2). The axis of symmetry is the vertical line that passes through the x-coordinate of the vertex. Hence, the axis of symmetry for the function f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2 is x=−yx = -y. This means that for any value of yy, the parabola will be symmetrical around the vertical line x=−yx = -y. The axis of symmetry provides a clear visual representation of the function's symmetry and is essential for graphing and analyzing the quadratic function. Understanding the axis of symmetry helps in predicting how the function will behave on either side of this line, which is particularly useful in optimization problems and real-world applications where symmetry plays a significant role. The axis of symmetry not only simplifies the graphing process but also gives critical insights into the function's minimum or maximum value, depending on whether the parabola opens upwards or downwards. In this case, since a=3a = 3 (which is positive), the parabola opens upwards, and the vertex represents the minimum point of the function. Thus, the axis of symmetry is a fundamental concept in understanding the overall behavior of quadratic functions and their applications.

Domain

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For polynomial functions, such as quadratic functions, the domain is typically all real numbers. This is because there are no restrictions on the values that can be substituted for xx in the function. In the given function, f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2, the input xx can take any real value without causing any undefined operations, such as division by zero or taking the square root of a negative number. Therefore, the domain of the function is all real numbers, which can be expressed in interval notation as (−∞,∞)(-\infty, \infty).

The domain being all real numbers signifies that the function is defined for every possible x-value, making it a continuous function across the entire number line. This is a crucial characteristic of quadratic functions and is often used in various mathematical and real-world applications. When analyzing functions, identifying the domain is one of the first steps, as it sets the boundaries within which the function operates. For quadratic functions, the infinite domain allows for a complete understanding of the function's behavior, including its minimum or maximum values, intercepts, and symmetry. The fact that the domain is unrestricted means that there are no specific values of xx that need to be excluded, simplifying the analysis and application of the function. In practical terms, this means that any input value can be used in the function, leading to a corresponding output value, which is essential in modeling various phenomena using quadratic equations. Moreover, the unrestricted domain of the function makes it easier to perform transformations, such as shifting, stretching, or reflecting the graph of the function, without having to worry about domain limitations. The domain essentially provides the foundation for understanding and working with the function, ensuring that the function is well-defined and can be applied in a wide range of contexts.

Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the quadratic function f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2, the range is determined by the vertex of the parabola and the direction in which the parabola opens. As we established earlier, the vertex of the parabola is (−y,−2)(-y, -2), and the coefficient of the (x+y)2(x+y)^2 term, which is 3, is positive. This indicates that the parabola opens upwards, meaning the vertex represents the minimum point of the function.

Since the parabola opens upwards, the minimum value of the function is the y-coordinate of the vertex, which is -2. The function will take on all values greater than or equal to this minimum value. Therefore, the range of the function is all real numbers greater than or equal to -2. In interval notation, this can be expressed as [−2,∞)[-2, \infty). The range is a critical aspect of understanding the function's output, providing insights into the limits of the function's values. For quadratic functions, the range is particularly influenced by the vertex and the leading coefficient, which determine whether the parabola opens upwards or downwards and the function's minimum or maximum value. In our case, the positive leading coefficient ensures that the range has a lower bound at the y-coordinate of the vertex, and the function extends infinitely upwards. Understanding the range is crucial in various applications, such as optimization problems, where the goal is to find the maximum or minimum value of a function. The range also helps in determining the feasibility of solutions in real-world scenarios, ensuring that the function's output falls within acceptable limits. For instance, in a problem involving the height of a projectile modeled by a quadratic function, the range would provide information about the maximum height the projectile can reach. Thus, the range is an essential element in the complete analysis of a quadratic function, complementing the understanding gained from the domain and the axis of symmetry.

In summary, for the function f(x)=3(x+y)2−2f(x) = 3(x+y)^2 - 2, the axis of symmetry is x=−yx = -y, the domain is all real numbers (−∞,∞)(-\infty, \infty), and the range is [−2,∞)[-2, \infty). These characteristics provide a comprehensive understanding of the function's behavior and graphical representation.