Graph Analysis Of F(x) = (x+6)(x+2) Vertex, Symmetry, Domain And Intervals

In the realm of mathematics, quadratic functions hold a special place, their graphs forming elegant parabolas that gracefully curve through the coordinate plane. Understanding the properties of these parabolas is crucial for solving a myriad of problems, from optimizing areas to modeling projectile motion. This article delves into the intricacies of the quadratic function f(x) = (x+6)(x+2), meticulously examining its graph and unraveling its key characteristics. We will explore the vertex, the axis of symmetry, the domain, and the intervals where the function is increasing or decreasing, providing a comprehensive understanding of this fundamental mathematical concept. Our journey will not only enhance your understanding of quadratic functions but also equip you with the tools to analyze and interpret other mathematical functions with confidence.

The Vertex The Minimum Point of the Parabola

In the context of our quadratic function, the vertex stands out as a pivotal point, marking either the minimum or maximum value of the function. Specifically, when dealing with a parabola that opens upwards, as is the case with f(x) = (x+6)(x+2), the vertex represents the minimum value. This point is the lowest point on the graph, and its y-coordinate indicates the smallest value that the function can attain. To determine whether the vertex represents a maximum or minimum, we examine the coefficient of the x² term. If the coefficient is positive, the parabola opens upwards, indicating a minimum. Conversely, a negative coefficient signifies a downward-opening parabola, with the vertex representing a maximum. In our case, expanding f(x) = (x+6)(x+2) yields x² + 8x + 12, revealing a positive coefficient of 1 for the x² term. This confirms that the parabola opens upwards, and the vertex indeed marks the minimum value of the function. The vertex is not just a point of minimum or maximum value; it's also a cornerstone for understanding the function's behavior. Its coordinates provide crucial information about the function's range and symmetry. In essence, the vertex serves as a key landmark on the parabolic landscape, guiding our understanding of the function's overall characteristics and behavior. Therefore, statement A, asserting that the vertex is the maximum value, is incorrect. The vertex represents the minimum value for this upward-opening parabola.

The Axis of Symmetry A Mirror for the Parabola

The axis of symmetry is an imaginary vertical line that gracefully bisects the parabola, dividing it into two perfectly symmetrical halves. This line acts like a mirror, reflecting one side of the parabola onto the other. The axis of symmetry always passes through the vertex of the parabola, making the vertex a central point in understanding the parabola's symmetry. The equation of the axis of symmetry takes the form x = h, where h represents the x-coordinate of the vertex. This equation tells us that the axis of symmetry is a vertical line that intersects the x-axis at the same point as the vertex. To find the axis of symmetry for the function f(x) = (x+6)(x+2), we first need to determine the x-coordinate of the vertex. This can be achieved by finding the midpoint of the roots of the quadratic equation. The roots are the x-values where the function equals zero, which in this case are x = -6 and x = -2. The midpoint, calculated as (-6 + -2) / 2, is -4. Therefore, the x-coordinate of the vertex is -4, and the equation of the axis of symmetry is x = -4. The axis of symmetry is not merely a line of reflection; it provides valuable insights into the function's behavior. It helps us understand how the function's values change symmetrically around the vertex. For instance, for any given y-value, there will be two x-values equidistant from the axis of symmetry that produce that y-value (except for the y-value at the vertex). This symmetry is a fundamental characteristic of parabolas and quadratic functions. Thus, statement B, which states that the axis of symmetry is x = -4, is correct. The axis of symmetry perfectly bisects the parabola of f(x) = (x+6)(x+2), passing through the vertex and reflecting the function's symmetrical nature.

The Domain All Real Numbers Welcome

The domain of a function encompasses all possible input values (x-values) for which the function is defined. In simpler terms, it's the set of all x-values that you can plug into the function and get a valid output (y-value). For quadratic functions, the domain is typically all real numbers. This means that you can input any real number into the function, and it will produce a real number output. There are no restrictions on the x-values you can use. This stems from the fact that quadratic functions involve only polynomial expressions, which are defined for all real numbers. There are no denominators that could become zero, no square roots of negative numbers, or any other operations that could lead to undefined results. In the case of f(x) = (x+6)(x+2), there are no restrictions on the x-values. You can substitute any real number for x, and the function will produce a real number output. This is a fundamental characteristic of quadratic functions, setting them apart from functions with more restrictive domains, such as rational functions or radical functions. Understanding the domain of a function is crucial for interpreting its graph and behavior. It tells us the range of x-values over which the function is defined and can help us identify any potential limitations or discontinuities. For quadratic functions, the domain being all real numbers simplifies the analysis, allowing us to focus on other key features like the vertex, axis of symmetry, and range. Therefore, statement C, stating that the domain is all real numbers, is correct. This reflects the inherent nature of quadratic functions, which are defined for all real number inputs.

Increasing and Decreasing Intervals The Function's Ascent and Descent

The concept of increasing and decreasing intervals helps us understand the direction in which a function's graph is moving as we traverse it from left to right. A function is said to be increasing over an interval if its y-values are rising as the x-values increase. Conversely, a function is decreasing over an interval if its y-values are falling as the x-values increase. For quadratic functions, the increasing and decreasing intervals are closely tied to the vertex and the axis of symmetry. Due to the parabolic shape, a quadratic function will either decrease up to the vertex and then increase after the vertex, or increase up to the vertex and then decrease after the vertex, depending on whether the parabola opens upwards or downwards. In the case of f(x) = (x+6)(x+2), the parabola opens upwards, meaning it has a minimum value at the vertex. As we determined earlier, the vertex has an x-coordinate of -4. Therefore, the function is decreasing for all x-values less than -4 and increasing for all x-values greater than -4. This can be visualized by tracing the graph from left to right. Before reaching the vertex, the graph slopes downwards, indicating a decreasing function. After passing the vertex, the graph slopes upwards, indicating an increasing function. The increasing and decreasing intervals provide valuable insights into the function's behavior and its relationship to the vertex. They help us understand how the function's output changes as the input varies and can be crucial for optimization problems and other applications. Understanding the increasing and decreasing intervals, along with other properties like the vertex and axis of symmetry, provides a comprehensive picture of the quadratic function's behavior. Therefore, statement D, which states that the function is increasing, is incomplete and requires further clarification. The function is increasing only for x-values greater than -4. It is decreasing for x-values less than -4. To accurately describe the function's behavior, we need to specify the intervals over which it is increasing or decreasing.

In conclusion, our exploration of the quadratic function f(x) = (x+6)(x+2) has unveiled its key properties. The vertex marks the minimum value, the axis of symmetry gracefully bisects the parabola, the domain encompasses all real numbers, and the increasing and decreasing intervals reveal the function's dynamic behavior. This comprehensive analysis equips us with a deeper understanding of quadratic functions and their graphical representations, empowering us to tackle a wide range of mathematical problems with confidence.