Analyzing True Statements About The Function F(x) = (x+4)(x-6)
Which of the following statements about the function f(x) = (x+4)(x-6) are true? Select two options.
In this article, we will delve into the analysis of the quadratic function f(x) = (x+4)(x-6). Our primary goal is to scrutinize the given graph and discern the true statements about the function. We will focus on key features such as the vertex and explore different methods to determine its coordinates accurately. By understanding the properties of quadratic functions and their graphical representations, we can confidently evaluate the provided options and select the correct ones. This exploration will not only enhance our understanding of this specific function but also equip us with valuable tools for analyzing a broader range of quadratic equations and their corresponding graphs. Let's embark on this mathematical journey to unravel the characteristics of f(x) and master the art of interpreting quadratic functions.
Understanding the Quadratic Function f(x) = (x+4)(x-6)
To accurately analyze the graph and determine the true statements about the function f(x) = (x+4)(x-6), a thorough understanding of quadratic functions is essential. A quadratic function is a polynomial function of degree two, typically expressed in the standard form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The key features of a parabola include its vertex, axis of symmetry, and x-intercepts (also known as roots or zeros). The vertex represents the minimum or maximum point of the parabola, depending on the sign of the coefficient a. If a > 0, the parabola opens upwards, and the vertex is the minimum point. Conversely, if a < 0, the parabola opens downwards, and the vertex is the maximum point. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The x-intercepts are the points where the parabola intersects the x-axis, and they correspond to the real roots of the quadratic equation f(x) = 0. In the given function, f(x) = (x+4)(x-6), we can observe that it is in factored form. This form allows us to easily identify the x-intercepts by setting each factor equal to zero and solving for x. The x-intercepts are crucial points that help us sketch the graph and determine the position of the vertex. By expanding the factored form, we can rewrite the function in standard form, which provides additional insights into the coefficients and the overall shape of the parabola. Understanding these fundamental concepts of quadratic functions is crucial for interpreting the graph and making accurate statements about the function's properties.
Determining the Vertex of the Parabola
The vertex of a parabola is a critical point that provides valuable information about the quadratic function. As mentioned earlier, it represents either the minimum or maximum value of the function, depending on the parabola's orientation. To determine the vertex of the function f(x) = (x+4)(x-6), we can employ several methods. One approach is to find the x-coordinate of the vertex using the formula x = -b / 2a, where a and b are the coefficients of the quadratic equation in standard form. To use this formula, we first need to expand the factored form of the function: f(x) = (x+4)(x-6) = x² - 6x + 4x - 24 = x² - 2x - 24. Now, we can identify a = 1 and b = -2. Plugging these values into the formula, we get x = -(-2) / (2 * 1) = 1. This gives us the x-coordinate of the vertex. To find the corresponding y-coordinate, we substitute x = 1 back into the function: f(1) = (1+4)(1-6) = (5)(-5) = -25. Therefore, the vertex of the parabola is located at the point (1, -25). Another method to find the vertex involves using the x-intercepts. The x-intercepts of the function are the solutions to the equation f(x) = 0, which are x = -4 and x = 6. The x-coordinate of the vertex lies exactly midway between the x-intercepts due to the symmetry of the parabola. Thus, the x-coordinate of the vertex is the average of the x-intercepts: (-4 + 6) / 2 = 1. Again, substituting x = 1 into the function gives us the y-coordinate of the vertex: f(1) = -25. Both methods confirm that the vertex of the parabola is at (1, -25). This understanding is crucial for evaluating the statements about the function's vertex.
Analyzing the Given Statements About the Function
Now that we have a solid understanding of the function f(x) = (x+4)(x-6) and have determined its vertex to be at (1, -25), we can proceed to analyze the given statements and identify the true ones. The statements primarily focus on the vertex of the function, which we have already calculated. Statement A asserts that the vertex of the function is at (1, -25). Comparing this with our calculated vertex, we can confidently conclude that Statement A is true. Statement B, on the other hand, claims that the vertex of the function is at (1, -24). This statement contradicts our calculated vertex of (1, -25). Therefore, Statement B is false. To further solidify our understanding, it's beneficial to consider the graphical representation of the function. The parabola opens upwards because the coefficient of the x² term is positive (a = 1). The vertex (1, -25) represents the minimum point of the parabola, and the graph extends upwards from this point. The x-intercepts, located at x = -4 and x = 6, provide additional reference points for visualizing the parabola. By accurately determining the vertex and understanding the properties of the quadratic function, we can confidently evaluate the statements and select the true ones. In this case, only Statement A aligns with our calculations and analysis. When analyzing such statements, it is crucial to rely on mathematical principles and methods rather than solely relying on visual interpretations of the graph, which might be subject to inaccuracies. The combination of algebraic techniques and graphical understanding provides a comprehensive approach to solving these types of problems.
Conclusion
In conclusion, by carefully analyzing the quadratic function f(x) = (x+4)(x-6) and applying the principles of quadratic equations and parabolas, we have successfully determined the vertex and evaluated the given statements. Through expanding the function, calculating the x-coordinate of the vertex using the formula x = -b / 2a, and substituting the value back into the function to find the y-coordinate, we accurately identified the vertex as (1, -25). This was further validated by considering the symmetry of the parabola and the relationship between the vertex and the x-intercepts. When presented with statements about the function, it is essential to employ a systematic approach, relying on established mathematical methods rather than solely on visual estimations. By calculating the vertex and comparing it with the information provided in the statements, we can confidently distinguish between true and false assertions. In this particular case, we found that Statement A, claiming the vertex is at (1, -25), is true, while Statement B, claiming the vertex is at (1, -24), is false. This exercise demonstrates the importance of a thorough understanding of quadratic functions, their properties, and their graphical representations. By mastering these concepts, we can effectively analyze and interpret quadratic equations and their corresponding graphs, leading to accurate conclusions and problem-solving skills. The ability to confidently determine key features such as the vertex is crucial for various applications of quadratic functions in mathematics and real-world scenarios. This analytical approach not only enhances our comprehension of this specific function but also equips us with a valuable skill set for tackling a wide range of mathematical challenges.