Analyzing The Parabola Of F(x) = -9(x + 6)^2 - 8 Key Features And Properties
In this comprehensive article, we will delve into the characteristics of the parabola represented by the quadratic function f(x) = -9(x + 6)^2 - 8. Understanding parabolas is crucial in mathematics, as they appear in various applications, from physics to engineering. Our primary focus will be to analyze the given function and determine which of the provided statements accurately describes its parabola. Specifically, we will explore whether the parabola opens upwards, identify the vertex, define the axis of symmetry, and calculate the y-intercept. By examining these key features, we can gain a thorough understanding of the parabola's behavior and graphical representation. This analysis will not only help in answering the specific question but also provide a solid foundation for tackling similar problems involving quadratic functions and their corresponding parabolas. So, let's embark on this mathematical journey to unravel the mysteries of the parabola.
To accurately describe the parabola represented by the function f(x) = -9(x + 6)^2 - 8, we need to break down the equation and understand how each component affects the shape and position of the parabola. The given function is in the vertex form of a quadratic equation, which is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola, and a determines the direction and stretch of the parabola. In our case, a = -9, h = -6, and k = -8. Understanding these values is crucial in determining the characteristics of the parabola.
Firstly, the value of a determines whether the parabola opens upwards or downwards. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards. In our function, a = -9, which is a negative value. Therefore, the parabola opens downwards, not upwards. This eliminates option (a). Secondly, the vertex form of the equation directly gives us the vertex of the parabola, which is (h, k). In our case, h = -6 and k = -8, so the vertex is (-6, -8). This contradicts option (b), which states the vertex is (6, -8). The axis of symmetry is a vertical line that passes through the vertex, and its equation is x = h. Since our vertex has an x-coordinate of -6, the axis of symmetry is x = -6. This confirms that option (c) is correct. Lastly, the y-intercept is the point where the parabola intersects the y-axis, which occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function: f(0) = -9(0 + 6)^2 - 8 = -9(36) - 8 = -324 - 8 = -332. This means the y-intercept is (0, -332), which contradicts option (d), which states the y-intercept is 8. By meticulously analyzing each component of the quadratic function, we have determined that only one of the provided statements is correct. The correct statement is that the axis of symmetry is x = -6.
To further solidify our understanding of the parabola represented by f(x) = -9(x + 6)^2 - 8, let's delve deeper into each of its key features. This comprehensive analysis will provide a clear picture of the parabola's behavior and graphical representation. We will revisit the direction of opening, the vertex, the axis of symmetry, and the y-intercept, providing detailed explanations and justifications for each characteristic.
Direction of Opening
The direction in which a parabola opens is determined by the coefficient a in the vertex form of the quadratic equation, f(x) = a(x - h)^2 + k. As previously mentioned, if a is positive, the parabola opens upwards, resembling a U shape. Conversely, if a is negative, the parabola opens downwards, resembling an inverted U shape. In our function, f(x) = -9(x + 6)^2 - 8, the coefficient a is -9, which is a negative value. This definitively tells us that the parabola opens downwards. The negative sign indicates a reflection across the x-axis, causing the parabola to open in the opposite direction compared to a parabola with a positive a value. Understanding this basic principle is essential for visualizing the parabola's shape and orientation. It also helps in predicting the maximum or minimum value of the function, which occurs at the vertex. Since our parabola opens downwards, it has a maximum value at its vertex.
Vertex
The vertex of a parabola is a critical point that represents either the maximum or minimum value of the quadratic function. In the vertex form of the equation, f(x) = a(x - h)^2 + k, the vertex is given by the coordinates (h, k). The vertex is the turning point of the parabola, where it changes direction. For a parabola that opens upwards, the vertex is the lowest point, representing the minimum value of the function. For a parabola that opens downwards, the vertex is the highest point, representing the maximum value of the function. In our function, f(x) = -9(x + 6)^2 - 8, we can identify h as -6 and k as -8. Therefore, the vertex of the parabola is (-6, -8). This point is crucial because it not only represents the maximum value of the function (-8) but also lies on the axis of symmetry. The vertex provides valuable information about the parabola's position and behavior on the coordinate plane.
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It passes through the vertex of the parabola, making the vertex a key reference point for determining the axis of symmetry. The equation of the axis of symmetry is given by x = h, where h is the x-coordinate of the vertex. In our function, f(x) = -9(x + 6)^2 - 8, the vertex is (-6, -8), so the x-coordinate h is -6. Therefore, the axis of symmetry is the vertical line x = -6. This line acts as a mirror, reflecting one half of the parabola onto the other half. The axis of symmetry is not only a geometrical feature but also a crucial element in understanding the symmetry properties of the parabola. It helps in sketching the graph of the parabola and in identifying corresponding points on either side of the vertex. Knowing the axis of symmetry simplifies the analysis of the parabola's behavior and its relationship to the quadratic function.
Y-intercept
The y-intercept is the point where the parabola intersects the y-axis. This occurs when the x-coordinate is 0. To find the y-intercept, we substitute x = 0 into the function and solve for f(x), which gives us the y-coordinate of the y-intercept. In our function, f(x) = -9(x + 6)^2 - 8, we substitute x = 0: f(0) = -9(0 + 6)^2 - 8. Simplifying this expression, we get f(0) = -9(6)^2 - 8 = -9(36) - 8 = -324 - 8 = -332. Therefore, the y-intercept is the point (0, -332). This point is significantly below the x-axis due to the large negative value. The y-intercept provides additional information about the parabola's position on the coordinate plane and helps in sketching its graph. It also gives us a specific point to reference when analyzing the function's behavior. In the context of real-world applications, the y-intercept can represent an initial value or a starting point of a process modeled by the quadratic function.
In summary, we have conducted a thorough analysis of the parabola represented by the quadratic function f(x) = -9(x + 6)^2 - 8. By examining the vertex form of the equation, we have determined the key features of the parabola, including its direction of opening, vertex, axis of symmetry, and y-intercept. We established that the parabola opens downwards due to the negative coefficient a (-9), has a vertex at (-6, -8), an axis of symmetry at x = -6, and a y-intercept at (0, -332). Among the given options, the correct statement is that the axis of symmetry is x = -6. This analysis underscores the importance of understanding the vertex form of a quadratic equation and how each component influences the parabola's characteristics. Parabolas are fundamental in mathematics and have numerous applications in various fields. A solid understanding of their properties is essential for solving problems and interpreting real-world phenomena modeled by quadratic functions. We encourage readers to continue exploring the fascinating world of parabolas and their applications in mathematics and beyond.
