Analyzing The Graph Of F(x) = 6(x+1)^2 - 9 Finding The Correct Statements
In this detailed exploration, we will delve into the characteristics of the quadratic function f(x) = 6(x+1)^2 - 9. Our primary focus will be on identifying the correct statements that accurately describe its graph. This involves understanding the fundamental properties of quadratic functions, including transformations, the direction of opening, the vertex, the axis of symmetry, and the intercepts. By meticulously analyzing each aspect, we aim to provide a comprehensive understanding of the graph of the given function. Quadratic functions are a cornerstone of algebra and calculus, and a thorough understanding of their properties is crucial for solving various mathematical problems and real-world applications. This analysis will not only help in answering the specific questions about the graph but also enhance the overall comprehension of quadratic functions and their graphical representations. We will dissect the equation, break down its components, and explain how each part contributes to the final shape and position of the graph. Let's embark on this analytical journey to uncover the key features of f(x) = 6(x+1)^2 - 9.
Direction of Opening: Determining the Graph's Orientation
The direction of opening is a crucial aspect of a quadratic function's graph. It tells us whether the parabola opens upwards or downwards. This is determined by the coefficient of the x^2 term in the quadratic function. In the given function, f(x) = 6(x+1)^2 - 9, the coefficient of the x^2 term is positive. To see this, we can expand the function: f(x) = 6(x^2 + 2x + 1) - 9 = 6x^2 + 12x + 6 - 9 = 6x^2 + 12x - 3. The coefficient of x^2 is 6, which is a positive number. When the coefficient of the x^2 term is positive, the parabola opens upwards, forming a U-shape. This means that the function has a minimum value. Conversely, if the coefficient were negative, the parabola would open downwards, forming an inverted U-shape, and the function would have a maximum value. Understanding the direction of opening is essential for sketching the graph and identifying the vertex as either a minimum or maximum point. This simple yet crucial observation provides a foundational understanding of the graph's overall shape and behavior. The positive coefficient of x^2 is a clear indicator that the graph will exhibit an upward trajectory, a key characteristic to consider in our comprehensive analysis.
Vertex of the Parabola: Locating the Minimum Point
The vertex is a critical point on the graph of a parabola, representing either the minimum or maximum value of the quadratic function. For the given function, f(x) = 6(x+1)^2 - 9, the vertex form of a quadratic function, f(x) = a(x-h)^2 + k, is particularly useful in identifying the vertex. In this form, the vertex is located at the point (h, k). Comparing our function to the vertex form, we can see that h = -1 and k = -9. Therefore, the vertex of the parabola is at the point (-1, -9). Since we've already determined that the parabola opens upwards, the vertex represents the minimum point of the function. This means that the lowest value the function reaches is -9, and it occurs when x = -1. The vertex not only gives us the minimum value but also helps in understanding the symmetry of the parabola. The parabola is symmetric about the vertical line that passes through the vertex. In this case, the axis of symmetry is the line x = -1. Locating the vertex is a fundamental step in sketching the graph of a quadratic function and understanding its behavior. It provides a reference point around which the rest of the graph is shaped. Understanding the vertex is crucial for solving optimization problems and analyzing the range of the function. The coordinates of the vertex offer valuable insights into the function's behavior and graphical representation.
Transformations: Understanding Shifts and Stretches
Transformations play a key role in understanding how the graph of f(x) = 6(x+1)^2 - 9 is related to the basic quadratic function f(x) = x^2. The given function is a transformed version of the basic parabola. The term (x+1) inside the square indicates a horizontal shift. Specifically, it shifts the graph 1 unit to the left. This is because replacing x with (x+1) moves the graph in the negative x-direction. The coefficient 6 in front of the squared term represents a vertical stretch. It stretches the graph vertically by a factor of 6, making the parabola narrower compared to the basic parabola. The constant term -9 represents a vertical shift. It shifts the entire graph 9 units downwards. This is because subtracting 9 from the function's value moves the graph in the negative y-direction. Combining these transformations, we can visualize how the graph of f(x) = x^2 is transformed into the graph of f(x) = 6(x+1)^2 - 9. First, it's shifted 1 unit left, then stretched vertically by a factor of 6, and finally shifted 9 units down. Understanding these transformations allows us to quickly sketch the graph and predict its behavior without plotting numerous points. It also provides a deeper insight into how changing the parameters in the quadratic function affects its graph. The ability to recognize and apply transformations is essential for analyzing and manipulating functions in various mathematical contexts. These shifts and stretches collectively define the final position and shape of the parabola on the coordinate plane.
Incorrect Statement A: Misinterpreting Vertical Shifts
Statement A, which claims that the graph is obtained by shifting the graph of f(x) = 6(x+1)^2 up 9 units, is incorrect. The given function, f(x) = 6(x+1)^2 - 9, is actually obtained by shifting the graph of f(x) = 6(x+1)^2 down 9 units. The -9 term at the end of the function indicates a downward vertical shift, not an upward shift. To understand why, consider the effect of subtracting 9 from the function's value. For every x, the value of f(x) is reduced by 9, which means the entire graph is shifted downwards. This is a common misconception, as the negative sign often leads to confusion about the direction of the shift. It's crucial to remember that adding a constant shifts the graph upwards, while subtracting a constant shifts it downwards. This understanding is fundamental to correctly interpreting transformations of functions. The incorrect statement highlights the importance of careful analysis and attention to detail when dealing with function transformations. By recognizing this error, we reinforce our understanding of how vertical shifts affect the graph of a function. The distinction between upward and downward shifts is essential for accurately sketching and analyzing quadratic functions.
Correct Statement B: The Parabola Opens Upward
Statement B, which asserts that the graph opens upward, is indeed correct. As we discussed earlier, the direction of opening is determined by the coefficient of the x^2 term. In the function f(x) = 6(x+1)^2 - 9, the coefficient of the x^2 term is 6, which is positive. A positive coefficient indicates that the parabola opens upwards, forming a U-shape. This means that the function has a minimum value, which occurs at the vertex. The upward opening is a fundamental characteristic of the graph and is directly related to the positive leading coefficient. This observation aligns with the general properties of quadratic functions, where a positive leading coefficient always results in an upward-opening parabola. The correct identification of the graph's opening direction is crucial for visualizing its shape and understanding its behavior. The upward trajectory of the parabola is a key feature that distinguishes it from downward-opening parabolas, which have a negative leading coefficient. This statement provides a clear and accurate description of the graph's orientation.
Correct Statement C: The Vertex at (-1, -9)
Statement C, which identifies the vertex of the graph, is also correct. As we determined earlier, the vertex of the function f(x) = 6(x+1)^2 - 9 is located at the point (-1, -9). This was found by analyzing the vertex form of the quadratic function, f(x) = a(x-h)^2 + k, where the vertex is at (h, k). By comparing the given function to this form, we identified h = -1 and k = -9. The vertex represents the minimum point of the parabola since the graph opens upwards. It is a crucial point for sketching the graph and understanding the function's behavior. The vertex also lies on the axis of symmetry, which is the vertical line x = -1. The accurate identification of the vertex is essential for solving optimization problems and analyzing the range of the function. This point provides a reference around which the rest of the parabola is shaped. The coordinates of the vertex offer valuable insights into the function's minimum value and its location on the coordinate plane. This statement correctly pinpoints the pivotal point of the parabola.
In conclusion, the analysis of the quadratic function f(x) = 6(x+1)^2 - 9 reveals several key features of its graph. The graph opens upwards due to the positive coefficient of the x^2 term. The vertex of the parabola is located at the point (-1, -9), representing the minimum value of the function. The graph is a transformation of the basic parabola f(x) = x^2, shifted 1 unit left, stretched vertically by a factor of 6, and shifted 9 units down. Statement A, which incorrectly states that the graph is shifted up 9 units, is false. Statements B and C, which correctly identify the upward opening and the vertex, are true. Understanding these properties allows us to accurately sketch the graph and analyze the behavior of the quadratic function. This comprehensive analysis provides a solid foundation for further exploration of quadratic functions and their applications in various mathematical contexts. The ability to dissect and interpret quadratic functions is a valuable skill in algebra and calculus. By mastering these concepts, we can confidently tackle a wide range of problems and gain a deeper appreciation for the elegance and power of mathematics. The combination of graphical and algebraic insights provides a holistic understanding of the function's characteristics and behavior.
